# The ”hit” problem of five variables in the generic degree
and its application

D-ặng Võ Phúc  Faculty of Education Studies, University of Khanh Hoa,
01 Nguyen Chanh, Nha Trang, Khanh Hoa, Viet Nam<EMAIL_ADDRESS>
###### Abstract

Let $P_{s}:=\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{s}]$ be the graded polynomial
algebra over the prime field of two elements, $\mathbb{F}_{2}$, in $s$
variables $x_{1},x_{2},\ldots,x_{s}$, each of degree $1$. We are interested in
the Peterson ”hit” problem of finding a minimal set of generators for $P_{s}$
as a graded left module over the mod-2 Steenrod algebra, $\mathscr{A}$. For
$s\geqslant 5,$ it is still open.

In this paper, we study the hit problem of five variables in a generic degree.
By using this result, we survey Singer’s conjecture [26] for the fifth
algebraic transfer in the respective degrees. This gives an efficient method
to study the algebraic transfer and it is different from the ones of Singer.

###### keywords:

Steenrod algebra, Peterson hit problem, Algebraic transfer. Steenrod squares,
invariant theory

###### MSC:

[2010] Primary 55S10; Secondary 55S05, 55T15.

##  1 Introduction and main results

Let $\mathscr{A}$ denote the mod-2 Steenrod algebra. This algebra is defined
to be the graded algebra over the prime field $\mathbb{F}_{2},$ generated by
the Steenrod squares $Sq^{i},$ in grading $i\geqslant 0,$ subject to the Adem
relations and $Sq^{0}=1$ (see Steenrod and Epstein [28]). From a topological
point of view, the Steenrod algebra is the algebra of stable cohomology
operations for ordinary cohomology $H^{*}$ over $\mathbb{F}_{2}.$

Let $P_{s}:=\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{s}]=\bigoplus_{d\geqslant
0}(P_{s})_{d}$ be the polynomial algebra viewed as a graded left module over
the Steenrod algebra at the prime $2.$ The grading is by the degree of the
homogeneous terms $(P_{s})_{d}$ of degree $d$ in $s$ variables with the degree
of each $x_{i}$ being $1.$ The algebra $P_{s}$ arises as the cohomology with
$\mathbb{F}_{2}$-coefficients of the product of $s$ copies of the infinite
real projective space. The Steenrod algebra acts by the composition of linear
operators on $P_{s}$ and the action of the Steenrod squares
$Sq^{i}:(P_{s})_{d}\to(P_{s})_{d+i}$ is determined by the Cartan formula and
its elementary properties (see [28, 45]).

A homogeneous element $f\in(P_{s})_{d}$ in $\mathscr{A}$-module $P_{s}$ is hit
if there is a finite sum $f=\sum_{j>0}Sq^{j}(f_{j}),$ where the homogeneous
elements $f_{j}\in P_{s}$ have grading strictly less than $d.$ We denote by
$QP_{s}:=\mathbb{F}_{2}\otimes_{\mathscr{A}}P_{s}$ the quotient of the left
$\mathscr{A}$-module $P_{s}$ by the hit elements in $\mathscr{A}^{+}\cdot
P_{s},$ where $\mathscr{A}^{+}$ is the augmentation ideal of $\mathscr{A}$ and
$\mathbb{F}_{2}$ is viewed as a right $\mathscr{A}$-module concentrated in
grading $0.$ Then, $QP_{s}$ is a graded vector space over $\mathbb{F}_{2}$ and
a basis for $QP_{s}$ lifts to a minimal generating set for $P_{s}$ as a module
over $\mathscr{A}.$

We study the hit problem of determination of a minimal set of generators for
$\mathscr{A}$-module $P_{s}$. The problem has first been studied by Peterson
[16], Wood [44], Singer [26], and Priddy [22], who show its relationship to
several classical problems in the homotopy theory such as cobordism theory of
manifolds, modular representation theory of the general linear group, and
stable homotopy type of classifying spaces of finite groups. Many authors then
investigated this problem (see Boardman [2], Bruner-Hà-Hưng [3], Crabb and
Hubbuck [4], Hưng [9], Kameko [10], Silverman [24], Silverman and Singer [25],
Walker and Wood [41, 42, 43], the present author [17]-[21], Sum [29]-[35] and
others).

For a non-negative integer $d,$ denote by $(QP_{s})_{d}$ the subspace of
$QP_{s}$ consisting of all the classes represented by the homogeneous
polynomials in $(P_{s})_{d}.$ One of the extremely useful tools for studying
the hit problem is Kameko’s homomorphism
$\widetilde{Sq^{0}_{*}}=(\widetilde{Sq^{0}_{*}})_{(s,2d+s)}:(QP_{s})_{2d+s}\to(QP_{s})_{d}$
(see [10]). This homomorphism is induced by the $\mathbb{F}_{2}$-linear map
$\psi:P_{s}\longrightarrow P_{s},$ given by

$\psi(x_{1}^{n_{1}}x_{2}^{n_{2}}\ldots
x_{s}^{n_{s}})=\begin{cases}{x_{1}^{\frac{n_{1}-1}{2}}x_{2}^{\frac{n_{2}-1}{2}}\ldots
x_{s}^{\frac{n_{s}-1}{2}}}&\text{if }n_{1},n_{2},\ldots,n_{s}\mbox{ odd},\\\
0&\text{otherwise}.\end{cases}$

for any monomial $x_{1}^{n_{1}}x_{2}^{n_{2}}\ldots x_{s}^{n_{s}}\in P_{s}.$
The map $\psi$ is not an $\mathscr{A}$-homomorphism. However, $\psi
Sq^{2i}=Sq^{i}\psi$ and $\psi Sq^{2i+1}=0$ for any $i\geqslant 0.$

The space $QP_{s}$ was explicitly determined for the cases $s\leqslant 4$ (see
[2, 10, 16, 32]). Based on the results of Wood [44] and Sum [35], the hit
problem is reduced to the calculation at the degrees $d$ of the form

$d=r(2^{t}-1)+2^{t}m,$ (1.1)

where $r,\,t,\,m$ are non-negative integers, $1\leqslant r=\mu(d)\leqslant s$
and $\mu(m)<r$ (see Subsection 2.2). Here, $\mu(n)$ denotes the smallest
number $k$ for which it is possible to write $n=\sum\limits_{1\leqslant
i\leqslant k}(2^{u_{i}}-1),$ where $u_{i}>0.$ For $r=s-1,$ the problem was
studied by Crabb and Hubbuck [4], Nam [14], Repka and Selick [23], Mothebe
[12], Sum [30, 32] and by us [17, 18]. For $r=s=5,$ it was explicitly
determined by Tín [37, 38, 39], Tín-Sum [40] with $m=1,2,3,$ by Sum [31, 35]
with $m=5,10$ and by the present author [19, 21] with $m=6,$ unknown in
general.

In the present paper, we study the hit problem of five variables in the
generic degree $d$ of the form (1.1) for $r=s-2=3$ and $m=1$. The following is
one of the main results.

######  Theorem 1.1.

Let $d=3(2^{t}-1)+2^{t}$ with $t$ a positive integer. The dimension of the
$\mathbb{F}_{2}$-vector space $(QP_{5})_{d}$ is determined by the following
table:

$d=3(2^{t}-1)+2^{t}$ $t=1$ $t=2$ $t=3$ $t=4$ $t=5$ $t\geqslant 6$
$\dim(QP_{5})_{d}$ $46$ $250$ $645$ $945$ $1115$ $1116$

Noting that if $t=0$ then $d=3(2^{t}-1)+2^{t}=1.$ Then, we can easily conclude
that $\dim(QP_{5})_{1}=5.$ The theorem has been proved by Sum [31, 35] for
$t=1$ and by Moetele-Mothebe [13] for $t=2.$ However, in this article, we
prove the case $t=2$ in another way (see Subsection 3.2). We know that
Kameko’s homomorphism

$\widetilde{Sq^{0}_{*}}=(\widetilde{Sq^{0}_{*}})_{(5,d)}:(QP_{5})_{d}\longrightarrow(QP_{5})_{\frac{d-5}{2}}$

is an epimorphism of $\mathbb{F}_{2}$-vector spaces. From this and our results
in [17, 18], to prove the theorem, we need only to determine the kernel of
$(\widetilde{Sq^{0}_{*}})_{(5,d)}.$ The calculations is long and very
technical. Our method is based on the Kameko homomorphism and our recent
results in [17, 18].

Let $GL_{s}:=GL(s,\mathbb{F}_{2})$ be the general linear group of rank $s$
over $\mathbb{F}_{2}.$ As it is known, the group $GL_{s}$ acts regularly on
$P_{s}$ by matrix substitution. Further, the two actions of $\mathscr{A}$ and
$GL_{s}$ upon $P_{s}$ commute with each other; hence there is an inherited
action of $GL_{s}$ on $QP_{s}.$ From this event, the hit problem becomes one
of the main tools for the studying of representations of the general linear
groups over $\mathbb{F}_{2}$. Recently, many authors showed their interest in
the study of this problem in conjunction with the transfer, which was defined
by Singer [26]. It is a homomorphism

$\varphi_{s}:\mbox{Tor}^{\mathscr{A}}_{s,s+*}(\mathbb{F}_{2},\mathbb{F}_{2})\longrightarrow(QP_{s})_{*}^{GL_{s}},$

where $\mbox{Tor}^{\mathscr{A}}_{s,s+*}(\mathbb{F}_{2},\mathbb{F}_{2})$ is
dual to Ext${}_{\mathscr{A}}^{s,s+*}(\mathbb{F}_{2},\mathbb{F}_{2})$, the
$E_{2}$ term of the Adams spectral sequence of spheres [1], and
$(QP_{s})_{*}^{GL_{s}}$ is the subspace of $(QP_{s})_{*}$ consisting of all
the $GL_{s}$-invariant classes. This transfer expected to be a useful tool in
describing the homology of the Steenrod algebra,
$\mbox{Tor}^{\mathscr{A}}_{*,*}(\mathbb{F}_{2},\mathbb{F}_{2}).$ Note that
computing explicitly the homology of the Steenrod algebra is one of the
problems of great importance in stable homotopy theory. But despite intensive
investigation for nearly half a century, the structure of this homology
algebra remains elusive.

Singer showed in [26] that $\varphi_{s}$ is an isomorphism for $s=1,2.$ Later,
Boardman [2], with additional calculations by Kameko [10], showed that
$\varphi_{3}$ is also an isomorphism. In [26], by using invariant theory,
Singer proved that this transfer is not a monomorphism in bidegree $(5,\,14)$
and made the following conjecture.

######  Conjecture 1.2 (Singer [26]).

The algebraic transfer $\varphi_{s}$ is an epimorphism for any $s>0.$

The conjecture is true for $s\leqslant 3$. It can be verified for $s=4$ by
using the results in Sum [29, 32]. The conjecture for $s\geqslant 5$ is an
open problem. Many authors have studied the conjecture for $s=4,5$ (see
Bruner-Hà-Hưng [3], Hưng [9], Chơn-Hà [6, 7], Hà [8], Nam [15], the present
author [19, 20], Sum [31, 33, 34, 35] and others). By using techniques of the
hit problem of five variables, we investigate Singer’s conjecture in bidegree
$(5,\,d+5),$ where $d$ is determined as in Theorem 1.1. This gives an
efficient method to study the Singer transfer and it is different from the
ones of Singer [26]. More explicitly, by using this method, we obtain the
following.

######  Theorem 1.3.

For any positive integer $t,$ we have
$(QP_{5})^{GL_{5}}_{3(2^{t}-1)+2^{t}}=0.$

Noting that the space $QP_{5}$ is a representation of $GL_{5}$ over
$\mathbb{F}_{2}.$ It is known that Kameko’s homomorphism

$\widetilde{Sq^{0}_{*}}=(\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})}:(QP_{5})_{3(2^{t}-1)+2^{t}}\longrightarrow(QP_{5})_{2^{t+1}-4}$

is an epimorphism of $GL_{5}$-modules. On the other hand,
$(QP_{5})^{GL_{5}}_{2^{t+1}-4}=0$ for all $t>0$ (see Sum [33]). Therefore, to
prove the theorem, we need only to determine all $GL_{5}$-invariants of ${\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})}).$ The computations are
based on Theorem 1.1.

From the results of Chen [5], Lin [11] and Tangora [36], we have ${\rm
Ext}_{\mathscr{A}}^{5,\,2^{t+2}+2}(\mathbb{F}_{2},\mathbb{F}_{2})=0$ for any
$t>0.$ By passing to the dual, one gets ${\rm
Tor}^{\mathscr{A}}_{5,2^{t+2}+2}(\mathbb{F}_{2},\mathbb{F}_{2})=0$ for all
$t>0.$ So, by Theorem 1.3, the transfer homomorphism $\varphi_{5}:{\rm
Tor}^{\mathscr{A}}_{5,2^{t+2}+2}(\mathbb{F}_{2},\mathbb{F}_{2})\to(QP_{5})^{GL_{5}}_{3(2^{t}-1)+2^{t}}$
is an isomorphism. Therefore, we get

######  Corollary 1.4.

Singer’s conjecture is true for the case $s=5$ and the degree
$3(2^{t}-1)+2^{t}$ with $t$ an arbitrary positive integer.

The paper contains five sections and is organized as follows: Background is
provided in Section 2. The proofs of Theorem 1.1 and 1.3 will be presented in
Section 3 and 4, respectively. All the admissible monomials of degree $d$ in
Theorem 1.1 are described in the Appendix.

Acknowledgment

This research is funded by Vietnam’s National Foundation for Science and
Technology Development (NAFOSTED) under grant number 101.04-2017.05.

The author appreciates anonymous referees for their careful corrections to,
valuable comments on, and helpful suggestions to the last version of this
paper.

The author would also like to express my profound gratitude to Professor
Nguyen Sum for pointing out some errors in the original paper and his help in
this work.

##  2 Background

In this section, we recall some needed information from Kameko [10], Phúc-Sum
[17], and Sum [32] on the weight vector of a monomial, the admissible
monomials and some homomorphisms.

###  2.1 The weight vector of a monomial

Let $\alpha_{j}(d)$ denote the $j$-th coefficients in dyadic expansion of a
non-negative integer $d.$ That means

$d=\alpha_{0}(d)2^{0}+\alpha_{1}(d)2^{1}+\cdots+\alpha_{j}(d)2^{j}+\cdots$

for $\alpha_{j}(d)\in\\{0,1\\}$ with $j\geqslant 0.$

For a monomial $u=x_{1}^{a_{1}}x_{2}^{a_{2}}\ldots x_{s}^{a_{s}}\in P_{s},$ we
define two sequences associated with $u$ by

$\displaystyle\omega(u)$ $\displaystyle=$
$\displaystyle(\omega_{1}(u),\omega_{2}(u),\ldots,\omega_{j}(u),\ldots),$
$\displaystyle\sigma(u)$ $\displaystyle=$
$\displaystyle(a_{1},a_{2},\ldots,a_{s}),$

where $\omega_{j}(u)=\sum\limits_{1\leqslant i\leqslant
s}\alpha_{j-1}(a_{i}).$ Noting that $\omega_{j}(u)\leqslant s$ for all $j.$
The sequence $\omega(u)$ is called the weight vector of the monomial $u$ and
$\sigma(u)$ called the exponent vector of the monomial $u.$

Let $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{i},\ldots)$ be a sequence of
non-negative integers. The sequence $\omega$ is called a weight vector if
$\omega_{i}=0$ for $i\gg 0.$ The sets of all the weight vectors and the
exponent vectors are given the left lexicographical order.

###  2.2 The admissible monomial

For a weight vector $\omega,$ we define $\deg\omega=\sum_{i\geqslant
1}2^{i-1}\omega_{i}.$ Denote by $P_{s}(\omega)$ the subspace of $P_{s}$
spanned by all monomials $x\in P_{s}$ such that $\deg(x)=\deg(\omega),\
\omega(x)\leqslant\omega,$ and by $P_{s}^{-}(\omega)$ the subspace of $P_{s}$
spanned by all monomials $x\in P_{s}$ such that $\deg(x)=\deg(\omega),\
\omega(x)<\omega.$

######  Definition 2.2.1 (see [10], [32]).

Let $\omega$ be a weight vector and $f,g$ two homogeneous polynomials of the
same degree in $P_{s}.$

  1. (i)

$f\equiv g$ if and only if $(f+g)\in\mathscr{A}^{+}\cdot P_{s}.$ If $f\equiv
0$ then $f$ is called hit.

  2. (ii)

$f\equiv_{\omega}g$ if and only if $f,g\in P_{s}(\omega)$ and
$(f+g)\in((\mathscr{A}^{+}\cdot P_{s}\cap P_{s}(\omega))+P_{s}^{-}(\omega)).$
If $f\equiv_{\omega}0$ then $f$ is called $\omega$-hit.

The Definition 2.2.1(ii) is a modification of a definition in Sum [32,
Definition 2.3]. Obviously, the relations $\equiv$ and $\equiv_{\omega}$ are
equivalence ones.

######  Definition 2.2.2 (see [10]).

Let $x,y$ be monomials of the same degree in $P_{s}.$ We say that $x<y$ if and
only if one of the following holds:

  1. (i)

$\omega(x)<\omega(y);$

  2. (ii)

$\omega(x)=\omega(y)$ and $\sigma(x)<\sigma(y).$

######  Definition 2.2.3 (see [10], [32]).

A monomial $x$ is said to be inadmissible if there exist monomials
$y_{1},y_{2},\ldots,y_{m}$ such that $y_{t}<x$ for $1\leqslant t\leqslant m$
and $x\equiv\sum\limits_{1\leqslant t\leqslant m}y_{t}.$

A monomial $x$ is said to be admissible if it is not inadmissible.

It is easy to see that the set of all the admissible monomials of degree $d$
in $P_{s}$ is a minimal set of $\mathscr{A}$-generators for $P_{s}$ in degree
$d.$ Therefore, $(QP_{s})_{d}$ is an $\mathbb{F}_{2}$-vector space with a
basis consisting of all the classes represent by the admissible monomials of
degree $d$ in $P_{s}.$

######  Definition 2.2.4 (see [10], [32]).

A monomial $x$ is said to be strictly inadmissible if and only if there exist
monomials $y_{1},y_{2},\ldots,y_{m}$ such that $y_{t}<x$ for $1\leqslant
t\leqslant m$ and

$x=\sum\limits_{1\leqslant t\leqslant m}y_{t}+\sum\limits_{1\leqslant
j\leqslant 2^{r}-1}Sq^{j}(h_{j})$

with $r={\rm max}\\{i\in\mathbb{Z}:\omega_{i}(x)>0\\}$ and suitable
polynomials $h_{j}\in P_{s}.$

Obviously, if $x$ is strictly inadmissible, then it is inadmissible. In
general, the converse is not true. For instance, the monomial
$x=x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{2}x_{5}^{2}x_{6}\in P_{6}$ is inadmissible,
but it is not strictly inadmissible.

######  Theorem 2.2.5 (see [10], [32]).

Let $x,y$ and $u$ be monomials in $P_{s}$ such that $\omega_{i}(x)=0$ for
$i>r>0,\omega_{t}(u)\neq 0$ and $\omega_{i}(u)=0$ for $i>t>0.$

  1. (i)

If $u$ is inadmissible, then $xu^{2^{r}}$ is also inadmissible.

  2. (ii)

If $u$ is strictly inadmissible, then $uy^{2^{t}}$ is also strictly
inadmissible.

Form now on, we use the following notations.

$\displaystyle P_{s}^{0}$ $\displaystyle=$
$\displaystyle\langle\\{x=x_{1}^{a_{1}}x_{2}^{a_{2}}\ldots x_{s}^{a_{s}}\in
P_{s}\;|\;a_{1}a_{2}\ldots a_{s}=0\\}\rangle,$ $\displaystyle P_{s}^{+}$
$\displaystyle=$ $\displaystyle\langle\\{x=x_{1}^{a_{1}}x_{2}^{a_{2}}\ldots
x_{s}^{a_{s}}\in P_{s}\;|\;a_{1}a_{2}\ldots a_{s}>0\\}\rangle.$

Then $P_{s}^{0}$ and $P_{s}^{+}$ are the $\mathscr{A}$-submodules of $P_{s}.$
Furthermore, we have a direct summand decomposition of the
$\mathbb{F}_{2}$-vector spaces

$QP_{s}=QP_{s}^{0}\,\bigoplus\,QP_{s}^{+}.$

Here $QP_{s}^{0}:=\mathbb{F}_{2}\otimes_{\mathscr{A}}P_{s}^{0}$ and
$QP_{s}^{+}:=\mathbb{F}_{2}\otimes_{\mathscr{A}}P_{s}^{+}.$

Denote by $QP_{s}(\omega)$ the quotient of $P_{s}(\omega)$ by the equivalence
relation $\equiv_{\omega}.$ According to Walker and Wood [43], we have a
filtration of $QP_{s}$:

$\\{0\\}\subseteq\cdots\subseteq P_{s}^{-}(\omega)/((\mathscr{A}^{+}\cdot
P_{s})\cap P_{s}^{-}(\omega))\subseteq P_{s}(\omega)/((\mathscr{A}^{+}\cdot
P_{s})\cap P_{s}(\omega))\subseteq\cdots\subseteq P_{s}/(\mathscr{A}^{+}\cdot
P_{s})=QP_{s}.$

Note that this is not only a filtration of $QP_{s}$ as a vector space, but
also as a $GL_{s}$-module. The inclusion of $P_{s}^{-}(\omega)$ into
$P_{s}(\omega)$ induces a monomorphism

$P_{s}^{-}(\omega)/((\mathscr{A}^{+}\cdot P_{s})\cap P_{s}^{-}(\omega))\to
P_{s}(\omega)/((\mathscr{A}^{+}\cdot P_{s})\cap P_{s}(\omega))$

and the sequence is short exact:

$0\to P_{s}^{-}(\omega)/((\mathscr{A}^{+}\cdot P_{s})\cap
P_{s}^{-}(\omega))\to P_{s}(\omega)/((\mathscr{A}^{+}\cdot P_{s})\cap
P_{s}(\omega))\to P_{s}(\omega)/(((\mathscr{A}^{+}\cdot P_{s})\cap
P_{s}(\omega))+P_{s}^{-}(\omega))\to 0.$

Combining this and Definition 2.2.1(ii), one gets

$QP_{s}(\omega)=P_{s}(\omega)/((\mathscr{A}^{+}\cdot P_{s}\cap
P_{s}(\omega))+P_{s}^{-}(\omega))\cong(P_{s}(\omega)/((\mathscr{A}^{+}\cdot
P_{s})\cap P_{s}(\omega)))/(P_{s}^{-}(\omega)/((\mathscr{A}^{+}\cdot
P_{s})\cap P_{s}^{-}(\omega))).$

By Sum [35], $QP_{s}(\omega)$ is also an $GL_{s}$-module, further this
isomorphism is also an isomorphism of $GL_{s}$-modules. Combining this and the
above filtration of $QP_{s}$, we have immediately:

$\dim((QP_{s})_{d})=\sum_{\deg(\omega)=d}\dim(QP_{s}(\omega)),\ \
\dim((QP_{s})^{GL_{s}}_{d})\leqslant\sum_{\deg(\omega)=d}\dim(QP_{s}(\omega)^{GL_{s}}).$

Denote by $\mathscr{B}_{s}(d)$ the set of all admissible monomials of degree
$d$ in $P_{s}.$ Set

$\mathscr{B}_{s}^{0}(d):=\mathscr{B}_{s}(d)\cap(P_{s}^{0})_{d},\,\,\mathscr{B}_{s}^{+}(d):=\mathscr{B}_{s}(d)\cap(P_{s}^{+})_{d}.$

If $\omega$ is a weight vector of degree $d,$ we set

$\mathscr{B}_{s}(\omega):=\mathscr{B}_{s}(d)\cap
P_{s}(\omega),\,\mathscr{B}_{s}^{0}(\omega):=\mathscr{B}_{s}(\omega)\cap(P^{0}_{s})_{d},\,\mathscr{B}_{s}^{+}(\omega):=\mathscr{B}_{s}(\omega)\cap(P^{+}_{s})_{d}.$

For a polynomial $f\in P_{s},$ we denote by $[f]$ the classes in $QP_{s}$
represented by $f.$ If $\omega$ is a weight vector and $f\in P_{s}(\omega),$
then denote by $[f]_{\omega}$ the classes in $QP_{s}(\omega)$ represented by
$f.$ For a subset $\mathscr{B}\subset P_{s},$ we denote
$[\mathscr{B}]=\\{[f]\,:\,f\in\mathscr{B}\\}.$ If $\mathscr{B}\subset
P_{s}(\omega),$ then we set
$[\mathscr{B}]_{\omega}=\\{[f]_{\omega}\,:\,f\in\mathscr{B}\\}.$ Thus, we see
that
$[\mathscr{B}_{s}(\omega)]_{\omega},\,[\mathscr{B}^{0}_{s}(\omega)]_{\omega}$
and $[\mathscr{B}_{s}^{+}(\omega)]_{\omega}$ are respectively the bases of the
$\mathbb{F}_{2}$-vector spaces $QP_{s}(\omega),\
QP_{s}^{0}(\omega):=QP_{s}(\omega)\cap(QP^{0}_{s})_{d}$ and
$QP_{s}^{+}(\omega):=QP_{s}(\omega)\cap(QP^{+}_{s})_{d}.$

For a natural number $d,$ we set $\mu(d)=\mbox{min}\big{\\{}r\in\mathbb{N}:\
d=\sum\limits_{1\leqslant i\leqslant r}(2^{u_{i}}-1),\ u_{i}>0\big{\\}}.$ By
Wood [44], $(QP_{s})_{d}=0$ if $\mu(d)>s.$ By Sum [35], $\mu(d)=r\leqslant s$
if and only if there exists uniquely a sequence of integers
$u_{1}>u_{2}>\cdots>u_{r-1}\geqslant u_{r}>0$ such that $d=\sum_{1\leqslant
i\leqslant r}(2^{u_{i}}-1).$ From these results, we need only to determine the
space $(QP_{s})_{d}$ in degrees $d$ with $1\leqslant\mu(d)=r\leqslant s$ and
$d$ has the form

$d=r(2^{t}-1)+2^{t}.m,$

where $t,\,m$ are non-negative integers and $\mu(m)<r.$ Indeed, if
$\mu(d)=r=1,$ then $d$ has the form

$d=2^{u}-1=1.(2^{u}-1)+2^{u}.0=r.(2^{u}-1)+2^{u}.m,$

where $u>0$ and $\mu(m=0)=0<1.$ If $\mu(d)=r\geqslant 2,$ then by Sum [35],
$d$ has the form $\sum_{1\leqslant i\leqslant r}(2^{u_{i}}-1),$ where
$u_{1}>u_{2}>\cdots>u_{r-1}\geqslant u_{r}>0.$ Consider $m=\sum_{1\leqslant
j\leqslant r-1}(2^{u_{j}-u_{r}}-1).$ According to Sum [35], we see that
$\mu(m)=r-2<r$ if $u_{r-1}=u_{r}$ and $\mu(m)=r-1<r$ if $u_{r-1}>u_{r}.$
Further, $d=r(2^{t}-1)+2^{t}.m$ with $t=u_{r}.$

We end this section by recalling some definitions in [32], which will be used
in the next sections.

For $1\leqslant i\leqslant s,$ define the homomorphism
$\rho_{i}:P_{s-1}\rightarrow P_{s}$ of algebras by substituting

$\rho_{i}(x_{j})=\left\\{\begin{array}[]{ll}{x_{j}}&\text{if }\;1\leqslant
j<i,\\\ x_{j+1}&\text{if}\;i\leqslant j<s.\end{array}\right.$

The maps $\rho_{i}$ are also the monomorphism of $\mathscr{A}$-modules.

We now denote

$\mathcal{N}_{s}:=\\{(i;I)\;|\;I=(i_{1},i_{2},\ldots,i_{r}),1\leqslant
i<i_{1}<i_{2}<\ldots<i_{r}\leqslant s,0\leqslant r<s\\}.$

Here, by convention, $I=\emptyset,$ if $r=0.$ Denote by $r=\ell(I)$ the length
of $I.$

######  Definition 2.2.6.

Let $(i;I)\in\mathcal{N}_{s},r=\ell(I),$ and let $u$ be an integer with
$1\leqslant u\leqslant r.$ A monomial $x=x_{1}^{a_{1}}x_{2}^{a_{2}}\ldots
x_{s-1}^{a_{s-1}}\in P_{s-1}$ is said to be $u$-compatible with $(i;I)$ if all
of the following hold:

  1. (i)

$a_{i_{1}-1}=a_{i_{2}-1}=\ldots=a_{i_{(u-1)}-1}=2^{r}-1,$

  2. (ii)

$a_{i_{u}-1}>2^{r}-1,$

  3. (iii)

$\alpha_{r-t}(a_{i_{u}-1})=1,\;\forall t,\ 1\leqslant t\leqslant u,$

  4. (iv)

$\alpha_{r-t}(a_{i_{t}-1})=1,\;\forall t,\ u<t\leqslant r.$

Clearly, a monomial $x$ can be $u$-compatible with a given
$(i;I)\in\mathcal{N}_{s}$ for at most one value of $u.$ By convention, $x$ is
$1$-compatible with $(i;\emptyset).$

######  Definition 2.2.7.

Let $(i;I)\in\mathcal{N}_{s},0<r<s.$ Denote

$x_{(I,u)}=x_{i_{u}}^{2^{r-1}+2^{r-2}+\,\cdots\,+2^{r-u}}\prod\limits_{u<t\leqslant
r}x_{i_{t}}^{2^{r-t}}$

for $1\leqslant u\leqslant r,\;x_{(\emptyset,1)}=1.$ For a monomial $x\in
P_{s-1},$ we define the monomial $\phi_{(i;I)}(x)$ in $P_{s}$ by setting

$\phi_{(i;I)}(x)=\left\\{\begin{array}[]{ll}\dfrac{x_{i}^{2^{r}-1}\rho_{i}(x)}{x_{(I,u)}}&\text{if
there exist $u$ such that $x$ is $u$-compatible with $(i;I)$},\\\
0&\text{otherwise}.\end{array}\right.$

Then we have an $\mathbb{F}_{2}$-linear map $\phi_{(i;I)}:P_{s-1}\rightarrow
P_{s}.$ Inparticular, $\phi_{(i;\emptyset)}=\rho_{i}.$ Noting that
$\phi_{(i;I)}$ is not an $\mathscr{A}$-homomorphism. Moreover, we see that if
$\phi_{(i;I)}(x)\neq 0,$ then $\omega(\phi_{(i;I)}(x))=\omega(x).$

For a subset $U\subset P_{s-1},$ denote

$\begin{array}[]{ll}\Phi^{0}(U)&=\bigcup\limits_{1\leqslant i\leqslant
s}\phi_{(i;\emptyset)}(U)=\bigcup\limits_{1\leqslant i\leqslant
s}\rho_{i}(U),\\\
\Phi^{+}(U)&=\bigcup\limits_{(i;I)\in\mathcal{N}_{s},\;0<\ell(I)<s}\phi_{(i;I)}(U)\setminus
P_{s}^{0}.\end{array}$

It is easy to see that if $U$ is a minimal set of generators for
$\mathscr{A}$-module $P_{s-1}$ in degree $d$, then $\Phi^{0}(U)$ is also a
minimal set of generators for $\mathscr{A}$-module $P_{s}^{0}$ in degree $d.$

######  Definition 2.2.8.

For any $(i;I)\in\mathcal{N}_{s},$ we define the homomorphism
$p_{(i;I)}:P_{s}\rightarrow P_{s-1}$ of algebras by substituting

$p_{(i;I)}(x_{j})=\left\\{\begin{array}[]{ll}{x_{j}}&\text{if }\;1\leqslant
j<i,\\\ \sum\limits_{k\in I}x_{k-1}&\text{if}\;j=i,\\\
x_{j-1}&\text{if}\;i<j\leqslant s.\end{array}\right.$

Noting that $p_{(i;I)}$ is also a homomorphism of $\mathscr{A}$-modules. In
particular, we have $p_{(i;\emptyset)}(x_{i})=0$ for $1\leqslant i\leqslant s$
and $p_{(i;I)}(\rho_{i}(u))=u$ for any $u\in P_{s-1}.$

######  Lemma 2.2.9 (see [17]).

If $x$ is a monomial in $P_{s},$ then $p_{(i;I)}(x)\in P_{s-1}(\omega(x)).$

This result implies that if $\omega$ is a weight vector and $x\in
P_{s}(\omega(x)),$ then $p_{(i;I)}(x)\in P_{s-1}(\omega).$ Futhermore,
$p_{(i;I)}$ passes to a homomorphism from $QP_{s}(\omega)$ to
$QP_{s-1}(\omega).$

In Section 3, we use Lemma 2.2.9 and the results in Sum [32] to prove a
certain subset of $QP_{s}$ is linearly independent. More precisely, let
$\mathscr{B}$ be a finite subset of $P_{s}$ consisting of some monomials of
degree $d.$ Denote by $|\mathscr{B}|$ the cardinal of $\mathscr{B}.$ To prove
the set $[\mathscr{B}]$ is linearly independent in $(QP_{s})_{d},$ we denote
the elements of $\mathscr{B}$ by $b_{d,\,k},\ 1\leqslant k\leqslant
n=|\mathscr{B}|$ and assume that there is a linear relation

$\mathcal{S}=\sum\limits_{k=1}^{n}\gamma_{k}b_{d,\,k}\equiv 0,$

with $\gamma_{k}\in\mathbb{F}_{2}$ for all $k,\,1\leqslant k\leqslant n.$ For
$(i;I)\in\mathcal{N}_{s},$ we explicitly compute $p_{(i;I)}(\mathcal{S})$ in
terms of the admissible monomials in $P_{s-1}\,({\rm mod}(\mathscr{A}^{+}\cdot
P_{s-1})).$ Computing from some relations $p_{(i;I)}(\mathcal{S})\equiv 0$
with $(i;I)\in\mathcal{N}_{s},$ we obtain $\gamma_{k}=0$ for all $k.$

##  3 $\mathscr{A}$-generators for $P_{5}$ in degree $3(2^{t}-1)+2^{t}$

In this section, we prove Theorem 1.1. More precisely, we explicitly determine
the admissible monomials of degree $3(2^{t}-1)+2^{t}$ in $P_{5}.$ We first
recall a result of Singer [27] on the hit monomials in $P_{s}.$

###  3.1 Singer’s criterion on the hit monomials

######  Definition 3.1.1.

A monomial $z=x_{1}^{b_{1}}x_{2}^{b_{2}}\ldots x_{s}^{b_{s}}$ in $P_{s}$ is
called a spike if $b_{i}=2^{t_{i}}-1$ for $t_{i}$ a non-negative integer and
$i=1,2,\ldots,s.$ If $z$ is a spike with $t_{1}>t_{2}>\ldots>t_{r-1}\geqslant
t_{r}>0$ and $t_{j}=0$ for $j>r,$ then it is called a minimal spike.

######  Lemma 3.1.2.

All the spikes in $P_{s}$ are admissible and their weight vectors are weakly
decreasing. Furthermore, if a weight vector
$\omega=(\omega_{1},\omega_{2},\ldots)$ is weakly decreasing and
$\omega_{1}\leqslant s,$ then there is a spike $\mathscr{Z}$ in $P_{s}$ such
that $\omega(\mathscr{Z})=\omega.$

###### Proof.

Let $\mathscr{Z}=\prod\limits_{1\leqslant i\leqslant s}x_{i}^{2^{d_{i}}-1}$ be
a spike monomial in $P_{s}.$ Suppose that $\mathscr{Z}$ is inadmissible. Then,
there exist monomials $y_{1},y_{2},\ldots,y_{m}\in P_{s}$ such that
$y_{t}<\mathscr{Z}$ for all $t$ and $\mathscr{Z}=\sum_{1\leqslant t\leqslant
m}y_{t}+\sum\limits_{j>0}Sq^{j}(w_{j}),$ with suitable monomial $w_{j}\in
P_{s}.$ This implies that there is a monomial $u\in P_{s}$ with $\deg
u<\deg\mathscr{Z}$ such that $Sq^{n}(u)=\mathscr{Z}+\sum_{i}X_{i},$ where
$n>0$ and $X_{i}\in P_{s},\ \deg X_{i}=\deg\mathscr{Z}.$ Suppose
$u=\prod\limits_{1\leqslant\ell\leqslant k}x_{\ell}^{a_{\ell}}.$ Since $\deg
u<\deg\mathscr{Z},$ ther is a number $t,\,1\leqslant t\leqslant s$ such that
$a_{t}<2^{d_{t}}-1.$ According to Cartan’s formula, we have

$\mathscr{Z}+f=Sq^{n}(u)=\sum\limits_{i\geqslant
0}Sq^{i}(x_{t}^{a_{t}})Sq^{n-i}(x_{1}^{a_{1}}\ldots
x_{t-1}^{a_{t-1}}x_{t+1}^{a_{t+1}}\ldots x_{s}^{a_{s}}).$

Since $2^{d_{t}}-1-a_{t}>0$ and $\mathscr{Z}$ is a term of
$\sum\limits_{i\geqslant 0}Sq^{i}(x_{t}^{a_{t}})Sq^{n-i}(x_{1}^{a_{1}}\ldots
x_{t-1}^{a_{t-1}}x_{t+1}^{a_{t+1}}\ldots x_{s}^{a_{s}}),$ hence $\mathscr{Z}$
is a term of
$Sq^{2^{d_{t}}-1-a_{t}}(x_{t}^{a_{t}})Sq^{n-(2^{d_{t}}-1-a_{t})}(x_{1}^{a_{1}}\ldots
x_{t-1}^{a_{t-1}}x_{t+1}^{a_{t+1}}\ldots x_{s}^{a_{s}}).$ On the other hand,
$2^{d_{t}}-1-a_{t}\leqslant 2^{d_{t}}-1,$ so there exist a number
$\ell\geqslant 0$ such that $\alpha_{\ell}(2^{d_{t}}-1-a_{t})=1.$ This implies
$\alpha_{\ell}(a_{t})=0.$ Since $\binom{a_{t}}{2^{d_{t}}-1-a_{t}}$ is even,
$Sq^{2^{d_{t}}-1-a_{t}}(x_{t}^{a_{t}})=0.$ So, one gets $\mathscr{Z}=0.$ This
contradicts the fact that $\mathscr{Z}=\prod\limits_{1\leqslant i\leqslant
s}x_{i}^{2^{d_{i}}-1}\neq 0.$ Obviously, if $\mathscr{Z}$ is a spike, then its
weight vector is weakly decreasing.

Now, suppose that $\omega=(\omega_{1},\omega_{2},\ldots)$ is weakly decreasing
and $\omega_{1}\leqslant s.$ For $j\geqslant s$, we denote by $a_{j}$ the non-
negative integer such that

$\alpha_{i}(a_{j})=\begin{cases}1,&\text{if }\omega_{i+1}\geqslant j,\\\
0,&\text{if }\omega_{i+1}<j.\end{cases}$

Denote by $s_{j}$ the greatest integer such that $\omega_{d_{j}}\geqslant j$.
Since $\omega$ is weakly decreasing, $\alpha_{i}(a_{j})=1$ for $1\leqslant
i<d_{j}$ and $\alpha_{i}(a_{j})=0$ with $i\geqslant d_{j}$. So,
$a_{j}=2^{d_{j}}-1$. On the other hand, we have
$\omega_{i+1}\leqslant\omega_{1}\leqslant s$ for all $i\geqslant 0.$ Hence, if
$j>s$, then $a_{j}=0$. Therefore,
$\mathscr{Z}=x_{1}^{a_{1}}x_{2}^{a_{2}}\ldots x_{s}^{a_{s}}$ is a spike in
$P_{s}$ and $\omega(\mathscr{Z})=\omega$. The lemma is proved. ∎

In [27], Singer showed that if $\mu(d)\leqslant s,$ then there exists uniquely
a minimal spike of degree $d$ in $P_{s}.$ Further, we have the following.

######  Theorem 3.1.3 (see Singer [27]).

Suppose that $x\in P_{s}$ is a monomial of degree $d,$ where $\mu(d)\leqslant
s.$ Let $z$ be the minimal spike of degree $d$ in $P_{s}.$ If
$\omega(x)<\omega(z),$ then $x$ is hit.

###  3.2 Proof of Theorem 1.1

For convenience, we use the following notations:

$\displaystyle\mathbb{N}_{s}$ $\displaystyle=$
$\displaystyle\\{1,2,\ldots,s\\},$ $\displaystyle X_{\mathbb{I}}$
$\displaystyle=$ $\displaystyle
X_{\\{i_{1},i_{2},\ldots,i_{r}\\}}=\prod\limits_{i\in\mathbb{N}_{s}\setminus\mathbb{I}}x_{i},\,\,\mathbb{I}=\\{i_{1},i_{2},\ldots,i_{r}\\}\subseteq\mathbb{N}_{s}.$

In particular, $X_{\mathbb{N}_{s}}=1,\ X_{\emptyset}=x_{1}x_{2}\ldots x_{s},$
and $X_{\\{i\\}}=x_{1}\ldots\hat{x}_{i}\ldots x_{s},\,\,i=1,2,\ldots,s.$ Let
$x=x_{1}^{a_{1}}x_{2}^{a_{2}}\ldots x_{s}^{a_{s}}\in P_{s}.$ Set
$\mathbb{I}_{j}(x)=\\{i\in\mathbb{N}_{s}:\;\alpha_{j}(a_{i})=0\\},$ for
$j\geqslant 0.$ Then we have $x=\prod\limits_{j\geqslant
0}X_{\mathbb{I}_{j}(x)}^{2^{j}}.$

It is known that Kameko’s homomorphism

$\widetilde{Sq^{0}_{*}}=(\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})}:(QP_{5})_{3(2^{t}-1)+2^{t}}\longrightarrow(QP_{5})_{2^{t+1}-4}$

is an epimorphism, so we have

$(QP_{5})_{3(2^{t}-1)+2^{t}}\cong{\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})})\,\bigoplus\,(QP_{5})_{2^{t+1}-4}.$

The space $(QP_{5})_{2^{t+1}-4}$ has been explicitly computed in [17, 18].

######  Theorem 3.2.1 (see [17, 18]).

Let $n=2^{t+1}-4$ with $t$ a positive integer. The dimension of the
$\mathbb{F}_{2}$-vector space $(QP_{5})_{n}$ is determined by the following
table:

$n=2^{t+1}-4$ $t=1$ $t=2$ $t=3$ $t=4$ $t=5$ $t\geqslant 6$ $\dim(QP_{5})_{n}$
$1$ $45$ $190$ $480$ $650$ $651$

Noting that for $t=1,$ $2^{t+1}-4=0$ and $(QP_{5})_{0}\cong\mathbb{F}_{2}.$
Hence, $\dim((QP_{5})_{0})=\dim(\mathbb{F}_{2})=1.$

Now, we determine ${\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})}).$ Denote by

$\omega_{(5,t)}=(3,3,\ldots,3,1),\,\mbox{and
}\,\omega^{*}_{(5,t)}=(3,3,\ldots,3)\ \mbox{($t$ times of $3$)}.$

######  Lemma 3.2.2.

If $x$ is an admissible monomial of degree $3(2^{t}-1)+2^{t}$ in $P_{5}$ and
$[x]$ belongs to ${\rm Ker}((\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})})$
then $\omega(x)=\omega_{(5,t)}.$

###### Proof.

We prove the lemma by induction on $t.$ By Sum [31, 35], we see that the lemma
holds for $t=1.$

Suppose $t>1$ and the lemma holds for $1,2,\ldots,t-1$. Observe that the
monomial $z=x_{1}^{2^{t+1}-1}x_{2}^{2^{t}-1}x_{3}^{2^{t}-1}$ is the minimal
spike of degree $3(2^{t}-1)+2^{t}$ in $P_{5}$ and $\omega(z)=\omega_{(5,t)}.$

Since $3(2^{t}-1)+2^{t}$ is odd and $[x]\neq[0],$ one gets either
$\omega_{1}(x)=3$ or $\omega_{1}(x)=5.$ If $\omega_{1}(x)=5,$ then
$x=X_{\emptyset}y^{2}$ with $y$ a monomial of degree $2^{t+1}-4$ in $P_{5}.$
Since $x$ is admissible, by Theorem 2.2.5, $y$ is admissible. Hence,
$(\widetilde{Sq}_{*}^{0})_{(5,3(2^{t}-1)+2^{t})}([x])=[y]\neq[0].$ This
contradicts the face that $[x]\in{\rm
Ker}((\widetilde{Sq_{*}^{0}})_{(5,3(2^{t}-1)+2^{t})}),$ so $\omega_{1}(x)=3.$
Then, we have $x=X_{\\{i,j\\}}u^{2}$ with $1\leqslant i<j\leqslant 5$ and $u$
an admissible monomial of degree $3(2^{t-1}-1)+2^{t-1}$ in $P_{5}.$ Now, the
lemma follows from the inductive hypothesis. ∎

From Lemma 3.2.2, we get

${\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})})=(QP_{5}^{0})_{3(2^{t}-1)+2^{t}}\,\bigoplus\,\mathbb{V},$

where $\mathbb{V}=({\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})})\cap(QP_{5}^{+})_{3(2^{t}-1)+2^{t}})$
and $\dim(\mathbb{V})=\dim(QP_{5}^{+}(\omega_{(5,t)})).$

The case $t=1$. We have $d=3(2^{t}-1)+2^{t}=5.$ According to Sum [31, 35], we
have $|\mathscr{B}_{5}^{0}(5)|=|\mathscr{B}_{5}^{0}(\omega_{(5,1)})|=45$ and
$|\mathscr{B}_{5}^{+}(\omega_{(5,1)})|=0.$ Combining this with Theorem 3.2.1,
one gets $\dim(QP_{5})_{5}=46.$

The case $t=2$. We have $3(2^{t}-1)+2^{t}=13.$ In [13], for $s\geqslant 1,$
Moetele and Mothebe showed that $\dim(QP_{s})_{13}=\sum\limits_{3\leqslant
j\leqslant 13}\binom{d}{j}C_{j},$ where $C_{j},\ 3\leqslant j\leqslant 13,$
are determined by the following table:

$j$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$ $13$ $C_{j}$ $3$ $23$ $105$
$268$ $415$ $438$ $322$ $164$ $55$ $11$ $1$

This implies $\dim(QP_{5})_{13}=250.$ Their methods are based on Kameko’s
homomorphism [10] and a result on the admissible monomials [13, Theorem 2.7].
In this section, we prove this result in another way. To compute
$(QP_{5})_{13},$ we determine all the admissible monomials of degree $13$ in
$P_{5}$ by using Lemma 2.2.9 and some homomorphisms in Subsection 2.2.

By Sum [32], we have
$|\mathscr{B}_{4}(13)|=|\mathscr{B}_{4}(\omega_{(5,2)})|=35.$ A direct
computation shows that

$\mathscr{B}_{5}^{0}(13)=\mathscr{B}_{5}^{0}(\omega_{(5,2)})=\Phi^{0}(\mathscr{B}_{4}(13))$

is the set consisting of $145$ admissible monomials: $q_{2,k},\ 1\leqslant
k\leqslant 145$ (see Subsection 5.1). Hence, one gets
$\dim(QP_{5}^{0})_{13}=145.$

Next, we compute $QP_{5}^{+}(\omega_{(5,2)}).$ By a direct computation, we
obtain

$\Phi^{+}(\mathscr{B}_{4}(\omega_{(5,2)}))\cup\\{x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{3}x_{5}^{3},\
\ x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}x_{5}^{2},\ \
x_{1}^{3}x_{2}^{4}x_{3}x_{4}^{3}x_{5}^{2},\ \
x_{1}^{3}x_{2}^{4}x_{3}x_{4}^{2}x_{5}^{3}\\}$

is the set consisting of 60 admissible monomials: $b_{13,\,k},\ 1\leqslant
k\leqslant 60$ (see Subsection 5.1).

######  Proposition 3.2.3.

The set $\\{[b_{13,\,k}]_{\omega_{(5,2)}},\ 1\leqslant k\leqslant 60\\}$ is a
basis of $\mathbb{F}_{2}$-vector space $QP_{5}^{+}(\omega_{(5,2)}).$

We need some lemmas for the proof of the proposition. The following lemma is
proved by a direct computation.

######  Lemma 3.2.4.

The following monomials are strictly inadmissible:

  1. (i)

$x_{i}^{2}x_{j}^{2}x_{k}x_{\ell}x_{m}^{3},\
x_{i}^{2}x_{j}x_{k}^{2}x_{\ell}x_{m}^{3},\
x_{i}^{2}x_{j}x_{k}x_{\ell}^{2}x_{m}^{3},\ i<j<k<\ell.$ Here $(i,j,k,\ell,m)$
is a permutation of $(1,\,2,\,3,\,4,\,5).$

  2. (ii)

$\rho_{i}(u),\ 1\leqslant i\leqslant 5,$ where $u$ is one of the following
monomials:

$x_{1}^{2}x_{2}x_{3}^{3}x_{4}^{3},\ x_{1}^{2}x_{2}^{3}x_{3}x_{4}^{3},\
x_{1}^{2}x_{2}^{3}x_{3}^{3}x_{4},\ x_{1}^{3}x_{2}^{2}x_{3}x_{4}^{3},\
x_{1}^{3}x_{2}^{2}x_{3}^{3}x_{4},\ x_{1}^{3}x_{2}^{3}x_{3}^{2}x_{4},\
x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}^{3}.$

######  Lemma 3.2.5.

The following monomials are strictly inadmissible:

$x_{1}x_{2}^{2}x_{3}^{2}x_{4}x_{5}^{7}$, |  $x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{3}x_{5}^{5}$, |  $x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{5}x_{5}^{3}$, |  $x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{7}x_{5}$, |  $x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{2}x_{5}^{5}$,
---|---|---|---|---
$x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{6}x_{5}$, |  $x_{1}x_{2}^{2}x_{3}^{6}x_{4}x_{5}^{3}$, |  $x_{1}x_{2}^{2}x_{3}^{6}x_{4}^{3}x_{5}$, |  $x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{2}x_{5}$, |  $x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{2}x_{5}^{5}$,
$x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{6}x_{5}$, |  $x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{2}x_{5}$, |  $x_{1}x_{2}^{6}x_{3}^{2}x_{4}x_{5}^{3}$, |  $x_{1}x_{2}^{6}x_{3}^{2}x_{4}^{3}x_{5}$, |  $x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{2}x_{5}$,
$x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{2}x_{5}$, |  $x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{2}x_{5}^{5}$, |  $x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{6}x_{5}$, |  $x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{2}x_{5}$, |  $x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{2}x_{5}$,
$x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{2}x_{5}.$ |  |  |  |

###### Proof.

We prove the lemma for $x=x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{3}x_{5}^{5}.$ The
others can be proven by a similar computation. By a direct computation using
the Cartan formula, we have

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0ptx&=x_{1}x_{2}^{2}x_{3}x_{4}^{3}x_{5}^{6}+Sq^{1}(x_{1}x_{2}x_{3}^{2}x_{4}^{3}x_{5}^{5}+x_{1}x_{2}x_{3}^{2}x_{4}^{5}x_{5}^{3}+x_{1}x_{2}x_{3}^{4}x_{4}^{3}x_{5}^{3}+x_{1}x_{2}^{2}x_{3}x_{4}^{3}x_{5}^{5}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}x_{2}^{2}x_{3}x_{4}^{5}x_{5}^{3}+x_{1}x_{2}^{4}x_{3}x_{4}^{3}x_{5}^{3}.+x_{1}^{2}x_{2}x_{3}x_{4}^{5}x_{5}^{3}+x_{1}^{4}x_{2}x_{3}x_{4}^{3}x_{5}^{3})\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+Sq^{2}(x_{1}x_{2}x_{3}x_{4}^{3}x_{5}^{5}+x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{3}x_{5}^{3}+x_{1}^{2}x_{2}x_{3}^{2}x_{4}^{3}x_{5}^{3}+x_{1}^{2}x_{2}^{2}x_{3}x_{4}^{3}x_{5}^{3})\\\
&\quad+Sq^{4}(x_{1}x_{2}x_{3}x_{4}^{3}x_{5}^{3})\ \rm
mod(P_{5}^{-}(\omega_{(5,2)})).\end{array}$

This equality implies that $x$ is strictly inadmissible. ∎

###### Proof of Proposition 3.2.3.

Let $x$ be an admissible monomial of degree $13$ in $P_{5}$ and
$\omega(x)=\omega_{(5,2)}.$ Then, $x=X_{\\{i,j\\}}u^{2}$ with $1\leqslant
i<j\leqslant 5$ and $u$ a monomial of degree $5$ in $P_{5}.$ Since $x$ is
admissible, according to Theorem 2.2.5, we get
$u\in\mathscr{B}_{5}(\omega_{(5,1)}).$

By a direct computation we see that for all
$y\in\mathscr{B}_{5}(\omega_{(5,1)}),$ such that $X_{\\{i,j\\}}y^{2}\neq
b_{13,\,k},\ \forall k,\ 1\leqslant k\leqslant 60,$ there is a monomial $w$
which is given in one of Lemmas 3.2.4, and 3.2.5 such that
$X_{\\{i,j\\}}y^{2}=wz^{2^{r}}$ with a monomial $z\in P_{5}$ and $r={\rm
max}\\{m\in\mathbb{Z}\ :\ \omega_{m}(w)>0\\}.$ By Theorem 2.2.5,
$X_{\\{i,j\\}}y^{2}$ is inadmissible. Since $x=X_{\\{i,j\\}}u^{2}$ is
admissible, one gets $x=b_{13,\,k}$ for suitable $k.$

We now prove the set $\\{[b_{13,\,k}]_{\omega_{(5,2)}}\ :\ 1\leqslant
k\leqslant 60\\}$ is linearly independent in $(QP_{5})_{13}.$ Suppose there is
a linear relation

$\mathcal{S}=\sum\limits_{1\leqslant k\leqslant
60}\gamma_{k}b_{13,\,k}\equiv_{\omega_{(5,2)}}0,$ (3.2.1)

where $\gamma_{k}\in\mathbb{F}_{2}.$

From a result in [32], $\dim QP_{4}^{+}(\omega_{(5,2)})=23,$ with the basis
$\\{[u_{i}]\ :\ 1\leqslant i\leqslant 23\\},$ where

$u_{1}.\ x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{7}$, |  $u_{2}.\ x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{3}$, |  $u_{3}.\ x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{7}$, |  $u_{4}.\ x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{6}$, |  $u_{5}.\ x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{3}$, |
---|---|---|---|---|---
$u_{6}.\ x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{2}$, |  $u_{7}.\ x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{3}$, |  $u_{8}.\ x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{3}$, |  $u_{9}.\ x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{2}$, |  $u_{10}.\ x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{7}$, |
$u_{11}.\ x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{6}$, |  $u_{12}.\ x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{3}$, |  $u_{13}.\ x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{2}$, |  $u_{14}.\ x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{6}$, |  $u_{15}.\ x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{4}$, |
$u_{16}.\ x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{3}$, |  $u_{17}.\ x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{2}$, |  $u_{18}.\ x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{3}$, |  $u_{19}.\ x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{2}$, |  $u_{20}.\ x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{2}$, |
$u_{21}.\ x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{3}$, |  $u_{22}.\ x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{2}$, | $u_{23}.\ x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{2}.$ |  |  |

We explicitly compute $p_{(i;I)}(\mathcal{S})$ in terms $u_{j},\ 1\leqslant
j\leqslant 23.$ By a direct computation using Lemma 2.2.9,Theorem 3.1.3, and
from the relations $p_{(i;j)}(\mathcal{S})\equiv_{\omega_{(5,2)}}0$ with
either $i=1,j=2,3$ or $i=2,j=3,4,$ one gets

$\left\\{\begin{array}[]{ll}\gamma_{k}=0,\ k\in\mathbb{J},\\\
\gamma_{27}=\gamma_{55},\ \gamma_{14}=\gamma_{49}=\gamma_{58},\
\gamma_{13}=\gamma_{20}=\gamma_{43}=\gamma_{59},\\\
\gamma_{30}+\gamma_{35}+\gamma_{58}=0,\
\gamma_{44}+\gamma_{55}+\gamma_{56}+\gamma_{58}=\gamma_{47}+\gamma_{53}+\gamma_{54}+\gamma_{59}=0.\end{array}\right.$

Here
$\mathbb{J}=\\{1,2,3,4,5,6,7,8,9,10,11,12,15,16,17,18,19,21,22,23,24,25,26,28,29,\\\
31,38,39,40,41,42,45,46,48,50,51,57,60\\}.$ Then, the relation (3.2.1) becomes

$\mathcal{S}=\sum\limits_{k\,\in\,\mathbb{N}_{60}\setminus\mathbb{J}}\gamma_{k}b_{13,\,k}\equiv_{\omega_{(5,2)}}0.$
(3.2.2)

Applying the homomorphisms
$p_{(1;4)},\,p_{(3;4)},\,p_{(3;5)},\,p_{(4;5)}:P_{5}\to P_{4}$ to (3.2.2), we
obtain $\gamma_{k}=0,\ 1\leqslant k\leqslant 60.$ This finishes the proof. ∎

Since $\dim(QP_{5}^{0})_{13}=145,$ from Theorem 3.2.1 and Proposition 3.2.3,
one gets $\dim(QP_{5})_{13}=250.$

For $t\geqslant 3,$ we have $|\mathscr{B}_{4}(3(2^{t}-1)+2^{t})|=45$ (see Sum
[32]). By a simple computation, we get

$\mathscr{B}_{5}^{0}(3(2^{t}-1)+2^{t})=\mathscr{B}_{5}^{0}(\omega_{(5,t)})=\Phi^{0}(\mathscr{B}_{4}(3(2^{t}-1)+2^{t}))=\\{q_{t,\,k}\,:\,1\leqslant
k\leqslant 195\\},$

for all $t\geqslant 3.$ Here, the monomials $q_{t,\,k},\,1\leqslant k\leqslant
195,$ are listed in Subsection 5.1.

We now compute $QP_{5}^{+}(\omega_{(5,t)})$ for $t\geqslant 3.$

The case $t=3$. We have $d=3(2^{t}-1)+2^{t}=29.$ We obtain the following.

######  Proposition 3.2.6.

$QP_{5}^{+}(\omega_{(5,3)})$ is the $\mathbb{F}_{2}$-vector space of dimension
$260$ with a basis consisting of all the classes represented by the monomials
$b_{29,\,k},\,1\leqslant k\leqslant 260,$ which are determined as in
Subsection 5.2.

To prove the proposition, we need some results. The following is a corollary
of a result in [32].

######  Lemma 3.2.7.

If $\bar{x}$ is one of the following monomials then
$\rho_{i}(\bar{x}),\,1\leqslant i\leqslant 5$ are strictly inadmissible:

$x_{1}^{6}x_{2}x_{3}^{7}x_{4}^{7}$, |  $x_{1}^{6}x_{2}^{7}x_{3}x_{4}^{7}$, |  $x_{1}^{6}x_{2}^{7}x_{3}^{7}x_{4}$, |  $x_{1}^{7}x_{2}^{6}x_{3}x_{4}^{7}$, |  $x_{1}^{7}x_{2}^{6}x_{3}^{7}x_{4}$, |  $x_{1}^{7}x_{2}^{7}x_{3}^{6}x_{4}$,
---|---|---|---|---|---
$x_{1}^{2}x_{2}^{5}x_{3}^{7}x_{4}^{7}$, |  $x_{1}^{2}x_{2}^{7}x_{3}^{5}x_{4}^{7}$, |  $x_{1}^{2}x_{2}^{7}x_{3}^{7}x_{4}^{5}$, |  $x_{1}^{7}x_{2}^{2}x_{3}^{5}x_{4}^{7}$, |  $x_{1}^{7}x_{2}^{2}x_{3}^{7}x_{4}^{5}$, |  $x_{1}^{7}x_{2}^{7}x_{3}^{2}x_{4}^{5}$,
$x_{1}^{3}x_{2}^{4}x_{3}^{7}x_{4}^{7}$, |  $x_{1}^{3}x_{2}^{7}x_{3}^{4}x_{4}^{7}$, |  $x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{4}^{4}$, |  $x_{1}^{7}x_{2}^{3}x_{3}^{4}x_{4}^{7}$, |  $x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{4}^{4}$, |  $x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{4}^{4}$,
$x_{1}^{3}x_{2}^{6}x_{3}^{5}x_{4}^{7}$, |  $x_{1}^{3}x_{2}^{6}x_{3}^{7}x_{4}^{5}$, |  $x_{1}^{3}x_{2}^{7}x_{3}^{6}x_{4}^{5}$, |  $x_{1}^{6}x_{2}^{3}x_{3}^{5}x_{4}^{7}$, |  $x_{1}^{6}x_{2}^{3}x_{3}^{7}x_{4}^{5}$, |  $x_{1}^{6}x_{2}^{7}x_{3}^{3}x_{4}^{5}$,
$x_{1}^{7}x_{2}^{3}x_{3}^{6}x_{4}^{5}$, |  $x_{1}^{7}x_{2}^{6}x_{3}^{3}x_{4}^{5}$, | $x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}^{7}.$ |  |  |

######  Lemma 3.2.8.

The following monomials are strictly inadmissible:

  1. (i)

$x_{k}^{2}x_{\ell}x_{m}^{5}x_{t}^{6}x_{u}^{7},\,k<\ell,\,x_{k}^{2}x_{\ell}x_{m}^{4}x_{t}^{7}x_{u}^{7},\,x_{k}^{2}x_{\ell}^{4}x_{m}x_{t}^{7}x_{u}^{7},\,x_{k}^{6}x_{\ell}x_{m}^{2}x_{t}^{5}x_{u}^{7},\,x_{k}^{6}x_{\ell}^{3}x_{m}^{4}x_{t}x_{u}^{7},\,k<\ell<m.$
Here $(k,\ell,m,t,u)$ is a permutation of $(1,\,2,\,3,\,4,\,5).$

  2. (ii)

$x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{6}x_{5}^{5}$, |  $x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{6}x_{5}^{5}$, |  $x_{1}x_{2}^{6}x_{3}^{6}x_{4}^{3}x_{5}^{5}$, |  $x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{6}x_{5}^{5}$,
---|---|---|---
$x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{6}x_{5}$, |  $x_{1}^{3}x_{2}^{6}x_{3}x_{4}^{5}x_{5}^{6}$, |  $x_{1}^{3}x_{2}^{6}x_{3}x_{4}^{6}x_{5}^{5}$, |  $x_{1}^{3}x_{2}^{6}x_{3}^{5}x_{4}x_{5}^{6}$,
$x_{1}^{3}x_{2}^{6}x_{3}^{5}x_{4}^{6}x_{5}$, |  $x_{1}^{3}x_{2}^{6}x_{3}^{6}x_{4}x_{5}^{5}$, | $x_{1}^{3}x_{2}^{6}x_{3}^{6}x_{4}^{5}x_{5}.$ |

The proof of Lemma 3.2.8 is straightforward.

######  Lemma 3.2.9.

If $(i,j,k,\ell,m)$ is a permutation of $(1,2,3,4,5)$ such that $i<j<k<\ell$
then the following monomials are strictly inadmissible:

$x_{i}x_{j}^{6}x_{k}^{6}x_{\ell}x_{m}^{7}$, |  $x_{i}^{6}x_{j}x_{k}x_{\ell}^{6}x_{m}^{7}$, |  $x_{i}^{6}x_{j}x_{k}^{6}x_{\ell}x_{m}^{7}$, |  $x_{i}^{6}x_{j}^{6}x_{k}x_{\ell}x_{m}^{7}$, |  $x_{i}x_{j}^{2}x_{k}^{6}x_{\ell}^{5}x_{m}^{7}$,
---|---|---|---|---
$x_{i}x_{j}^{6}x_{k}^{2}x_{\ell}^{5}x_{m}^{7}$, |  $x_{i}^{3}x_{j}^{4}x_{k}x_{\ell}^{6}x_{m}^{7}$, |  $x_{i}^{3}x_{j}^{4}x_{k}^{6}x_{\ell}x_{m}^{7}$, |  $x_{i}^{3}x_{j}^{6}x_{k}^{4}x_{\ell}x_{m}^{7}$, |  $x_{i}^{3}x_{j}^{6}x_{k}x_{\ell}^{4}x_{m}^{7}$,
$x_{i}^{6}x_{j}x_{k}^{3}x_{\ell}^{5}x_{m}^{6}$, |  $x_{i}^{6}x_{j}^{3}x_{k}x_{\ell}^{5}x_{m}^{6}$, |  $x_{i}^{6}x_{j}^{3}x_{k}^{5}x_{\ell}x_{m}^{6}$, |  $x_{i}^{2}x_{j}^{2}x_{k}^{5}x_{\ell}^{5}x_{m}^{7}$, |  $x_{i}^{2}x_{j}^{5}x_{k}^{2}x_{\ell}^{5}x_{m}^{7}$,
$x_{i}^{3}x_{j}^{4}x_{k}^{3}x_{\ell}^{4}x_{m}^{7}$, |  $x_{i}^{2}x_{j}^{3}x_{k}^{4}x_{\ell}^{5}x_{m}^{7}$, |  $x_{i}^{2}x_{j}^{3}x_{k}^{5}x_{\ell}^{4}x_{m}^{7}$, |  $x_{i}^{2}x_{j}^{4}x_{k}^{3}x_{\ell}^{5}x_{m}^{7}$, |  $x_{i}^{2}x_{j}^{5}x_{k}^{3}x_{\ell}^{4}x_{m}^{7}$,
$x_{i}^{3}x_{j}^{2}x_{k}^{4}x_{\ell}^{5}x_{m}^{7}$, |  $x_{i}^{3}x_{j}^{2}x_{k}^{5}x_{\ell}^{4}x_{m}^{7}$, |  $x_{i}^{3}x_{j}^{4}x_{k}^{2}x_{\ell}^{5}x_{m}^{7}$, |  $x_{i}^{2}x_{j}^{3}x_{k}^{5}x_{\ell}^{5}x_{m}^{6}$, |  $x_{i}^{2}x_{j}^{5}x_{k}^{3}x_{\ell}^{5}x_{m}^{6}$,
$x_{i}^{3}x_{j}^{2}x_{k}^{5}x_{\ell}^{5}x_{m}^{6}$, |  $x_{i}^{3}x_{j}^{5}x_{k}^{2}x_{\ell}^{5}x_{m}^{6}$. |  $x_{i}^{3}x_{j}^{3}x_{k}^{4}x_{\ell}^{5}x_{m}^{6}$, |  $x_{i}^{3}x_{j}^{3}x_{k}^{5}x_{\ell}^{4}x_{m}^{6}$, |  $x_{i}^{3}x_{j}^{4}x_{k}^{3}x_{\ell}^{5}x_{m}^{6}$,
$x_{i}^{3}x_{j}^{5}x_{k}^{3}x_{\ell}^{4}x_{m}^{6}$. |  |  |  |

###### Proof.

We prove the lemma for the monomial
$x_{i}x_{j}^{6}x_{k}^{6}x_{\ell}x_{m}^{7}.$ The others can be proved by a
similar computation. By using the Cartan formula, we have

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0ptx_{i}x_{j}^{6}x_{k}^{6}x_{\ell}x_{m}^{7}&=Sq^{1}(x_{i}^{2}x_{j}^{5}x_{k}^{5}x_{\ell}x_{m}^{7})+Sq^{2}(x_{i}x_{j}^{5}x_{k}^{5}x_{\ell}x_{m}^{7}+x_{i}x_{j}^{3}x_{k}^{5}x_{\ell}x_{m}^{9})\\\
&\quad+x_{i}x_{j}^{6}x_{k}^{5}x_{\ell}^{2}x_{m}^{7}+x_{i}x_{j}^{5}x_{k}^{6}x_{\ell}^{2}x_{m}^{7}\
\ {\rm mod}(P_{5}^{-}(\omega^{*}_{(5,3)})).\end{array}$

This relation implies that $x_{i}x_{j}^{6}x_{k}^{6}x_{\ell}x_{m}^{7}$ is
strictly inadmissible. The lemma is proved. ∎

######  Lemma 3.2.10.

The following monomials are strictly inadmissible:

$x_{1}x_{2}^{6}x_{3}^{7}x_{4}^{8}x_{5}^{7}$, |  $x_{1}x_{2}^{7}x_{3}^{6}x_{4}^{8}x_{5}^{7}$, |  $x_{1}x_{2}^{7}x_{3}^{10}x_{4}^{4}x_{5}^{7}$, |  $x_{1}x_{2}^{7}x_{3}^{10}x_{4}^{7}x_{5}^{4}$, |  $x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{12}x_{5}^{7}$,
---|---|---|---|---
$x_{1}^{3}x_{2}^{3}x_{3}^{12}x_{4}^{4}x_{5}^{7}$, |  $x_{1}^{3}x_{2}^{3}x_{3}^{12}x_{4}^{7}x_{5}^{4}$, |  $x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{3}x_{5}^{12}$, |  $x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{11}x_{5}^{4}$, |  $x_{1}^{3}x_{2}^{5}x_{3}^{8}x_{4}^{6}x_{5}^{7}$,
$x_{1}^{3}x_{2}^{5}x_{3}^{8}x_{4}^{7}x_{5}^{6}$, |  $x_{1}^{3}x_{2}^{5}x_{3}^{9}x_{4}^{6}x_{5}^{6}$, |  $x_{1}^{3}x_{2}^{5}x_{3}^{14}x_{4}^{3}x_{5}^{4}$, |  $x_{1}^{3}x_{2}^{12}x_{3}^{3}x_{4}^{5}x_{5}^{6}$, |  $x_{1}^{3}x_{2}^{13}x_{3}^{6}x_{4}^{3}x_{5}^{4}$,
$x_{1}^{7}x_{2}x_{3}^{6}x_{4}^{8}x_{5}^{7}$, |  $x_{1}^{7}x_{2}x_{3}^{10}x_{4}^{4}x_{5}^{7}$, |  $x_{1}^{7}x_{2}x_{3}^{10}x_{4}^{7}x_{5}^{4}$, |  $x_{1}^{7}x_{2}^{8}x_{3}^{3}x_{4}^{5}x_{5}^{6}$, |  $x_{1}^{7}x_{2}^{9}x_{3}^{2}x_{4}^{4}x_{5}^{7}$,
$x_{1}^{7}x_{2}^{9}x_{3}^{2}x_{4}^{7}x_{5}^{4}$, | $x_{1}^{7}x_{2}^{9}x_{3}^{7}x_{4}^{2}x_{5}^{4}.$ |  |  |

###### Proof.

We prove the lemma for the monomial
$x=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{3}x_{5}^{12}.$ The others can be proved
by a similar computation. By a direct computation, we have

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0ptx&=x_{1}^{2}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{13}+x_{1}^{2}x_{2}^{3}x_{3}^{5}x_{4}^{12}x_{5}^{7}+x_{1}^{2}x_{2}^{5}x_{3}^{5}x_{4}^{6}x_{5}^{11}+x_{1}^{2}x_{2}^{5}x_{3}^{5}x_{4}^{10}x_{5}^{7}+x_{1}^{2}x_{2}^{5}x_{3}^{9}x_{4}^{6}x_{5}^{7}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{2}x_{2}^{11}x_{3}^{4}x_{4}^{5}x_{5}^{7}+x_{1}^{2}x_{2}^{11}x_{3}^{5}x_{4}^{4}x_{5}^{7}+x_{1}^{2}x_{2}^{13}x_{3}^{3}x_{4}^{4}x_{5}^{7}+x_{1}^{2}x_{2}^{13}x_{3}^{4}x_{4}^{3}x_{5}^{7}+x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{6}x_{5}^{13}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{12}x_{5}^{7}+x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{12}+x_{1}^{3}x_{2}^{3}x_{3}^{6}x_{4}^{5}x_{5}^{12}+x_{1}^{3}x_{2}^{3}x_{3}^{12}x_{4}^{4}x_{5}^{7}+x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}^{6}x_{5}^{13}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}^{12}x_{5}^{7}+x_{1}^{3}x_{2}^{4}x_{3}^{5}x_{4}^{6}x_{5}^{11}+x_{1}^{3}x_{2}^{4}x_{3}^{5}x_{4}^{10}x_{5}^{7}+x_{1}^{3}x_{2}^{4}x_{3}^{6}x_{4}^{3}x_{5}^{13}+x_{1}^{3}x_{2}^{4}x_{3}^{6}x_{4}^{9}x_{5}^{7}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{4}x_{3}^{12}x_{4}^{3}x_{5}^{7}+x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{5}x_{5}^{14}+x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{6}x_{5}^{13}+x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{4}x_{5}^{14}+x_{1}^{3}x_{2}^{5}x_{3}^{5}x_{4}^{6}x_{5}^{10}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{2}x_{5}^{13}+x_{1}^{5}x_{2}^{5}x_{3}^{5}x_{4}^{5}x_{5}^{9}+Sq^{1}(h_{1})+Sq^{2}(h_{2})+Sq^{4}(h_{4})\
\ {\rm mod}(P_{5}^{-}(\omega_{(5,3)})),\ \mbox{where}\\\ \vskip 6.0pt plus
2.0pt minus
2.0pth_{1}&=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{11}+x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{10}x_{5}^{7}+x_{1}^{3}x_{2}^{3}x_{3}^{9}x_{4}^{2}x_{5}^{11}+x_{1}^{3}x_{2}^{3}x_{3}^{9}x_{4}^{3}x_{5}^{10}+x_{1}^{3}x_{2}^{3}x_{3}^{9}x_{4}^{6}x_{5}^{7}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{5}x_{5}^{13}+x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{4}x_{5}^{13}+x_{1}^{3}x_{2}^{5}x_{3}^{4}x_{4}^{5}x_{5}^{11}+x_{1}^{3}x_{2}^{5}x_{3}^{4}x_{4}^{9}x_{5}^{7}+x_{1}^{3}x_{2}^{5}x_{3}^{9}x_{4}^{4}x_{5}^{7}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{6}x_{3}^{5}x_{4}^{5}x_{5}^{9}+x_{1}^{3}x_{2}^{9}x_{3}^{2}x_{4}^{3}x_{5}^{11}+x_{1}^{3}x_{2}^{11}x_{3}^{3}x_{4}^{4}x_{5}^{7}+x_{1}^{3}x_{2}^{11}x_{3}^{4}x_{4}^{3}x_{5}^{7}+x_{1}^{10}x_{2}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{9}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{10}x_{2}^{3}x_{3}^{3}x_{4}^{5}x_{5}^{7}+x_{1}^{10}x_{2}^{3}x_{3}^{5}x_{4}^{3}x_{5}^{7}+x_{1}^{10}x_{2}^{5}x_{3}^{3}x_{4}^{3}x_{5}^{7},\\\
\vskip 6.0pt plus 2.0pt minus
2.0pth_{2}&=x_{1}^{2}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{11}+x_{1}^{2}x_{2}^{3}x_{3}^{5}x_{4}^{10}x_{5}^{7}+x_{1}^{2}x_{2}^{3}x_{3}^{9}x_{4}^{6}x_{5}^{7}+x_{1}^{2}x_{2}^{11}x_{3}^{3}x_{4}^{4}x_{5}^{7}+x_{1}^{2}x_{2}^{11}x_{3}^{4}x_{4}^{3}x_{5}^{7}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{6}x_{5}^{11}+x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{10}x_{5}^{7}+x_{1}^{3}x_{2}^{3}x_{3}^{6}x_{4}^{4}x_{5}^{11}+x_{1}^{3}x_{2}^{3}x_{3}^{6}x_{4}^{8}x_{5}^{7}+x_{1}^{3}x_{2}^{3}x_{3}^{8}x_{4}^{6}x_{5}^{7}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{3}x_{3}^{10}x_{4}^{4}x_{5}^{7}+x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}^{6}x_{5}^{11}+x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}^{10}x_{5}^{7}+x_{1}^{3}x_{2}^{4}x_{3}^{6}x_{4}^{3}x_{5}^{11}+x_{1}^{3}x_{2}^{4}x_{3}^{10}x_{4}^{3}x_{5}^{7}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{5}x_{3}^{5}x_{4}^{5}x_{5}^{9}+x_{1}^{3}x_{2}^{6}x_{3}^{2}x_{4}^{5}x_{5}^{11}+x_{1}^{3}x_{2}^{6}x_{3}^{2}x_{4}^{9}x_{5}^{7}+x_{1}^{3}x_{2}^{6}x_{3}^{3}x_{4}^{4}x_{5}^{11}+x_{1}^{3}x_{2}^{6}x_{3}^{3}x_{4}^{8}x_{5}^{7}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{8}x_{3}^{3}x_{4}^{6}x_{5}^{7}+x_{1}^{3}x_{2}^{8}x_{3}^{6}x_{4}^{3}x_{5}^{7}+x_{1}^{3}x_{2}^{10}x_{3}^{2}x_{4}^{5}x_{5}^{7}+x_{1}^{3}x_{2}^{10}x_{3}^{3}x_{4}^{4}x_{5}^{7}+x_{1}^{5}x_{2}^{3}x_{3}^{6}x_{4}^{2}x_{5}^{11}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{5}x_{2}^{3}x_{3}^{6}x_{4}^{3}x_{5}^{10}+x_{1}^{5}x_{2}^{3}x_{3}^{6}x_{4}^{6}x_{5}^{7}+x_{1}^{5}x_{2}^{10}x_{3}^{2}x_{4}^{3}x_{5}^{7}+x_{1}^{9}x_{2}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{9}+x_{1}^{9}x_{2}^{3}x_{3}^{3}x_{4}^{5}x_{5}^{7}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{9}x_{2}^{3}x_{3}^{5}x_{4}^{3}x_{5}^{7}+x_{1}^{9}x_{2}^{5}x_{3}^{3}x_{4}^{3}x_{5}^{7}+x_{1}^{9}x_{2}^{6}x_{3}^{2}x_{4}^{3}x_{5}^{7},\\\
\vskip 6.0pt plus 2.0pt minus
2.0pth_{4}&=x_{1}^{3}x_{2}^{3}x_{3}^{6}x_{4}^{2}x_{5}^{11}+x_{1}^{3}x_{2}^{3}x_{3}^{6}x_{4}^{3}x_{5}^{10}+x_{1}^{3}x_{2}^{3}x_{3}^{6}x_{4}^{6}x_{5}^{7}+x_{1}^{3}x_{2}^{4}x_{3}^{5}x_{4}^{6}x_{5}^{7}+x_{1}^{3}x_{2}^{4}x_{3}^{6}x_{4}^{5}x_{5}^{7}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{3}x_{2}^{6}x_{3}^{5}x_{4}^{4}x_{5}^{7}+x_{1}^{3}x_{2}^{10}x_{3}^{2}x_{4}^{3}x_{5}^{7}+x_{1}^{5}x_{2}^{3}x_{3}^{4}x_{4}^{6}x_{5}^{7}+x_{1}^{5}x_{2}^{3}x_{3}^{6}x_{4}^{4}x_{5}^{7}+x_{1}^{5}x_{2}^{4}x_{3}^{3}x_{4}^{6}x_{5}^{7}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{5}x_{2}^{4}x_{3}^{6}x_{4}^{3}x_{5}^{7}+x_{1}^{5}x_{2}^{6}x_{3}^{2}x_{4}^{5}x_{5}^{7}+x_{1}^{5}x_{2}^{6}x_{3}^{3}x_{4}^{4}x_{5}^{7}.\end{array}$

Now, let $g=x_{1}x_{2}x_{3}x_{4}x_{5}^{2}$ and $h=x_{1}x_{2}x_{3}x_{4}x_{5}.$
Then, we have

$u:=x_{1}^{5}x_{2}^{5}x_{3}^{5}x_{4}^{5}x_{5}^{9}=h.g^{4}=hSq^{12}(g^{2}).$

On the other hand, by the formula $\chi$-trick (see [43]), one gets

$u=g^{2}\chi(Sq^{12})(h)+\sum_{1\leqslant i\leqslant
12}Sq^{i}(g^{2}\chi(Sq^{12-i})(h)).$

Since $\chi(Sq^{12})=Sq^{8}Sq^{4}+Sq^{8}Sq^{3}Sq^{1},$ by a direct
computation, we coclude that

$g^{2}\chi(Sq^{12})(h)\in P_{5}^{-}(\omega_{(5,3)}).$

This implies that $x$ is strictly inadmissible. The lemma follows. ∎

###### Proof of Proposition 3.2.6.

Let $x$ be an admissible monomial in $(P_{5}^{+})_{29}$ such that
$\omega(x)=\omega_{(5,3)}.$ Then $x=X_{\\{i,j\\}}y^{2}$ with
$y\in\mathscr{B}_{5}(\omega_{(5,2)}),$ and $1\leqslant i<j\leqslant 5.$

Let $u\in\mathscr{B}_{5}(\omega_{(5,2)})$ such that $X_{\\{i,j\\}}u^{2}\in
P_{5}^{+}.$ By a direct computation using Proposition 3.2.3, we see that if
$X_{\\{i,j\\}}u^{2}\neq b_{29,\,k},\ \forall t,\ 1\leqslant k\leqslant 260,$
then there is a monomial $w$ which is given in one of Lemmas 3.2.7, 3.2.8,
3.2.9 and 3.2.10 such that $X_{\\{i,j\\}}u^{2}=wu_{1}^{2^{\ell}}$ with
suitable monomial $u_{1}\in P_{5},$ and $\ell={\rm
max}\\{r\in\mathbb{Z}\,\,:\,\,\omega_{r}(w)>0\\}.$ By Theorem 2.2.5,
$X_{\\{i,j\\}}u^{2}$ is inadmissible. Since $x=X_{\\{i,j\\}}y^{2}$ with
$y\in\mathscr{B}_{5}(\omega_{(5,2)}),$ and $x$ is admissible, one can see that
$x=b_{29,\,k}.$ for some $k,\ 1\leqslant k\leqslant 260.$ This implies
$\mathscr{B}_{5}^{+}(\omega_{(5,3)})\subseteq\\{b_{29,\,k}\,:\,1\leqslant
k\leqslant 260\\}.$

Now, we prove the set $\\{[b_{29,\,k}]\ :\ 1\leqslant k\leqslant 260\\}$ is
linearly independent in $(QP_{5})_{29}.$ Suppose there is a linear relation

$\mathcal{S}=\sum\limits_{1\leqslant k\leqslant
260}\gamma_{k}b_{29,\,k}\equiv_{\omega_{(5,3)}}0,$

where $\gamma_{k}\in\mathbb{F}_{2}$ for all $k,\ 1\leqslant k\leqslant 260.$

From Theorem 3.1.3, for $(i;I)\in\mathcal{N}_{5}$, we explicitly compute
$p_{(i;I)}(\mathcal{S})$ in terms of a given minimal set of
$\mathscr{A}$-generators in $P_{4}\ ({\rm mod}(\mathscr{A}^{+}\cdot P_{4})).$
Based on the relations $p_{(i;I)}(\mathcal{S})\equiv_{\omega_{(5,3)}}0,\
\forall(i;I)\in\mathcal{N}_{5},\ \ell(I)>0,$ and some other simple
computations, we get $\gamma_{k}=0,\,k=1,2,\ldots,260.$ The proof is
completed. ∎

Recall that $\dim(QP_{5}^{0})_{29}=195.$ Combining this with Theorem 3.2.1 and
Proposition 3.2.6, we obtain $\dim(QP_{5})_{29}=645.$

The case $t\geqslant 4$. We have the following.

######  Proposition 3.2.11.

There exist exactly $270$ admissible monomials in $P_{5}^{+}$ such that their
weight vectors are $\omega_{(5,t)},$ for every $t\geqslant 4.$ Consequently
$\dim(QP_{5}^{+}(\omega_{(5,t)}))=270$ for $t\geqslant 4.$

We prove the proposition by showing that

$\mathscr{B}_{5}^{+}(\omega_{(5,t)})=\\{b_{3(2^{t}-1)+2^{t},k}:\ 1\leqslant
k\leqslant 270\\},$

for all $t\geqslant 4.$ Here, the monomials $b_{3(2^{t}-1)+2^{t},k},\
1\leqslant k\leqslant 270$, are listed in Subsection 5.2.

######  Lemma 3.2.12.

If $(i,j,k,\ell,m)$ is a permutation of $(1,2,3,4,5)$ such that $i<j<k<\ell$
then the following monomials are strictly inadmissible:

$x_{i}^{6}x_{j}x_{k}x_{\ell}^{6}x_{m}^{7},$ | $x_{i}x_{j}^{2}x_{k}^{6}x_{\ell}^{5}x_{m}^{7},$ | $x_{i}x_{j}^{6}x_{k}^{2}x_{\ell}^{5}x_{m}^{7},$ | $x_{i}^{6}x_{j}x_{k}^{2}x_{\ell}^{5}x_{m}^{7},$
---|---|---|---
$x_{i}x_{j}^{6}x_{k}^{3}x_{\ell}^{4}x_{m}^{7},$ | $x_{i}^{3}x_{j}^{6}x_{k}x_{\ell}^{4}x_{m}^{7},$ | $x_{i}^{6}x_{j}x_{k}^{3}x_{\ell}^{4}x_{m}^{7},$ | $x_{i}^{6}x_{j}^{3}x_{k}x_{\ell}^{4}x_{m}^{7},$
$x_{i}^{6}x_{j}x_{k}^{3}x_{\ell}^{5}x_{m}^{6},$ | $x_{i}^{6}x_{j}^{3}x_{k}^{5}x_{\ell}x_{m}^{6},$ | $x_{i}^{6}x_{j}^{3}x_{k}x_{\ell}^{5}x_{m}^{6},$ | $x_{i}^{2}x_{j}^{2}x_{k}^{5}x_{\ell}^{5}x_{m}^{7},$
$x_{i}^{2}x_{j}^{5}x_{k}^{2}x_{\ell}^{5}x_{m}^{7},$ | $x_{i}^{2}x_{j}^{3}x_{k}^{4}x_{\ell}^{5}x_{m}^{7},$ | $x_{i}^{2}x_{j}^{3}x_{k}^{5}x_{\ell}^{4}x_{m}^{7},$ | $x_{i}^{2}x_{j}^{4}x_{k}^{3}x_{\ell}^{5}x_{m}^{7},$
$x_{i}^{2}x_{j}^{5}x_{k}^{3}x_{\ell}^{4}x_{m}^{7},$ | $x_{i}^{3}x_{j}^{2}x_{k}^{4}x_{\ell}^{5}x_{m}^{7},$ | $x_{i}^{3}x_{j}^{2}x_{k}^{5}x_{\ell}^{4}x_{m}^{7},$ | $x_{i}^{3}x_{j}^{4}x_{k}^{2}x_{\ell}^{5}x_{m}^{7},$
$x_{i}^{2}x_{j}^{5}x_{k}^{3}x_{\ell}^{5}x_{m}^{6},$ | $x_{i}^{3}x_{j}^{2}x_{k}^{5}x_{\ell}^{5}x_{m}^{6},$ | $x_{i}^{3}x_{j}^{4}x_{k}^{3}x_{\ell}^{4}x_{m}^{7},$ | $x_{i}x_{j}^{7}x_{k}^{10}x_{\ell}^{12}x_{m}^{15},$
$x_{i}^{7}x_{j}x_{k}^{10}x_{\ell}^{12}x_{m}^{15},$ | $x_{i}^{3}x_{j}^{3}x_{k}^{12}x_{\ell}^{12}x_{m}^{15},$ | $x_{i}^{3}x_{j}^{5}x_{k}^{8}x_{\ell}^{14}x_{m}^{15},$ | $x_{i}^{3}x_{j}^{5}x_{k}^{14}x_{\ell}^{8}x_{m}^{15},$
$x_{i}^{7}x_{j}^{7}x_{k}^{8}x_{\ell}^{8}x_{m}^{15}.$ |  |  |

###### Proof.

We prove the lemma for the monomial
$x=x_{i}^{7}x_{j}^{7}x_{k}^{8}x_{\ell}^{8}x_{m}^{15}.$ The others can be
proved by a similar computation.

Applying the Cartan formula, we have

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0ptx&=x_{i}^{7}x_{j}^{5}x_{k}^{10}x_{\ell}^{8}x_{m}^{15}+x_{i}^{5}x_{j}^{11}x_{k}^{6}x_{\ell}^{8}x_{m}^{15}+x_{i}^{5}x_{j}^{7}x_{k}^{10}x_{\ell}^{8}x_{m}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{i}^{5}x_{j}^{11}x_{k}^{10}x_{\ell}^{4}x_{m}^{15}+x_{i}^{5}x_{j}^{7}x_{k}^{10}x_{\ell}^{8}x_{m}^{15}+x_{i}^{5}x_{j}^{10}x_{k}^{11}x_{\ell}^{4}x_{m}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{i}^{5}x_{j}^{6}x_{k}^{11}x_{\ell}^{8}x_{m}^{15}+x_{i}^{5}x_{j}^{3}x_{k}^{14}x_{\ell}^{8}x_{m}^{15}+x_{i}^{4}x_{j}^{11}x_{k}^{11}x_{\ell}^{4}x_{m}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{i}^{4}x_{j}^{7}x_{k}^{11}x_{\ell}^{8}x_{m}^{15}+Sq^{1}(x_{i}^{7}x_{j}^{7}x_{k}^{11}x_{\ell}^{4}x_{m}^{15}+x_{i}^{7}x_{j}^{5}x_{k}^{13}x_{\ell}^{4}x_{m}^{15})\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+Sq^{2}(y)+Sq^{4}(z)+Sq^{8}(x_{i}^{7}x_{j}^{5}x_{k}^{6}x_{\ell}^{4}x_{m}^{15})\
\ {\rm mod}(P_{5}^{-}(\omega^{*}_{(5,4)})),\ \mbox{where}\\\ \vskip 6.0pt plus
2.0pt minus
2.0pty&=x_{i}^{7}x_{j}^{7}x_{k}^{6}x_{\ell}^{8}x_{m}^{15}+x_{i}^{7}x_{j}^{3}x_{k}^{6}x_{\ell}^{4}x_{m}^{23}+x_{i}^{7}x_{j}^{7}x_{k}^{10}x_{\ell}^{4}x_{m}^{15}\
+x_{i}^{7}x_{j}^{6}x_{k}^{11}x_{\ell}^{4}x_{m}^{15}+x_{i}^{7}x_{j}^{3}x_{k}^{14}x_{\ell}^{4}x_{m}^{15},\\\
\vskip 6.0pt plus 2.0pt minus
2.0ptz&=x_{i}^{5}x_{j}^{7}x_{k}^{6}x_{\ell}^{8}x_{m}^{15}+x_{i}^{11}x_{j}^{5}x_{k}^{6}x_{\ell}^{4}x_{m}^{15}+x_{i}^{5}x_{j}^{7}x_{k}^{10}x_{\ell}^{4}x_{m}^{15}\\\
&\quad+x_{i}^{4}x_{j}^{7}x_{k}^{11}x_{\ell}^{4}x_{m}^{15}+x_{i}^{5}x_{j}^{6}x_{k}^{11}x_{\ell}^{4}x_{m}^{15}+x_{i}^{5}x_{j}^{3}x_{k}^{14}x_{\ell}^{4}x_{m}^{15}.\end{array}$

The above equalities imply that $x$ is strictly inadmissible. ∎

The following lemma is proved by a direct computation.

######  Lemma 3.2.13.

The following monomials are strictly inadmissible:

  1. (i)

$x_{i}^{6}x_{j}^{6}x_{k}x_{l}x_{m}^{7},\
x_{i}^{2}x_{j}x_{k}^{4}x_{l}^{7}x_{m}^{7},\
x_{i}^{2}x_{j}^{4}x_{k}x_{l}^{7}x_{m}^{7},\
x_{i}^{2}x_{j}^{5}x_{k}x_{l}^{6}x_{m}^{7},\
x_{i}^{2}x_{j}x_{k}^{5}x_{l}^{6}x_{m}^{7},\
x_{i}^{3}x_{j}^{4}x_{k}x_{l}^{6}x_{m}^{7}$ with $i<j<k.$ Here $(i,j,k,l,m)$ is
a permutation of $(1,2,3,4,5).$

  2. (ii)

$x_{1}^{3}x_{2}^{3}x_{3}^{6}x_{4}^{4}x_{5}^{5}$, |  $x_{1}^{3}x_{2}^{3}x_{3}^{6}x_{4}^{5}x_{5}^{4}$, |  $x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}^{6}x_{5}^{5}$, |  $x_{1}^{3}x_{2}^{4}x_{3}^{6}x_{4}^{3}x_{5}^{5}$,
---|---|---|---
$x_{1}^{3}x_{2}^{6}x_{3}^{3}x_{4}^{4}x_{5}^{5}$, |  $x_{1}^{3}x_{2}^{6}x_{3}^{3}x_{4}^{5}x_{5}^{4}$, |  $x_{1}^{3}x_{2}^{6}x_{3}^{4}x_{4}^{3}x_{5}^{5}$, |  $x_{1}^{3}x_{2}^{6}x_{3}^{5}x_{4}^{3}x_{5}^{4}$,
$x_{1}^{6}x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{5}$, |  $x_{1}^{6}x_{2}^{3}x_{3}^{3}x_{4}^{5}x_{5}^{4}$, |  $x_{1}^{6}x_{2}^{3}x_{3}^{4}x_{4}^{3}x_{5}^{5}$, |  $x_{1}^{6}x_{2}^{3}x_{3}^{5}x_{4}^{3}x_{5}^{4}$,
$x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{6}x_{5}^{5}$, |  $x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{2}x_{5}^{5}$, |  $x_{1}^{3}x_{2}^{6}x_{3}^{5}x_{4}^{2}x_{5}^{5}$, |  $x_{1}^{6}x_{2}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{5}$,
$x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{6}x_{5}^{5}$, |  $x_{1}^{3}x_{2}^{5}x_{3}^{9}x_{4}^{14}x_{5}^{14}$, | $x_{1}^{3}x_{2}^{5}x_{3}^{14}x_{4}^{9}x_{5}^{14}.$ |

######  Lemma 3.2.14.

The following monomials are strictly inadmissible:

  1. (i)

$\rho_{i}(f),\ 1\leqslant i\leqslant 5,$ where $f$ is one of the following
monomials:

$x_{1}^{2}x_{2}^{5}x_{3}^{7}x_{4}^{7}$, |  $x_{1}^{2}x_{2}^{7}x_{3}^{5}x_{4}^{7}$, |  $x_{1}^{2}x_{2}^{7}x_{3}^{7}x_{4}^{5}$, |  $x_{1}^{7}x_{2}^{2}x_{3}^{5}x_{4}^{7}$,
---|---|---|---
$x_{1}^{7}x_{2}^{2}x_{3}^{7}x_{4}^{5}$, |  $x_{1}^{7}x_{2}^{7}x_{3}^{2}x_{4}^{5}$, |  $x_{1}^{3}x_{2}^{4}x_{3}^{7}x_{4}^{7}$, |  $x_{1}^{3}x_{2}^{7}x_{3}^{4}x_{4}^{7}$,
$x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{4}^{4}$, |  $x_{1}^{7}x_{2}^{3}x_{3}^{4}x_{4}^{7}$, |  $x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{4}^{4}$, |  $x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{4}^{4}$,
$x_{1}^{3}x_{2}^{6}x_{3}^{5}x_{4}^{7}$, |  $x_{1}^{3}x_{2}^{6}x_{3}^{7}x_{4}^{5}$, |  $x_{1}^{3}x_{2}^{7}x_{3}^{6}x_{4}^{5}$, |  $x_{1}^{6}x_{2}^{3}x_{3}^{5}x_{4}^{7}$,
$x_{1}^{6}x_{2}^{3}x_{3}^{7}x_{4}^{5}$, |  $x_{1}^{6}x_{2}^{7}x_{3}^{3}x_{4}^{5}$, |  $x_{1}^{7}x_{2}^{3}x_{3}^{6}x_{4}^{5}$, |  $x_{1}^{7}x_{2}^{6}x_{3}^{3}x_{4}^{5}$,
$x_{1}^{6}x_{2}x_{3}^{7}x_{4}^{7}$, |  $x_{1}^{6}x_{2}^{7}x_{3}x_{4}^{7}$, |  $x_{1}^{6}x_{2}^{7}x_{3}^{7}x_{4}$, |  $x_{1}^{7}x_{2}^{6}x_{3}x_{4}^{7}$,
$x_{1}^{7}x_{2}^{6}x_{3}^{7}x_{4}$, |  $x_{1}^{7}x_{2}^{7}x_{3}^{6}x_{4}$. |  |

  2. (ii)

$x_{1}x_{2}^{15}x_{3}^{15}x_{4}^{18}x_{5}^{12}$, |  $x_{1}^{15}x_{2}x_{3}^{15}x_{4}^{18}x_{5}^{12}$, |  $x_{1}^{15}x_{2}^{15}x_{3}x_{4}^{18}x_{5}^{12}$, |  $x_{1}x_{2}^{14}x_{3}^{15}x_{4}^{15}x_{5}^{16}$,
---|---|---|---
$x_{1}x_{2}^{15}x_{3}^{14}x_{4}^{15}x_{5}^{16}$, |  $x_{1}^{15}x_{2}x_{3}^{14}x_{4}^{15}x_{5}^{16}$, |  $x_{1}x_{2}^{15}x_{3}^{15}x_{4}^{14}x_{5}^{16}$, |  $x_{1}^{15}x_{2}x_{3}^{15}x_{4}^{14}x_{5}^{16}$,
$x_{1}^{15}x_{2}^{15}x_{3}x_{4}^{14}x_{5}^{16}$, |  $x_{1}^{15}x_{2}^{15}x_{3}^{17}x_{4}^{2}x_{5}^{12}$, |  $x_{1}^{3}x_{2}^{13}x_{3}^{14}x_{4}^{19}x_{5}^{12}$, |  $x_{1}^{3}x_{2}^{13}x_{3}^{15}x_{4}^{18}x_{5}^{12}$,
$x_{1}^{3}x_{2}^{15}x_{3}^{13}x_{4}^{18}x_{5}^{12}$, |  $x_{1}^{15}x_{2}^{3}x_{3}^{13}x_{4}^{18}x_{5}^{12}$, |  $x_{1}^{3}x_{2}^{13}x_{3}^{14}x_{4}^{15}x_{5}^{16}$, |  $x_{1}^{3}x_{2}^{13}x_{3}^{15}x_{4}^{14}x_{5}^{16}$,
$x_{1}^{3}x_{2}^{15}x_{3}^{13}x_{4}^{14}x_{5}^{16}$, |  $x_{1}^{15}x_{2}^{3}x_{3}^{13}x_{4}^{14}x_{5}^{16}$, |  $x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}^{9}x_{5}^{30}$, |  $x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{8}x_{5}^{30}$,
$x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{30}x_{5}^{8}$, |  $x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{14}x_{5}^{24}$, |  $x_{1}^{7}x_{2}^{11}x_{3}^{13}x_{4}^{16}x_{5}^{14}$, |  $x_{1}^{7}x_{2}^{15}x_{3}^{17}x_{4}^{10}x_{5}^{12}$,
$x_{1}^{15}x_{2}^{7}x_{3}^{17}x_{4}^{10}x_{5}^{12}$, |  $x_{1}^{15}x_{2}^{19}x_{3}^{5}x_{4}^{10}x_{5}^{12}$. |  |

###### Proof.

The first part of the lemma is an immediate corollary from a result in [32].
We prove the second part of this lemma for the monomials
$x=x_{1}x_{2}^{14}x_{3}^{15}x_{4}^{15}x_{5}^{16}$ and
$y=x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{8}x_{5}^{30}.$ The others can be proved
by a similar computation. By a direct computation, one gets

$\begin{array}[]{ll}x&=x_{1}x_{2}^{10}x_{3}^{12}x_{4}^{15}x_{5}^{23}+x_{1}x_{2}^{10}x_{3}^{12}x_{4}^{23}x_{5}^{15}+x_{1}x_{2}^{10}x_{3}^{14}x_{4}^{13}x_{5}^{23}+x_{1}x_{2}^{10}x_{3}^{14}x_{4}^{21}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}x_{2}^{10}x_{3}^{20}x_{4}^{15}x_{5}^{15}+x_{1}x_{2}^{10}x_{3}^{22}x_{4}^{13}x_{5}^{15}+x_{1}x_{2}^{12}x_{3}^{15}x_{4}^{10}x_{5}^{23}+x_{1}x_{2}^{12}x_{3}^{15}x_{4}^{15}x_{5}^{18}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}x_{2}^{12}x_{3}^{18}x_{4}^{15}x_{5}^{15}+x_{1}x_{2}^{12}x_{3}^{26}x_{4}^{7}x_{5}^{15}+x_{1}x_{2}^{14}x_{3}^{12}x_{4}^{15}x_{5}^{19}+x_{1}x_{2}^{14}x_{3}^{12}x_{4}^{19}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}x_{2}^{14}x_{3}^{14}x_{4}^{13}x_{5}^{19}+x_{1}x_{2}^{14}x_{3}^{14}x_{4}^{17}x_{5}^{15}+x_{1}x_{2}^{14}x_{3}^{15}x_{4}^{8}x_{5}^{23}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+Sq^{1}(f_{1})+Sq^{2}(f_{2})+Sq^{4}(f_{4})+Sq^{8}(f_{8})\ \ {\rm
mod}(P_{5}^{-}(\omega_{(5,4)})),\ \mbox{where}\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\par
f_{1}&=x_{1}^{2}x_{2}^{11}x_{3}^{15}x_{4}^{15}x_{5}^{17}+x_{1}^{2}x_{2}^{11}x_{3}^{15}x_{4}^{17}x_{5}^{15}+x_{1}^{2}x_{2}^{11}x_{3}^{17}x_{4}^{15}x_{5}^{15}+x_{1}^{2}x_{2}^{11}x_{3}^{23}x_{4}^{7}x_{5}^{17}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{2}x_{2}^{11}x_{3}^{23}x_{4}^{9}x_{5}^{15}+x_{1}^{2}x_{2}^{11}x_{3}^{25}x_{4}^{7}x_{5}^{15}+x_{1}^{2}x_{2}^{13}x_{3}^{15}x_{4}^{15}x_{5}^{15}+x_{1}^{2}x_{2}^{13}x_{3}^{23}x_{4}^{7}x_{5}^{15},\\\
\vskip 6.0pt plus 2.0pt minus
2.0ptf_{2}&=x_{1}x_{2}^{11}x_{3}^{15}x_{4}^{15}x_{5}^{17}+x_{1}x_{2}^{11}x_{3}^{15}x_{4}^{17}x_{5}^{15}+x_{1}x_{2}^{11}x_{3}^{17}x_{4}^{15}x_{5}^{15}+x_{1}x_{2}^{11}x_{3}^{23}x_{4}^{7}x_{5}^{17}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}x_{2}^{11}x_{3}^{23}x_{4}^{9}x_{5}^{15}+x_{1}x_{2}^{11}x_{3}^{25}x_{4}^{7}x_{5}^{15}+x_{1}x_{2}^{13}x_{3}^{15}x_{4}^{15}x_{5}^{15}+x_{1}x_{2}^{13}x_{3}^{23}x_{4}^{7}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}x_{2}^{14}x_{3}^{18}x_{4}^{11}x_{5}^{15}+x_{1}x_{2}^{22}x_{3}^{15}x_{4}^{6}x_{5}^{15},\\\
\vskip 6.0pt plus 2.0pt minus
2.0ptf_{4}&=x_{1}x_{2}^{14}x_{3}^{12}x_{4}^{15}x_{5}^{15}+x_{1}x_{2}^{14}x_{3}^{14}x_{4}^{13}x_{5}^{15}+x_{1}x_{2}^{14}x_{3}^{20}x_{4}^{7}x_{5}^{15}+x_{1}x_{2}^{20}x_{3}^{15}x_{4}^{6}x_{5}^{15},\\\
\vskip 6.0pt plus 2.0pt minus
2.0ptf_{8}&=x_{1}x_{2}^{10}x_{3}^{12}x_{4}^{15}x_{5}^{15}+x_{1}x_{2}^{10}x_{3}^{14}x_{4}^{13}x_{5}^{15}+x_{1}x_{2}^{12}x_{3}^{15}x_{4}^{10}x_{5}^{15}+x_{1}x_{2}^{14}x_{3}^{15}x_{4}^{8}x_{5}^{15}.\end{array}$

The above relations show that $x$ is strictly inadmissible.

By a similar computation, we obtain

$\begin{array}[]{ll}y&=x_{1}^{4}x_{2}^{11}x_{3}^{12}x_{4}^{19}x_{5}^{15}+x_{1}^{4}x_{2}^{11}x_{3}^{19}x_{4}^{12}x_{5}^{15}+x_{1}^{4}x_{2}^{20}x_{3}^{11}x_{4}^{11}x_{5}^{15}+x_{1}^{4}x_{2}^{24}x_{3}^{7}x_{4}^{11}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{5}x_{2}^{3}x_{3}^{12}x_{4}^{26}x_{5}^{15}+x_{1}^{5}x_{2}^{3}x_{3}^{26}x_{4}^{12}x_{5}^{15}+x_{1}^{5}x_{2}^{7}x_{3}^{8}x_{4}^{14}x_{5}^{27}+x_{1}^{5}x_{2}^{7}x_{3}^{8}x_{4}^{26}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{5}x_{2}^{7}x_{3}^{10}x_{4}^{12}x_{5}^{27}+x_{1}^{5}x_{2}^{7}x_{3}^{10}x_{4}^{24}x_{5}^{15}+x_{1}^{5}x_{2}^{10}x_{3}^{12}x_{4}^{19}x_{5}^{15}+x_{1}^{5}x_{2}^{10}x_{3}^{19}x_{4}^{12}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{5}x_{2}^{11}x_{3}^{5}x_{4}^{10}x_{5}^{30}+x_{1}^{5}x_{2}^{11}x_{3}^{6}x_{4}^{12}x_{5}^{27}+x_{1}^{5}x_{2}^{11}x_{3}^{8}x_{4}^{14}x_{5}^{23}+x_{1}^{5}x_{2}^{11}x_{3}^{8}x_{4}^{22}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{5}x_{2}^{11}x_{3}^{9}x_{4}^{6}x_{5}^{30}+x_{1}^{5}x_{2}^{11}x_{3}^{10}x_{4}^{20}x_{5}^{15}+x_{1}^{5}x_{2}^{11}x_{3}^{16}x_{4}^{14}x_{5}^{15}+x_{1}^{5}x_{2}^{11}x_{3}^{18}x_{4}^{12}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{5}x_{2}^{18}x_{3}^{11}x_{4}^{12}x_{5}^{15}+x_{1}^{7}x_{2}^{3}x_{3}^{12}x_{4}^{24}x_{5}^{15}+x_{1}^{7}x_{2}^{3}x_{3}^{24}x_{4}^{12}x_{5}^{15}+x_{1}^{7}x_{2}^{5}x_{3}^{9}x_{4}^{10}x_{5}^{30}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{7}x_{2}^{5}x_{3}^{10}x_{4}^{12}x_{5}^{27}+x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}^{12}x_{5}^{27}+x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}^{14}x_{5}^{25}+x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}^{24}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+Sq^{1}(g_{1})+Sq^{2}(g_{2})+Sq^{4}(g_{4})+Sq^{8}(g_{8})\ \ {\rm
mod}(P_{5}^{-}(\omega_{(5,4)})),\ \mbox{where}\\\ \vskip 6.0pt plus 2.0pt
minus
2.0ptg_{1}&=x_{1}^{7}x_{2}^{5}x_{3}^{12}x_{4}^{21}x_{5}^{15}+x_{1}^{7}x_{2}^{5}x_{3}^{21}x_{4}^{12}x_{5}^{15}+x_{1}^{7}x_{2}^{7}x_{3}^{5}x_{4}^{12}x_{5}^{29}+x_{1}^{7}x_{2}^{7}x_{3}^{12}x_{4}^{19}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{7}x_{2}^{7}x_{3}^{19}x_{4}^{12}x_{5}^{15}+x_{1}^{7}x_{2}^{9}x_{3}^{8}x_{4}^{13}x_{5}^{23}+x_{1}^{7}x_{2}^{9}x_{3}^{8}x_{4}^{21}x_{5}^{15}+x_{1}^{7}x_{2}^{9}x_{3}^{16}x_{4}^{13}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{7}x_{2}^{17}x_{3}^{9}x_{4}^{12}x_{5}^{15}+x_{1}^{7}x_{2}^{20}x_{3}^{7}x_{4}^{11}x_{5}^{15},\\\
\vskip 6.0pt plus 2.0pt minus
2.0ptg_{2}&=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{38}+x_{1}^{7}x_{2}^{3}x_{3}^{6}x_{4}^{20}x_{5}^{23}+x_{1}^{7}x_{2}^{3}x_{3}^{12}x_{4}^{22}x_{5}^{15}+x_{1}^{7}x_{2}^{3}x_{3}^{22}x_{4}^{12}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{7}x_{2}^{6}x_{3}^{12}x_{4}^{19}x_{5}^{15}+x_{1}^{7}x_{2}^{6}x_{3}^{19}x_{4}^{12}x_{5}^{15}+x_{1}^{7}x_{2}^{7}x_{3}^{5}x_{4}^{10}x_{5}^{30}+x_{1}^{7}x_{2}^{7}x_{3}^{6}x_{4}^{12}x_{5}^{27}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}^{14}x_{5}^{23}+x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}^{22}x_{5}^{15}+x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{6}x_{5}^{30}+x_{1}^{7}x_{2}^{7}x_{3}^{10}x_{4}^{20}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{7}x_{2}^{7}x_{3}^{16}x_{4}^{14}x_{5}^{15}+x_{1}^{7}x_{2}^{7}x_{3}^{18}x_{4}^{12}x_{5}^{15}+x_{1}^{7}x_{2}^{18}x_{3}^{7}x_{4}^{12}x_{5}^{15}+x_{1}^{7}x_{2}^{18}x_{3}^{8}x_{4}^{11}x_{5}^{15},\\\
\vskip 6.0pt plus 2.0pt minus
2.0ptg_{4}&=x_{1}^{4}x_{2}^{7}x_{3}^{12}x_{4}^{19}x_{5}^{15}+x_{1}^{4}x_{2}^{7}x_{3}^{19}x_{4}^{12}x_{5}^{15}+x_{1}^{4}x_{2}^{20}x_{3}^{7}x_{4}^{11}x_{5}^{15}+x_{1}^{5}x_{2}^{3}x_{3}^{12}x_{4}^{22}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{5}x_{2}^{3}x_{3}^{22}x_{4}^{12}x_{5}^{15}+x_{1}^{5}x_{2}^{6}x_{3}^{12}x_{4}^{19}x_{5}^{15}+x_{1}^{5}x_{2}^{6}x_{3}^{19}x_{4}^{12}x_{5}^{15}+x_{1}^{5}x_{2}^{7}x_{3}^{5}x_{4}^{10}x_{5}^{30}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{5}x_{2}^{7}x_{3}^{6}x_{4}^{12}x_{5}^{27}+x_{1}^{5}x_{2}^{7}x_{3}^{8}x_{4}^{14}x_{5}^{23}+x_{1}^{5}x_{2}^{7}x_{3}^{8}x_{4}^{22}x_{5}^{15}+x_{1}^{5}x_{2}^{7}x_{3}^{9}x_{4}^{6}x_{5}^{30}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{5}x_{2}^{7}x_{3}^{10}x_{4}^{20}x_{5}^{15}+x_{1}^{5}x_{2}^{7}x_{3}^{16}x_{4}^{14}x_{5}^{15}+x_{1}^{5}x_{2}^{7}x_{3}^{18}x_{4}^{12}x_{5}^{15}+x_{1}^{5}x_{2}^{18}x_{3}^{7}x_{4}^{12}x_{5}^{15}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{11}x_{2}^{5}x_{3}^{5}x_{4}^{6}x_{5}^{30}+x_{1}^{11}x_{2}^{5}x_{3}^{6}x_{4}^{12}x_{5}^{23},\\\
g_{8}&=x_{1}^{7}x_{2}^{5}x_{3}^{5}x_{4}^{6}x_{5}^{30}+x_{1}^{7}x_{2}^{5}x_{3}^{6}x_{4}^{12}x_{5}^{23}+x_{1}^{7}x_{2}^{9}x_{3}^{10}x_{4}^{12}x_{5}^{15}+x_{1}^{7}x_{2}^{10}x_{3}^{8}x_{4}^{13}x_{5}^{15}.\end{array}$

By Definition 2.2.4, we see that $y$ is strictly inadmissible. The lemma
follows. ∎

######  Lemma 3.2.15.

For any $t\geqslant 4,$ the following monomials are strictly inadmissible:

$x_{1}^{3}x_{2}^{2^{t}-4}x_{3}^{3}x_{4}^{2^{t}-3}x_{5}^{2^{t+1}-2}$, |  $x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t+1}-2}x_{4}^{3}x_{5}^{2^{t}-4}$, |  $x_{1}^{3}x_{2}^{2^{t}-4}x_{3}^{3}x_{4}^{2^{t+1}-3}x_{5}^{2^{t}-2}$,
---|---|---
$x_{1}^{3}x_{2}^{2^{t+1}-3}x_{3}^{2^{t}-2}x_{4}^{3}x_{5}^{2^{t}-4}$, |  $x_{1}^{3}x_{2}^{2^{t+1}-4}x_{3}^{3}x_{4}^{2^{t}-3}x_{5}^{2^{t}-2}$, |  $x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t}-2}x_{4}^{3}x_{5}^{2^{t+1}-4}$,
$x_{1}^{3}x_{2}^{5}x_{3}^{2^{t+1}-2}x_{4}^{2^{t}-5}x_{5}^{2^{t}-4}$, |  $x_{1}^{3}x_{2}^{5}x_{3}^{2^{t}-2}x_{4}^{2^{t}-5}x_{5}^{2^{t+1}-4}$, |  $x_{1}^{3}x_{2}^{5}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-5}x_{5}^{2^{t}-4}$,
$x_{1}^{3}x_{2}^{5}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-7}x_{5}^{2^{t}-2}$, |  $x_{1}^{3}x_{2}^{5}x_{3}^{2^{t+1}-7}x_{4}^{2^{t}-2}x_{5}^{2^{t}-2}$, |  $x_{1}^{3}x_{2}^{2^{t+1}-3}x_{3}^{6}x_{4}^{2^{t}-5}x_{5}^{2^{t}-4}$,
$x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{6}x_{4}^{2^{t}-5}x_{5}^{2^{t+1}-4}$, |  $x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{6}x_{4}^{2^{t+1}-5}x_{5}^{2^{t}-4}$, |  $x_{1}^{3}x_{2}^{2^{t+1}-3}x_{3}^{7}x_{4}^{2^{t}-6}x_{5}^{2^{t}-4}$,
$x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{7}x_{4}^{2^{t}-6}x_{5}^{2^{t+1}-4}$, |  $x_{1}^{3}x_{2}^{7}x_{3}^{2^{t+1}-5}x_{4}^{2^{t}-4}x_{5}^{2^{t}-4}$, |  $x_{1}^{7}x_{2}^{3}x_{3}^{2^{t+1}-5}x_{4}^{2^{t}-4}x_{5}^{2^{t}-4}$,
$x_{1}^{7}x_{2}^{2^{t+1}-5}x_{3}^{3}x_{4}^{2^{t}-4}x_{5}^{2^{t}-4}$, |  $x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{7}x_{4}^{2^{t+1}-6}x_{5}^{2^{t}-4}$, |  $x_{1}^{3}x_{2}^{7}x_{3}^{2^{t+1}-7}x_{4}^{2^{t}-2}x_{5}^{2^{t}-4}$,
$x_{1}^{7}x_{2}^{3}x_{3}^{2^{t+1}-7}x_{4}^{2^{t}-2}x_{5}^{2^{t}-4}$, |  $x_{1}^{3}x_{2}^{13}x_{3}^{2^{t+1}-9}x_{4}^{2^{t}-6}x_{5}^{2^{t}-4}$, |  $x_{1}^{3}x_{2}^{13}x_{3}^{2^{t+1}-10}x_{4}^{2^{t}-5}x_{5}^{2^{t}-4}$,
$x_{1}^{7}x_{2}^{2^{t}-5}x_{3}^{5}x_{4}^{2^{t}-8}x_{5}^{2^{t+1}-2}$, |  $x_{1}^{7}x_{2}^{2^{t}-5}x_{3}^{5}x_{4}^{2^{t+1}-2}x_{5}^{2^{t}-8}$, |  $x_{1}^{7}x_{2}^{2^{t+1}-5}x_{3}^{5}x_{4}^{2^{t}-8}x_{5}^{2^{t}-2}$,
$x_{1}^{7}x_{2}^{2^{t+1}-5}x_{3}^{5}x_{4}^{2^{t}-2}x_{5}^{2^{t}-8}$, |  $x_{1}^{7}x_{2}^{2^{t}-5}x_{3}^{5}x_{4}^{2^{t}-2}x_{5}^{2^{t+1}-8}$, |  $x_{1}^{7}x_{2}^{2^{t}-5}x_{3}^{5}x_{4}^{2^{t+1}-8}x_{5}^{2^{t}-2}$,
$x_{1}^{7}x_{2}^{7}x_{3}^{2^{t}-7}x_{4}^{2^{t}-6}x_{5}^{2^{t+1}-4}$, |  $x_{1}^{7}x_{2}^{9}x_{3}^{2^{t}-9}x_{4}^{2^{t}-6}x_{5}^{2^{t+1}-4}$, |  $x_{1}^{7}x_{2}^{7}x_{3}^{2^{t}-7}x_{4}^{2^{t+1}-6}x_{5}^{2^{t}-4}$,
$x_{1}^{7}x_{2}^{9}x_{3}^{2^{t}-9}x_{4}^{2^{t+1}-6}x_{5}^{2^{t}-4}$, |  $x_{1}^{7}x_{2}^{7}x_{3}^{2^{t+1}-7}x_{4}^{2^{t}-6}x_{5}^{2^{t}-4}$, |  $x_{1}^{7}x_{2}^{11}x_{3}^{2^{t+1}-11}x_{4}^{2^{t}-8}x_{5}^{2^{t}-2}$,
$x_{1}^{7}x_{2}^{11}x_{3}^{2^{t+1}-11}x_{4}^{2^{t}-2}x_{5}^{2^{t}-8}$, |  $x_{1}^{7}x_{2}^{9}x_{3}^{2^{t+1}-9}x_{4}^{2^{t}-6}x_{5}^{2^{t}-4}$. |

###### Proof.

We prove the lemma for
$u=x_{1}^{7}x_{2}^{9}x_{3}^{2^{t+1}-9}x_{4}^{2^{t}-6}x_{5}^{2^{t}-4}.$ The
others can be proven by a similar computation. By a direct computation using
the Cartan formula, we have

$\begin{array}[]{ll}u&=x_{1}^{4}x_{2}^{7}x_{3}^{2^{t+1}-5}x_{4}^{2^{t}-5}x_{5}^{2^{t}-4}+x_{1}^{4}x_{2}^{11}x_{3}^{2^{t+1}-9}x_{4}^{2^{t}-5}x_{5}^{2^{t}-4}+x_{1}^{5}x_{2}^{7}x_{3}^{2^{t+1}-5}x_{4}^{2^{t}-6}x_{5}^{2^{t}-4}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{5}x_{2}^{11}x_{3}^{2^{t+1}-9}x_{4}^{2^{t}-6}x_{5}^{2^{t}-4}+x_{1}^{7}x_{2}^{7}x_{3}^{2^{t+1}-8}x_{4}^{2^{t}-5}x_{5}^{2^{t}-4}+x_{1}^{7}x_{2}^{7}x_{3}^{2^{t+1}-7}x_{4}^{2^{t}-6}x_{5}^{2^{t}-4}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+x_{1}^{7}x_{2}^{8}x_{3}^{2^{t+1}-9}x_{4}^{2^{t}-5}x_{5}^{2^{t}-4}+Sq^{1}(x_{1}^{7}x_{2}^{7}x_{3}^{2^{t+1}-9}x_{4}^{2^{t}-5}x_{5}^{2^{t}-4})+Sq^{2}(x_{1}^{7}x_{2}^{7}x_{3}^{2^{t+1}-9}x_{4}^{2^{t}-6}x_{5}^{2^{t}-4})\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+Sq^{4}(x_{1}^{4}x_{2}^{7}x_{3}^{2^{t+1}-9}x_{4}^{2^{t}-5}x_{5}^{2^{t}-4}+x_{1}^{5}x_{2}^{7}x_{3}^{2^{t+1}-9}x_{4}^{2^{t}-6}x_{5}^{2^{t}-4})\
\ {\rm mod}(P_{5}^{-}(\omega_{(5,t)})).\end{array}$

This equality shows that $u$ is strictly inadmissible. The lemma is proved. ∎

The proof of the following lemma is analogous to the proof of Lemma 3.2.15.

######  Lemma 3.2.16.

For any $t\geqslant 5,$ the following monomials are strictly inadmissible:

$x_{1}^{3}x_{2}^{13}x_{3}^{2^{t}-10}x_{4}^{2^{t}-5}x_{5}^{2^{t+1}-4}$, |  $x_{1}^{3}x_{2}^{13}x_{3}^{2^{t}-10}x_{4}^{2^{t+1}-5}x_{5}^{2^{t}-4}$, |  $x_{1}^{3}x_{2}^{13}x_{3}^{2^{t}-9}x_{4}^{2^{t}-6}x_{5}^{2^{t+1}-4}$,
---|---|---
$x_{1}^{3}x_{2}^{13}x_{3}^{2^{t}-9}x_{4}^{2^{t+1}-6}x_{5}^{2^{t}-4}$, |  $x_{1}^{3}x_{2}^{15}x_{3}^{2^{t}-11}x_{4}^{2^{t}-6}x_{5}^{2^{t+1}-4}$, |  $x_{1}^{15}x_{2}^{3}x_{3}^{2^{t}-11}x_{4}^{2^{t}-6}x_{5}^{2^{t+1}-4}$,
$x_{1}^{3}x_{2}^{15}x_{3}^{2^{t}-11}x_{4}^{2^{t+1}-6}x_{5}^{2^{t}-4}$, |  $x_{1}^{15}x_{2}^{3}x_{3}^{2^{t}-11}x_{4}^{2^{t+1}-6}x_{5}^{2^{t}-4}$, |  $x_{1}^{7}x_{2}^{11}x_{3}^{2^{t}-11}x_{4}^{2^{t}-8}x_{5}^{2^{t+1}-2}$,
$x_{1}^{7}x_{2}^{11}x_{3}^{2^{t}-11}x_{4}^{2^{t+1}-2}x_{5}^{2^{t}-8}$, |  $x_{1}^{7}x_{2}^{11}x_{3}^{2^{t}-11}x_{4}^{2^{t}-2}x_{5}^{2^{t+1}-8}$, |  $x_{1}^{7}x_{2}^{11}x_{3}^{2^{t}-11}x_{4}^{2^{t+1}-8}x_{5}^{2^{t}-2}$.

###### Proof of Proposition 3.2.11.

Let $x$ be an admissible monomial in $(P_{5}^{+})_{3(2^{t}-1)+2^{t}}$ such
that $\omega(x)=\omega_{(5,t)}$ with $t\geqslant 4.$ By induction on $t,$ we
see that $x\neq b_{3(2^{t}-1)+2^{t},k}$ for $k=1,2,\ldots,270,$ then there is
a monomial $z$, which is given in one of Lemmas 3.2.12, 3.2.13, 3.2.14,
3.2.15, and 3.2.16 such that $x=zy^{2^{\alpha}}$ for some monomial $y$ and
positive integer $\alpha.$ By Theorem 2.2.5, $x$ is inadmissible. Hence
$\mathscr{B}_{5}^{+}(\omega_{(5,t)})\subseteq\\{b_{3(2^{t}-1)+2^{t},k}:\
1\leqslant k\leqslant 270\\}$ for every $t\geqslant 4.$

Now we prove that the classes
$[b_{3(2^{t}-1)+2^{t},k}]_{\omega_{(5,t)}}\neq[0]_{\omega_{(5,t)}}$ with
$1\leqslant k\leqslant 270,$ and $t\geqslant 4.$ Suppose there is a linear
relation

$\mathcal{S}=\sum\limits_{1\leqslant k\leqslant
270}\gamma_{k}b_{3(2^{t}-1)+2^{t},k}\equiv_{\omega_{(5,t)}}0,$

where $\gamma_{k}\in\mathbb{F}_{2}$ for all $k,\ 1\leqslant k\leqslant 270.$

For $t\geqslant 4,$ according to Sum [32],
$\mathscr{B}_{4}^{+}(3(2^{t}-1)+2^{t})$ is the set consisting of $33$
monomials, namely:

$v_{1}.\ x_{1}x_{2}^{2^{t}-2}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-1}$, |  $v_{2}.\ x_{1}x_{2}^{2^{t}-2}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-1}$, |  $v_{3}.\ x_{1}x_{2}^{2^{t}-1}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-1}$, |
---|---|---|---
$v_{4}.\ x_{1}x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-2}$, |  $v_{5}.\ x_{1}x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-2}x_{4}^{2^{t}-1}$, |  $v_{6}.\ x_{1}x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-1}x_{4}^{2^{t}-2}$, |
$v_{7}.\ x_{1}^{2^{t}-1}x_{2}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-1}$, |  $v_{8}.\ x_{1}^{2^{t}-1}x_{2}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-2}$, |  $v_{9}.\ x_{1}^{2^{t}-1}x_{2}^{2^{t+1}-1}x_{3}x_{4}^{2^{t}-2}$, |
$v_{10}.\ x_{1}^{2^{t+1}-1}x_{2}x_{3}^{2^{t}-2}x_{4}^{2^{t}-1}$, |  $v_{11}.\ x_{1}^{2^{t+1}-1}x_{2}x_{3}^{2^{t}-1}x_{4}^{2^{t}-2}$, |  $v_{12}.\ x_{1}^{2^{t+1}-1}x_{2}^{2^{t}-1}x_{3}x_{4}^{2^{t}-2}$, |
$v_{13}.\ x_{1}x_{2}^{2^{t}-1}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-2}$, |  $v_{14}.\ x_{1}x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-2}x_{4}^{2^{t}-1}$, |  $v_{15}.\ x_{1}x_{2}^{2^{t+1}-2}x_{3}^{2^{t}-1}x_{4}^{2^{t}-1}$, |
$v_{16}.\ x_{1}^{2^{t}-1}x_{2}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-2}$, |  $v_{17}.\ x_{1}^{2^{t}-1}x_{2}x_{3}^{2^{t+1}-2}x_{4}^{2^{t}-1}$, |  $v_{18}.\ x_{1}^{2^{t}-1}x_{2}^{2^{t}-1}x_{3}x_{4}^{2^{t+1}-2}$, |
$v_{19}.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-1}$, |  $v_{20}.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-2}$, |  $v_{21}.\ x_{1}^{3}x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-3}x_{4}^{2^{t}-2}$, |
$v_{22}.\ x_{1}^{2^{t+1}-1}x_{2}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t}-2}$, |  $v_{23}.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-2}$, |  $v_{24}.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t+1}-2}x_{4}^{2^{t}-1}$, |
$v_{25}.\ x_{1}^{3}x_{2}^{2^{t}-1}x_{3}^{2^{t}-3}x_{4}^{2^{t+1}-2}$, |  $v_{26}.\ x_{1}^{2^{t}-1}x_{2}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t+1}-2}$, |  $v_{27}.\ x_{1}^{3}x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-3}x_{4}^{2^{t}-2}$, |
$v_{28}.\ x_{1}^{3}x_{2}^{2^{t+1}-3}x_{3}^{2^{t}-2}x_{4}^{2^{t}-1}$, |  $v_{29}.\ x_{1}^{3}x_{2}^{2^{t+1}-3}x_{3}^{2^{t}-1}x_{4}^{2^{t}-2}$, |  $v_{30}.\ x_{1}^{2^{t}-1}x_{2}^{3}x_{3}^{2^{t+1}-3}x_{4}^{2^{t}-2}$, |
$v_{31}.\ x_{1}^{7}x_{2}^{2^{t+1}-5}x_{3}^{2^{t}-3}x_{4}^{2^{t}-2}$, |  $v_{32}.\ x_{1}^{7}x_{2}^{2^{t}-5}x_{3}^{2^{t}-3}x_{4}^{2^{t+1}-2}$, |  $v_{33}.\ x_{1}^{7}x_{2}^{2^{t}-5}x_{3}^{2^{t+1}-3}x_{4}^{2^{t}-2}$. |

By a direct computation using Theorem 3.1.3, we express
$p_{(i;I)}(\mathcal{S}),$ in terms of $v_{j},\ 1\leqslant j\leqslant 33.$
Computing directly from the relations

$p_{(i;I)}(\mathcal{S})\equiv_{\omega_{(5,t)}}0,\
\forall(i;I)\in\mathcal{N}_{5},\ \ell(I)=1,$

we obtain $\gamma_{k}=0,\ k=1,2,\ldots,270.$ The proposition follows. ∎

We knew that

$\begin{array}[]{ll}(QP_{5})_{3(2^{t}-1)+2^{t}}&={\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})})\,\bigoplus\,(QP_{5})_{2^{t+1}-4},\\\
{\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})})&=(QP_{5}^{0})_{3(2^{t}-1)+2^{t}}\,\bigoplus\,QP_{5}^{+}(\omega_{(5,t)}),\end{array}$

and $\dim(QP_{5}^{0})_{3(2^{t}-1)+2^{t}}=195$ for any $t\geqslant 3.$ So, from
Theorem 3.2.1 and Proposition 3.2.11, we get

$\dim(QP_{5})_{3(2^{t}-1)+2^{t}}=\left\\{\begin{array}[]{ll}945&\mbox{if\
}t=4,\\\ 1115&\mbox{if\ }t=5,\\\ 1116&\mbox{if\ }t\geqslant
6.\end{array}\right.$

The proof of Theorem 1.1 is completed.

##  4 Proof of Theorem 1.3

In this section, we prove Theorem 1.3 by using the results in Section 3.
Firstly, we need some notations and definitions for the proof of the theorem.

For $1\leqslant i\leqslant s,$ define the $\mathscr{A}$-homomorphism
$\tau_{i}:P_{s}\longrightarrow P_{s},$ which is determined by
$\tau_{i}(x_{i})=x_{i+1},\;\tau_{i}(x_{i+1})=x_{i},\;\tau_{i}(x_{j})=x_{j}$
for $j\neq i,i+1,\;1\leqslant i<s$ and
$\tau_{s}(x_{1})=x_{1}+x_{2},\;\tau_{s}(x_{j})=x_{j}$ for $j>1.$ Observe that
the general linear group $GL_{s}$ is generated by the matrices associated with
by $\tau_{i},\;0\leqslant i\leqslant s$ and the symmetric group
$\Sigma_{s}\subset GL_{s}$ is generated by the ones associated with
$\tau_{i},\;1\leqslant i<s.$ Hence, a class $[f]_{\omega}$ represented by a
homogeneous polynomial $f\in P_{s}$ is an $GL_{s}$-invariant if and only if
$\tau_{i}(f)\equiv_{\omega}f$ for $1\leqslant i\leqslant s.$ $[f]_{\omega}$ is
an $\Sigma_{s}$-invariant if and only if $\tau_{i}(f)\equiv_{\omega}f$ for
$1\leqslant i<s.$

For the weight vector $\omega_{(5,t)}=(3,3,\ldots,3,1)$ ($t$ times of $3$), we
have $\deg\omega_{(5,t)}=3(2^{t}-1)+2^{t}.$ For any monomials
$g_{1},\,g_{2},\,\ldots,\,g_{m}$ in $P_{5}(\omega_{(5,t)})$ and for a subgroup
$G\subset GL_{5},$ we denote $G(g_{1},\,\ldots,\,g_{m})$ the $G$-submodule of
$QP_{5}(\omega_{(5,t)})$ generated by the set
$\\{[g_{i}]_{\omega_{(5,t)}}\,:\,1\leqslant i\leqslant m\\}.$ We see that
$\omega_{(5,t)}$ is the weight vector of the mimimal spike
$x_{1}^{2^{t+1}-1}x_{2}^{2^{t}-1}x_{3}^{2^{t}-1}\in(P_{5})_{3(2^{t}-1)+2^{t}};$
hence $[g_{i}]_{\omega_{(5,t)}}=[g_{i}]$ for all $i,\,1\leqslant i\leqslant
m.$

Recall that Kameko’s homomorphism

$\widetilde{Sq^{0}_{*}}=(\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})}:(QP_{5})_{3(2^{t}-1)+2^{t}}\longrightarrow(QP_{5})_{2^{t+1}-4}$

is an epimorphism of $GL_{5}$-modules. Then, as mentioned in Section 1, we
need only to determine ${\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})})^{GL_{5}}.$

From the results in Section 3, we get

$\dim({\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})}))=\dim(QP_{5}(\omega_{(5,t)})).$

Note that
$QP_{5}(\omega_{(5,t)})=QP_{5}^{0}(\omega_{(5,t)})\,\oplus\,QP_{5}^{+}(\omega_{(5,t)})$
and

$\dim({\rm
Ker}((\widetilde{Sq^{0}_{*}})_{(5,3(2^{t}-1)+2^{t})})^{GL_{5}})\leqslant\dim(QP_{5}(\omega_{(5,t)})^{GL_{5}}).$

Now, for $t=1,$ $3(2^{t}-1)+2^{t}=5.$ From a result of Sum [31, 35], we have
$QP_{5}(\omega_{(5,1)})^{GL_{5}}=0.$

We now compute $QP_{5}(\omega_{(5,t)})^{GL_{5}}$ for $t\geqslant 2.$

###  4.1 Computation of $QP_{5}(\omega_{(5,t)})^{\Sigma_{5}}$

As it is known,
$\mathscr{B}_{5}^{0}(3(2^{t}-1)+2^{t})=\mathscr{B}_{5}^{0}(\omega_{(5,t)})$ is
the set consisting of the admissible monomials $q_{t,k}$ (see Subsection 5.1).
By a simple computation, we see that the following subspaces are
$\Sigma_{5}$-submodules of $QP_{5}^{0}(\omega_{(5,t)})$:

For $t\geqslant 2,$

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\Sigma_{5}(q_{t,1})&=\langle[q_{t,k}]\,:\,1\leqslant k\leqslant
30\rangle,\\\ \Sigma_{5}(q_{t,31})&=\langle[q_{t,k}]\,:\,31\leqslant
k\leqslant 90\rangle.\end{array}$

For $t=2,$

$\mathcal{M}_{1}=\langle[q_{2,k}]\,:\,91\leqslant k\leqslant 145\rangle.$

For $t\geqslant 3,$

$\Sigma_{5}(q_{t,91})=\langle[q_{t,k}]\,:\,91\leqslant k\leqslant 110\rangle.$

For $t=3,$

$\mathcal{M}_{2}=\langle[q_{3,k}]\,:\,111\leqslant k\leqslant 195\rangle.$

For $t\geqslant 4,$

$\mathcal{M}^{*}_{t,1}=\langle[q_{t,k}]\,:\,111\leqslant k\leqslant
195\rangle.$

######  Lemma 4.1.1.

Let $t$ be an integer. Then, we obtain the following:

For $t\geqslant 2,$

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\Sigma_{5}(q_{t,1})^{\Sigma_{5}}&=\langle[p_{t,1}:=q_{t,1}+q_{t,2}+\cdots+q_{t,30}]\rangle,\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\Sigma_{5}(q_{t,31})^{\Sigma_{5}}&=\langle[p_{t,2}:=q_{t,31}+q_{t,32}+\cdots+q_{t,90}]\rangle.\\\
\end{array}$

For $t\geqslant 3,$

$\Sigma_{5}(q_{t,91})^{\Sigma_{5}}=\langle[p_{t,3}:=q_{t,91}+q_{t,92}+\cdots+q_{t,110}]\rangle.$

###### Outline of the proof.

The set $B:=\\{[q_{t,k}]\,:\,1\leqslant k\leqslant 30\\}$ is a basis of
$\Sigma_{5}(q_{t,1})$ for $t\geqslant 2.$ The action of $\Sigma_{5}$ on
$QP_{5}$ induces the one of it on $B.$ Furthermore, this action is transitive.
Hence, if $g\equiv\sum_{k=1}^{30}\gamma_{k}q_{t,k}$ with
$\gamma_{t}\in\mathbb{F}_{2}$ and $[g]\in\Sigma_{5}(q_{t,1})^{\Sigma_{5}},$
then the relations $\tau_{i}(g)\equiv g,\,i=1,\,2,\,3,\,4,$ imply
$\gamma_{k}=\gamma_{1},\,\forall k,\,1\leqslant k\leqslant 30.$ Hence,
$\Sigma_{5}(q_{t,1})^{\Sigma_{5}}=\langle[p_{t,1}]\rangle$ with
$p_{t,1}=\sum_{k=1}^{30}q_{t,k}.$

By a similar computation, we obtain
$\Sigma_{5}(q_{t,31})^{\Sigma_{5}}=\langle[p_{t,2}]\rangle$ with
$p_{t,2}=\sum_{k=31}^{90}q_{t,k},$ and
$\Sigma_{5}(q_{t,91})^{\Sigma_{5}}=\langle[p_{t,3}]\rangle$ with
$p_{t,3}=\sum_{k=91}^{110}q_{t,k}.$ The lemma follows. ∎

######  Lemma 4.1.2.

We have $\mathcal{M}_{1}^{\Sigma_{5}}=0$ and
$\dim(\mathcal{M}_{2}^{\Sigma_{5}})=\dim((\mathcal{M}^{*}_{t,1})^{\Sigma_{5}})=1$
for every $t\geqslant 4.$

###### Proof.

Using a result in Section 3, we have $\dim(\mathcal{M}_{1})=55.$ Suppose
$h\equiv\sum_{k=91}^{145}\gamma_{k}q_{2,k}$ with $\gamma_{k}\in\mathbb{F}_{2}$
and $[h]\in\mathcal{M}_{1}^{\Sigma_{5}}.$ By computing $\tau_{j}(h)+h$ in
terms of $q_{2,k},\,91\leqslant k\leqslant 145$ and using the relations
$\tau_{j}(h)+h\equiv 0,\,1\leqslant j\leqslant 4,$ we obtain $\gamma_{k}=0$
for all $k,$ $91\leqslant k\leqslant 145.$

Now suppose $h\equiv\sum_{k=111}^{195}\gamma_{k}q_{3,k}$ and
$[h]\in\mathcal{M}_{2}^{\Sigma_{5}}.$ We compute $\tau_{j}(h)+h$ in terms of
$q_{3,k},\,111\leqslant k\leqslant 195.$ By computing directly from the
relations $\tau_{j}(h)+h\equiv 0,\,j=1,2,3,4,$ one gets $\gamma_{k}=0$ for
$111\leqslant k\leqslant 135$ and $\gamma_{k}=\gamma_{136}$ for $136\leqslant
k\leqslant 195.$ This implies
$\mathcal{M}_{2}^{\Sigma_{5}}=\langle[p_{3,4}]\rangle$ with
$p_{3,4}=\sum_{i=136}^{195}q_{3,i}.$

Similarly, we have
$(\mathcal{M}^{*}_{t,1})^{\Sigma_{5}}=\langle[q^{*}_{t,1}]\rangle$ with
$q^{*}_{t,1}=\sum_{i=136}^{195}q_{t,i}$ for all $t\geqslant 4.$ The lemma is
proved. ∎

We now denote by $b_{t,k}=b_{3(2^{t}-1)+2^{t},\,k}$ the admissible monomials
in $(P_{5}^{+})_{3(2^{t}-1)+2^{t}}$ as given in Subsection 5.2.

The case $t=2$. By a direct computation using the results in Section 3, we
have the direct summand decomposition of the $\Sigma_{5}$-modules:

$QP_{5}^{+}(\omega_{(5,2)})=\Sigma_{5}(b_{2,1})\,\bigoplus\,\mathcal{M}_{3},$

where $\Sigma_{5}(b_{2,1})\ =\langle[b_{2,k}]:\,1\leqslant k\leqslant
10\rangle$ and $\mathcal{M}_{3}=\langle[b_{2,k}]:\,11\leqslant k\leqslant
60\rangle.$

The following lemma is proved by a direct computation.

######  Lemma 4.1.3.

$\Sigma_{5}(b_{2,1})^{\Sigma_{5}}=0$ and
$\mathcal{M}_{3}^{\Sigma_{5}}=\langle[p_{2,4}:=b_{2,31}+b_{2,32}+\cdots+b_{2,60}]\rangle.$

From Lemmas 4.1.1 and 4.1.3, we obtain the following.

######  Proposition 4.1.4.

$QP_{5}(\omega_{(5,2)})^{\Sigma_{5}}=\langle[p_{2,1}],[p_{2,2}],[p_{2,4}]\rangle.$

The case $t=3$. By a simple computation, we see that

$\begin{array}[]{ll}\Sigma_{5}(b_{3,1})&=\langle[b_{3,k}]\,:\,1\leqslant
k\leqslant 45\rangle,\\\
\Sigma_{5}(b_{3,46})&=\langle[b_{3,k}]\,:\,46\leqslant k\leqslant
65\rangle,\\\ \end{array}$

and $\mathcal{M}_{4}=\langle[b_{3,k}]\,:\,66\leqslant k\leqslant 260\rangle$
are $\Sigma_{5}$-submodules of $QP_{5}^{+}(\omega_{(5,3)}).$ Hence, we have a
direct summand decomposition of $\Sigma_{5}$-modules:

$QP_{5}^{+}(\omega_{(5,3)})=\Sigma_{5}(b_{t,1})\,\oplus\,\Sigma_{5}(b_{t,46})\,\oplus\,\mathcal{M}_{4}.$

######  Lemma 4.1.5.

We have
$\dim(\Sigma_{5}(b_{3,1})^{\Sigma_{5}})=\dim(\Sigma_{5}(b_{3,46})^{\Sigma_{5}})=1$
and $\mathcal{M}_{4}^{\Sigma_{5}}=0.$

###### Proof.

We have $\dim\Sigma_{5}(b_{3,1})=45$ and the set $\\{[b_{3,k}]\,:\,1\leqslant
k\leqslant 45\\}$ is a basis of $\Sigma_{5}(b_{3,1}).$ Suppose
$f\equiv\sum_{k=1}^{45}\gamma_{k}b_{3,k}$ with $\gamma_{k}\in\mathbb{F}_{2}$
and $[f]\in\Sigma_{5}(b_{3,1})^{\Sigma_{5}}.$ By computing $\tau_{j}(f)+f$ in
terms of $b_{3,k},\ 1\leqslant k\leqslant 45$ and using the relations
$\tau_{j}(f)+f\equiv 0,\ j=1,2,3,4,$ one gets

$\begin{array}[]{lll}\gamma_{1}&=\gamma_{k},&\text{for }\,2\leqslant
k\leqslant 10,\\\ \gamma_{k}&=0,&\text{for }11\leqslant k\leqslant
45.\end{array}$

Hence $\Sigma_{5}(b_{3,1})^{\Sigma_{5}}=\langle[p_{3,5}]\rangle$ with
$p_{3,5}=\sum_{k=1}^{10}b_{3,k}.$ By a similar computation, we get
$\Sigma_{5}(b_{3,46})^{\Sigma_{5}}=\langle[p_{3,6}]\rangle$ with
$p_{3,6}=\sum_{k=46}^{65}b_{3,k}$ and $\mathcal{M}_{4}^{\Sigma_{5}}=0.$ The
lemma follows. ∎

Based on Lemmas 4.1.1, 4.1.2 and 4.1.5, one gets the following.

######  Proposition 4.1.6.

$QP_{5}(\omega_{(5,3)})^{\Sigma_{5}}=\langle[p_{3,1}],[p_{3,2}],\ldots,[p_{3,6}]\rangle.$

The case $t\geqslant 4$. We set

$b_{t,1}=x_{1}x_{2}x_{3}^{2^{t}-2}x_{4}^{2^{t}-2}x_{5}^{2^{t+1}-1},\ \
b_{t,46}=x_{1}x_{2}^{2}x_{3}^{2^{t}-4}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-1}.$

Computing from the results in Section 3, one gets

$\dim(\Sigma_{5}(b_{t,1}))=45,\ \ \dim(\Sigma_{5}(b_{t,46}))=20,$

where

$\begin{array}[]{ll}\Sigma_{5}(b_{t,1})&=\langle[b_{t,k}]\,:\,1\leqslant
k\leqslant 45\rangle,\\\
\Sigma_{5}(b_{t,46})&=\langle[b_{t,k}]\,:\,46\leqslant k\leqslant
65\rangle.\end{array}$

######  Lemma 4.1.7.

For any $t\geqslant 4,$
$\Sigma_{5}(b_{t,46})^{\Sigma_{5}}=\langle[q^{*}_{t,2}:=b_{t,46}+b_{t,47}+\cdots+b_{t,65}]\rangle.$

The proof of this lemma is straightforward.

For $t\geqslant 4,$ by an easy computation, we have a direct summand
decomposition of $\Sigma_{5}$-modules:

$QP_{5}^{+}(\omega_{(5,t)})=\Sigma_{5}(b_{t,1})\,\oplus\,\Sigma_{5}(b_{t,46})\,\oplus\,\mathcal{M}^{*}_{t,2},$

where $\mathcal{M}^{*}_{t,2}=\langle[b_{t,k}]\,:\,66\leqslant k\leqslant
270\rangle.$

######  Lemma 4.1.8.

For any integer $t\geqslant 4,$ we have

$\Sigma_{5}(b_{t,1})^{\Sigma_{5}}=\langle[q^{*}_{t,3}],[q^{*}_{t,4}]\rangle,\
\ (\mathcal{M}^{*}_{t,2})^{\Sigma_{5}}=0,$

with $q^{*}_{t,3}:=\sum_{k=1}^{15}b_{t,k},\ \
q^{*}_{t,4}:=\sum_{k=16}^{45}b_{t,k}.$

###### Proof.

Suppose $g\equiv\sum_{k=1}^{45}\gamma_{k}b_{t,k}$ with
$\gamma_{k}\in\mathbb{F}_{2}$ and $[g]\in\Sigma_{5}(b_{t,1})^{\Sigma_{5}}.$ By
computing $\tau_{i}(g)+g$ in terms of $b_{t,k},\ 1\leqslant i\leqslant 45$ and
using the relations $\tau_{i}(g)+g\equiv 0,\ i=1,2,3,4,$ we get

$\begin{array}[]{lll}\gamma_{1}&=\gamma_{k},&\text{for }\,2\leqslant
k\leqslant 15,\\\ \gamma_{16}&=\gamma_{k},&\text{for }17\leqslant k\leqslant
45.\end{array}$

This implies
$\Sigma_{5}(b_{t,1})^{\Sigma_{5}}=\langle[q^{*}_{t,3}],[q^{*}_{t,4}]\rangle$
with $q^{*}_{t,3}=\sum_{k=1}^{15}b_{t,k}$ and
$q^{*}_{t,4}=\sum_{k=16}^{45}b_{t,k}.$

Now suppose $g\equiv\sum_{k=66}^{270}\gamma_{k}b_{t,k}$ and
$[g]\in(\mathcal{M}^{*}_{t,2})^{\Sigma_{5}}.$ We compute $\tau_{j}(g)+g$ in
terms of $b_{t,k},\,66\leqslant k\leqslant 270.$ By computing directly from
the relations $\tau_{j}(g)+g\equiv 0,\,j=1,2,3,4,$ we obtain $\gamma_{k}=0$
for $66\leqslant k\leqslant 270.$ The lemma is proved. ∎

Combining this result and Lemmas 4.1.1, 4.1.2, 4.1.7, 4.1.8, one gets the
following.

######  Proposition 4.1.9.

For any $t\geqslant 4,$

$QP_{5}(\omega_{(5,t)})^{\Sigma_{5}}=\langle[p_{t,1}],[p_{t,2}],[p_{t,3}],[q^{*}_{t,1}],[q^{*}_{t,2}],[q^{*}_{t,3}],[q^{*}_{t,4}]\rangle.$

###  4.2 Computation of $QP_{5}(\omega_{(5,t)})^{GL_{5}}$

Let $g\in(P_{5})_{3(2^{t}-1)+2^{t}}$ such that
$[g]\in(QP_{5})_{3(2^{t}-1)+2^{t}}^{GL_{5}}.$ Then we have $[g]\in
QP_{5}(\omega_{(5,t)})^{GL_{5}}.$

For $t=1,$ we have $3(2^{t}-1)+2^{t}=5$ and $(QP_{5})_{5}^{GL_{5}}=0$ (see Sum
[31, 35]). So, the theorem is true for $t=1.$

For $t=2,$ from Proposition 4.1.4, we have
$g\equiv\gamma_{1}p_{2,1}+\gamma_{2}p_{2,2}+\gamma_{3}p_{2,3}$ with
$\gamma_{i}\in\mathbb{F}_{2},\ i=1,2,3.$ By computing $\tau_{5}(g)+g$ in terms
of the admissible monomials, we get

$\tau_{5}(g)+g\equiv(\gamma_{1}+\gamma_{2})x_{1}^{3}x_{2}^{3}x_{4}^{3}x_{5}^{4}+\gamma_{2}x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{3}x_{5}^{2}+(\gamma_{2}+\gamma_{3})x_{1}x_{2}^{3}x_{3}^{2}x_{4}x_{5}^{6}+\
\text{other terms}\equiv 0.$

This relation implies $\gamma_{i}=0$ for $1\leqslant i\leqslant 3.$ The
theorem is proved for $t=2.$

For $t=3,$ from Proposition 4.1.6, we have
$g\equiv\sum_{j=1}^{6}\beta_{j}p_{3,j}$ with
$\beta_{j}\in\mathbb{F}_{2},j=1,2,\ldots,6.$ A direct computation shows that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\tau_{5}(g)+g&\equiv\beta_{1}x_{1}^{7}x_{2}^{15}x_{5}^{7}+\beta_{2}x_{2}x_{3}^{6}x_{4}^{7}x_{5}^{15}+(\beta_{2}+\beta_{3})x_{1}x_{2}^{7}x_{4}^{6}x_{5}^{15}+(\beta_{2}+\beta_{4})x_{1}x_{2}x_{3}^{14}x_{4}^{7}x_{5}^{6}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+\beta_{5}x_{1}^{3}x_{2}^{15}x_{3}^{3}x_{4}^{4}x_{5}^{4}+\beta_{6}x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{2}x_{5}^{4}+\mbox{other
terms\, }\equiv 0.\end{array}$

From this relation, we obtain $\beta_{j}=0,\,1\leqslant j\leqslant 6.$ The
theorem holds for $t=3.$

For $t\geqslant 4,$ by Proposition 4.1.9, we have

$g\equiv\lambda_{1}p_{t,1}+\lambda_{2}p_{t,2}+\lambda_{3}p_{t,3}+\sum_{j=1}^{4}\lambda_{3+j}q^{*}_{t,j},$

where $\lambda_{k}\in\mathbb{F}_{2},\ 1\leqslant k\leqslant 7.$ By computing
$\tau_{5}(g)+g$ in terms of the admissible monomials, we get

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\tau_{5}(g)+g&\equiv\lambda_{1}x_{2}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-2}+(\lambda_{2}+\lambda_{4})x_{1}^{2^{t}-1}x_{3}^{3}x_{4}^{2^{t}-3}x_{5}^{2^{t+1}-2}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+\lambda_{3}x_{1}^{3}x_{2}^{2^{t+1}-3}x_{3}^{2^{t}-1}x_{4}^{2^{t}-2}+\lambda_{4}x_{1}^{7}x_{2}^{2^{t+1}-5}x_{4}^{2^{t}-3}x_{5}^{2^{t}-2}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\lambda_{1}+\lambda_{2}+\lambda_{5})x_{1}^{3}x_{2}^{5}x_{3}^{2^{t}-6}x_{4}^{2^{t+1}-3}x_{5}^{2^{t}-2}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+\lambda_{6}x_{1}^{3}x_{2}^{5}x_{3}^{2^{t+1}-6}x_{4}^{2^{t}-1}x_{5}^{2^{t}-4}+\lambda_{7}x_{1}^{7}x_{2}^{2^{t}-5}x_{3}^{5}x_{4}^{2^{t}-6}x_{5}^{2^{t+1}-4}+\mbox{other
terms\, }\equiv 0.\end{array}$

The last equality implies $\gamma_{k}=0,\ k=1,2,\ldots,7.$ The theorem is
completely proved.

##  5 Appendix

In this section, we list all the admissible monomials of degree
$d=3(2^{t}-1)+2^{t}$ in $P_{5}.$

###  5.1 The admissible monomials of degree $3(2^{t}-1)+2^{t}$ in $P_{5}^{0}$

Recall that

$\mathscr{B}_{5}^{0}(3(2^{t}-1)+2^{t})=\mathscr{B}_{5}^{0}(\omega_{(5,t)})=\Phi^{0}(\mathscr{B}_{4}(3(2^{t}-1)+2^{t}))$

with $\omega_{(5,t)}=\underset{\mbox{{$t$ times of
$3$}}}{\underbrace{(3,3,\ldots,3}},1).$ By Sum [32], we have

$|\mathscr{B}_{4}(3(2^{t}-1)+2^{t})|=\left\\{\begin{array}[]{ll}15&\text{if}\
t=1,\\\ 35&\text{if}\ t=2,\\\ 45&\text{if}\ t\geqslant 3.\end{array}\right.$

Set $u_{t}=|\mathscr{B}_{5}^{0}(3(2^{t}-1)+2^{t})|$ and
$\mathscr{B}_{5}^{0}(3(2^{t}-1)+2^{t})=\\{q_{t,k}:\ 1\leqslant k\leqslant
u_{t}\\}.$ We have $u_{1}=45,\ u_{2}=145,\ u_{t}=195$ for $t\geqslant 3.$

The admissible monomials $q_{t,k},\ 1\leqslant k\leqslant u_{t},$ are
determined as follows:

For $t\geqslant 1,$

$1.\ x_{3}^{2^{t}-1}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-1}$ | $2.\ x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-1}x_{5}^{2^{t}-1}$ | $3.\ x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-1}x_{5}^{2^{t}-1}$
---|---|---
$4.\ x_{2}^{2^{t}-1}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-1}$ | $5.\ x_{2}^{2^{t}-1}x_{4}^{2^{t+1}-1}x_{5}^{2^{t}-1}$ | $6.\ x_{2}^{2^{t}-1}x_{3}^{2^{t}-1}x_{5}^{2^{t+1}-1}$
$7.\ x_{2}^{2^{t}-1}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-1}$ | $8.\ x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-1}x_{5}^{2^{t}-1}$ | $9.\ x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-1}$
$10.\ x_{2}^{2^{t+1}-1}x_{4}^{2^{t}-1}x_{5}^{2^{t}-1}$ | $11.\ x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-1}x_{5}^{2^{t}-1}$ | $12.\ x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-1}x_{4}^{2^{t}-1}$
$13.\ x_{1}^{2^{t}-1}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-1}$ | $14.\ x_{1}^{2^{t}-1}x_{4}^{2^{t+1}-1}x_{5}^{2^{t}-1}$ | $15.\ x_{1}^{2^{t}-1}x_{3}^{2^{t}-1}x_{5}^{2^{t+1}-1}$
$16.\ x_{1}^{2^{t}-1}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-1}$ | $17.\ x_{1}^{2^{t}-1}x_{3}^{2^{t+1}-1}x_{5}^{2^{t}-1}$ | $18.\ x_{1}^{2^{t}-1}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-1}$
$19.\ x_{1}^{2^{t}-1}x_{2}^{2^{t}-1}x_{5}^{2^{t+1}-1}$ | $20.\ x_{1}^{2^{t}-1}x_{2}^{2^{t}-1}x_{4}^{2^{t+1}-1}$ | $21.\ x_{1}^{2^{t}-1}x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-1}$
$22.\ x_{1}^{2^{t}-1}x_{2}^{2^{t+1}-1}x_{5}^{2^{t}-1}$ | $23.\ x_{1}^{2^{t}-1}x_{2}^{2^{t+1}-1}x_{4}^{2^{t}-1}$ | $24.\ x_{1}^{2^{t}-1}x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-1}$
$25.\ x_{1}^{2^{t+1}-1}x_{4}^{2^{t}-1}x_{5}^{2^{t}-1}$ | $26.\ x_{1}^{2^{t+1}-1}x_{3}^{2^{t}-1}x_{5}^{2^{t}-1}$ | $27.\ x_{1}^{2^{t+1}-1}x_{3}^{2^{t}-1}x_{4}^{2^{t}-1}$
$28.\ x_{1}^{2^{t+1}-1}x_{2}^{2^{t}-1}x_{5}^{2^{t}-1}$ | $29.\ x_{1}^{2^{t+1}-1}x_{2}^{2^{t}-1}x_{4}^{2^{t}-1}$ |  $30.\ x_{1}^{2^{t+1}-1}x_{2}^{2^{t}-1}x_{3}^{2^{t}-1}$.

For $t=1,$

$31.\ x_{2}x_{3}x_{4}x_{5}^{2}$ | $32.\ x_{2}x_{3}x_{4}^{2}x_{5}$ | $33.\ x_{2}x_{3}^{2}x_{4}x_{5}$ | $34.\ x_{1}x_{3}x_{4}x_{5}^{2}$ | $35.\ x_{1}x_{3}x_{4}^{2}x_{5}$
---|---|---|---|---
$36.\ x_{1}x_{3}^{2}x_{4}x_{5}$ | $37.\ x_{1}x_{2}x_{4}x_{5}^{2}$ | $38.\ x_{1}x_{2}x_{4}^{2}x_{5}$ | $39.\ x_{1}x_{2}x_{3}x_{5}^{2}$ | $40.\ x_{1}x_{2}x_{3}x_{4}^{2}$
$41.\ x_{1}x_{2}x_{3}^{2}x_{5}$ | $42.\ x_{1}x_{2}x_{3}^{2}x_{4}$ | $43.\ x_{1}x_{2}^{2}x_{4}x_{5}$ | $44.\ x_{1}x_{2}^{2}x_{3}x_{5}$ |  $45.\ x_{1}x_{2}^{2}x_{3}x_{4}$.

For $t\geqslant 2,$

$31.\ x_{1}x_{2}^{2^{t}-2}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-1}$ | $32.\ x_{1}x_{2}^{2^{t}-2}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-1}$ | $33.\ x_{1}x_{2}^{2^{t}-1}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-1}$
---|---|---
$34.\ x_{1}x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-2}$ | $35.\ x_{1}x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-2}x_{4}^{2^{t}-1}$ | $36.\ x_{1}x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-1}x_{4}^{2^{t}-2}$
$37.\ x_{1}^{2^{t}-1}x_{2}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-1}$ | $38.\ x_{1}^{2^{t}-1}x_{2}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-2}$ | $39.\ x_{1}^{2^{t}-1}x_{2}^{2^{t+1}-1}x_{3}x_{4}^{2^{t}-2}$
$40.\ x_{1}^{2^{t+1}-1}x_{2}x_{3}^{2^{t}-2}x_{4}^{2^{t}-1}$ | $41.\ x_{1}^{2^{t+1}-1}x_{2}x_{3}^{2^{t}-1}x_{4}^{2^{t}-2}$ | $42.\ x_{1}^{2^{t+1}-1}x_{2}^{2^{t}-1}x_{3}x_{4}^{2^{t}-2}$
$43.\ x_{2}^{2^{t}-2}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-1}$ | $44.\ x_{2}^{2^{t}-2}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-1}$ | $45.\ x_{2}^{2^{t}-1}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-1}$
$46.\ x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-2}$ | $47.\ x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-2}x_{4}^{2^{t}-1}$ | $48.\ x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-1}x_{4}^{2^{t}-2}$
$49.\ x_{1}^{2^{t}-2}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-1}$ | $50.\ x_{1}^{2^{t}-2}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-1}$ | $51.\ x_{1}^{2^{t}-2}x_{2}^{2^{t}-1}x_{4}^{2^{t+1}-1}$
$52.\ x_{1}^{2^{t}-2}x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-1}$ | $53.\ x_{1}^{2^{t}-2}x_{2}^{2^{t+1}-1}x_{4}^{2^{t}-1}$ | $54.\ x_{1}^{2^{t}-2}x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-1}$
$55.\ x_{1}^{2^{t}-1}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-1}$ | $56.\ x_{1}^{2^{t}-1}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-2}$ | $57.\ x_{1}^{2^{t}-1}x_{2}^{2^{t}-2}x_{4}^{2^{t+1}-1}$
$58.\ x_{1}^{2^{t}-1}x_{2}^{2^{t}-2}x_{3}^{2^{t+1}-1}$ | $59.\ x_{1}^{2^{t}-1}x_{2}^{2^{t+1}-1}x_{4}^{2^{t}-2}$ | $60.\ x_{1}^{2^{t}-1}x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-2}$
$61.\ x_{1}^{2^{t+1}-1}x_{3}^{2^{t}-2}x_{4}^{2^{t}-1}$ | $62.\ x_{1}^{2^{t+1}-1}x_{3}^{2^{t}-1}x_{4}^{2^{t}-2}$ | $63.\ x_{1}^{2^{t+1}-1}x_{2}^{2^{t}-2}x_{4}^{2^{t}-1}$
$64.\ x_{1}^{2^{t+1}-1}x_{2}^{2^{t}-2}x_{3}^{2^{t}-1}$ | $65.\ x_{1}^{2^{t+1}-1}x_{2}^{2^{t}-1}x_{4}^{2^{t}-2}$ | $66.\ x_{1}^{2^{t+1}-1}x_{2}^{2^{t}-1}x_{3}^{2^{t}-2}$
$67.\ x_{2}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-1}$ | $68.\ x_{2}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-2}$ | $69.\ x_{2}^{2^{t+1}-1}x_{3}x_{4}^{2^{t}-2}$
$70.\ x_{1}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-1}$ | $71.\ x_{1}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-2}$ | $72.\ x_{1}x_{2}^{2^{t}-2}x_{4}^{2^{t+1}-1}$
$73.\ x_{1}x_{2}^{2^{t}-2}x_{3}^{2^{t+1}-1}$ | $74.\ x_{1}x_{2}^{2^{t+1}-1}x_{4}^{2^{t}-2}$ | $75.\ x_{1}x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-2}$
$76.\ x_{1}^{2^{t+1}-1}x_{3}x_{4}^{2^{t}-2}$ | $77.\ x_{1}^{2^{t+1}-1}x_{2}x_{4}^{2^{t}-2}$ | $78.\ x_{1}^{2^{t+1}-1}x_{2}x_{3}^{2^{t}-2}$
$79.\ x_{2}x_{3}^{2^{t}-2}x_{4}^{2^{t}-1}$ | $80.\ x_{2}x_{3}^{2^{t}-1}x_{4}^{2^{t}-2}$ | $81.\ x_{2}^{2^{t}-1}x_{3}x_{4}^{2^{t}-2}$
$82.\ x_{1}x_{3}^{2^{t}-2}x_{4}^{2^{t}-1}$ | $83.\ x_{1}x_{3}^{2^{t}-1}x_{4}^{2^{t}-2}$ | $84.\ x_{1}x_{2}^{2^{t}-2}x_{4}^{2^{t}-1}$
$85.\ x_{1}x_{2}^{2^{t}-2}x_{3}^{2^{t}-1}$ | $86.\ x_{1}x_{2}^{2^{t}-1}x_{4}^{2^{t}-2}$ | $87.\ x_{1}x_{2}^{2^{t}-1}x_{3}^{2^{t}-2}$
$88.\ x_{1}^{2^{t}-1}x_{3}x_{4}^{2^{t}-2}$ | $89.\ x_{1}^{2^{t}-1}x_{2}x_{4}^{2^{t}-2}$ |  $90.\ x_{1}^{2^{t}-1}x_{2}x_{3}^{2^{t}-2}$.

For $t=2,$

$91.\ x_{2}x_{3}^{3}x_{4}^{3}x_{5}^{6}$ | $92.\ x_{2}x_{3}^{3}x_{4}^{6}x_{5}^{3}$ | $93.\ x_{2}x_{3}^{6}x_{4}^{3}x_{5}^{3}$ | $94.\ x_{2}^{3}x_{3}x_{4}^{3}x_{5}^{6}$ | $95.\ x_{2}^{3}x_{3}x_{4}^{6}x_{5}^{3}$
---|---|---|---|---
$96.\ x_{2}^{3}x_{3}^{3}x_{4}x_{5}^{6}$ | $97.\ x_{1}x_{3}^{3}x_{4}^{3}x_{5}^{6}$ | $98.\ x_{1}x_{3}^{3}x_{4}^{6}x_{5}^{3}$ | $99.\ x_{1}x_{3}^{6}x_{4}^{3}x_{5}^{3}$ | $100.\ x_{1}x_{2}^{3}x_{4}^{3}x_{5}^{6}$
$101.\ x_{1}x_{2}^{3}x_{4}^{6}x_{5}^{3}$ | $102.\ x_{1}x_{2}^{3}x_{3}^{3}x_{5}^{6}$ | $103.\ x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{6}$ | $104.\ x_{1}x_{2}^{3}x_{3}^{6}x_{5}^{3}$ | $105.\ x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{3}$
$106.\ x_{1}x_{2}^{6}x_{4}^{3}x_{5}^{3}$ | $107.\ x_{1}x_{2}^{6}x_{3}^{3}x_{5}^{3}$ | $108.\ x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{3}$ | $109.\ x_{1}^{3}x_{3}x_{4}^{3}x_{5}^{6}$ | $110.\ x_{1}^{3}x_{3}x_{4}^{6}x_{5}^{3}$
$111.\ x_{1}^{3}x_{3}^{3}x_{4}x_{5}^{6}$ | $112.\ x_{1}^{3}x_{2}x_{4}^{3}x_{5}^{6}$ | $113.\ x_{1}^{3}x_{2}x_{4}^{6}x_{5}^{3}$ | $114.\ x_{1}^{3}x_{2}x_{3}^{3}x_{5}^{6}$ | $115.\ x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{6}$
$116.\ x_{1}^{3}x_{2}x_{3}^{6}x_{5}^{3}$ | $117.\ x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{3}$ | $118.\ x_{1}^{3}x_{2}^{3}x_{4}x_{5}^{6}$ | $119.\ x_{1}^{3}x_{2}^{3}x_{3}x_{5}^{6}$ | $120.\ x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{6}$
$121.\ x_{2}^{3}x_{3}^{3}x_{4}^{5}x_{5}^{2}$ | $122.\ x_{2}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{3}$ | $123.\ x_{2}^{3}x_{3}^{5}x_{4}^{3}x_{5}^{2}$ | $124.\ x_{1}^{3}x_{3}^{3}x_{4}^{5}x_{5}^{2}$ | $125.\ x_{1}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{3}$
$126.\ x_{1}^{3}x_{3}^{5}x_{4}^{3}x_{5}^{2}$ | $127.\ x_{1}^{3}x_{2}^{3}x_{4}^{5}x_{5}^{2}$ | $128.\ x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{5}^{2}$ | $129.\ x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{2}$ | $130.\ x_{1}^{3}x_{2}^{5}x_{4}^{2}x_{5}^{3}$
$131.\ x_{1}^{3}x_{2}^{5}x_{4}^{3}x_{5}^{2}$ | $132.\ x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{5}^{3}$ | $133.\ x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{3}$ | $134.\ x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{5}^{2}$ | $135.\ x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{2}$
$136.\ x_{2}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{4}$ | $137.\ x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{3}$ | $138.\ x_{1}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{4}$ | $139.\ x_{1}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{3}$ | $140.\ x_{1}^{3}x_{2}^{3}x_{4}^{3}x_{5}^{4}$
$141.\ x_{1}^{3}x_{2}^{3}x_{4}^{4}x_{5}^{3}$ | $142.\ x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{5}^{4}$ | $143.\ x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{4}$ | $144.\ x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{5}^{3}$ |  $145.\ x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{3}$.

For $t\geqslant 3,$

$91.\ x_{2}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t}-2}x_{5}^{2^{t+1}-1}$ | $92.\ x_{2}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t+1}-1}x_{5}^{2^{t}-2}$ | $93.\ x_{2}^{3}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-3}x_{5}^{2^{t}-2}$
---|---|---
$94.\ x_{2}^{2^{t+1}-1}x_{3}^{3}x_{4}^{2^{t}-3}x_{5}^{2^{t}-2}$ | $95.\ x_{1}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t}-2}x_{5}^{2^{t+1}-1}$ | $96.\ x_{1}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t+1}-1}x_{5}^{2^{t}-2}$
$97.\ x_{1}^{3}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-3}x_{5}^{2^{t}-2}$ | $98.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{4}^{2^{t}-2}x_{5}^{2^{t+1}-1}$ | $99.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{4}^{2^{t+1}-1}x_{5}^{2^{t}-2}$
$100.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t}-2}x_{5}^{2^{t+1}-1}$ | $101.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t}-2}x_{4}^{2^{t+1}-1}$ | $102.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t+1}-1}x_{5}^{2^{t}-2}$
$103.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t+1}-1}x_{4}^{2^{t}-2}$ | $104.\ x_{1}^{3}x_{2}^{2^{t+1}-1}x_{4}^{2^{t}-3}x_{5}^{2^{t}-2}$ | $105.\ x_{1}^{3}x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-3}x_{5}^{2^{t}-2}$
$106.\ x_{1}^{3}x_{2}^{2^{t+1}-1}x_{3}^{2^{t}-3}x_{4}^{2^{t}-2}$ | $107.\ x_{1}^{2^{t+1}-1}x_{3}^{3}x_{4}^{2^{t}-3}x_{5}^{2^{t}-2}$ | $108.\ x_{1}^{2^{t+1}-1}x_{2}^{3}x_{4}^{2^{t}-3}x_{5}^{2^{t}-2}$
$109.\ x_{1}^{2^{t+1}-1}x_{2}^{3}x_{3}^{2^{t}-3}x_{5}^{2^{t}-2}$ |  $110.\ x_{1}^{2^{t+1}-1}x_{2}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t}-2}$. |

For $t=3,$

$111.\ x_{2}x_{3}^{7}x_{4}^{7}x_{5}^{14}$ | $112.\ x_{2}^{7}x_{3}x_{4}^{7}x_{5}^{14}$ | $113.\ x_{1}x_{3}^{7}x_{4}^{7}x_{5}^{14}$ | $114.\ x_{1}x_{2}^{7}x_{4}^{7}x_{5}^{14}$ | $115.\ x_{1}x_{2}^{7}x_{3}^{7}x_{5}^{14}$
---|---|---|---|---
$116.\ x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{14}$ | $117.\ x_{1}^{7}x_{3}x_{4}^{7}x_{5}^{14}$ | $118.\ x_{1}^{7}x_{2}x_{4}^{7}x_{5}^{14}$ | $119.\ x_{1}^{7}x_{2}x_{3}^{7}x_{5}^{14}$ | $120.\ x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{14}$
$121.\ x_{2}^{3}x_{3}^{5}x_{4}^{14}x_{5}^{7}$ | $122.\ x_{2}^{3}x_{3}^{7}x_{4}^{5}x_{5}^{14}$ | $123.\ x_{1}^{3}x_{3}^{5}x_{4}^{14}x_{5}^{7}$ | $124.\ x_{1}^{3}x_{3}^{7}x_{4}^{5}x_{5}^{14}$ | $125.\ x_{1}^{3}x_{2}^{5}x_{4}^{14}x_{5}^{7}$
$126.\ x_{1}^{3}x_{2}^{5}x_{3}^{14}x_{5}^{7}$ | $127.\ x_{1}^{3}x_{2}^{5}x_{3}^{14}x_{4}^{7}$ | $128.\ x_{1}^{3}x_{2}^{7}x_{4}^{5}x_{5}^{14}$ | $129.\ x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{5}^{14}$ | $130.\ x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{14}$
$131.\ x_{2}^{3}x_{3}^{7}x_{4}^{13}x_{5}^{6}$ | $132.\ x_{1}^{3}x_{3}^{7}x_{4}^{13}x_{5}^{6}$ | $133.\ x_{1}^{3}x_{2}^{7}x_{4}^{13}x_{5}^{6}$ | $134.\ x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{5}^{6}$ | $135.\ x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{4}^{6}$
$136.\ x_{2}x_{3}^{7}x_{4}^{14}x_{5}^{7}$ | $137.\ x_{2}x_{3}^{14}x_{4}^{7}x_{5}^{7}$ | $138.\ x_{2}^{7}x_{3}x_{4}^{14}x_{5}^{7}$ | $139.\ x_{2}^{7}x_{3}^{7}x_{4}x_{5}^{14}$ | $140.\ x_{1}x_{3}^{7}x_{4}^{14}x_{5}^{7}$
$141.\ x_{1}x_{3}^{14}x_{4}^{7}x_{5}^{7}$ | $142.\ x_{1}x_{2}^{7}x_{4}^{14}x_{5}^{7}$ | $143.\ x_{1}x_{2}^{7}x_{3}^{14}x_{5}^{7}$ | $144.\ x_{1}x_{2}^{7}x_{3}^{14}x_{4}^{7}$ | $145.\ x_{1}x_{2}^{14}x_{4}^{7}x_{5}^{7}$
$146.\ x_{1}x_{2}^{14}x_{3}^{7}x_{5}^{7}$ | $147.\ x_{1}x_{2}^{14}x_{3}^{7}x_{4}^{7}$ | $148.\ x_{1}^{7}x_{3}x_{4}^{14}x_{5}^{7}$ | $149.\ x_{1}^{7}x_{3}^{7}x_{4}x_{5}^{14}$ | $150.\ x_{1}^{7}x_{2}x_{4}^{14}x_{5}^{7}$
$151.\ x_{1}^{7}x_{2}x_{3}^{14}x_{5}^{7}$ | $152.\ x_{1}^{7}x_{2}x_{3}^{14}x_{4}^{7}$ | $153.\ x_{1}^{7}x_{2}^{7}x_{4}x_{5}^{14}$ | $154.\ x_{1}^{7}x_{2}^{7}x_{3}x_{5}^{14}$ | $155.\ x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{14}$
$156.\ x_{2}^{3}x_{3}^{5}x_{4}^{7}x_{5}^{14}$ | $157.\ x_{2}^{7}x_{3}^{3}x_{4}^{5}x_{5}^{14}$ | $158.\ x_{1}^{3}x_{3}^{5}x_{4}^{7}x_{5}^{14}$ | $159.\ x_{1}^{3}x_{2}^{5}x_{4}^{7}x_{5}^{14}$ | $160.\ x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{5}^{14}$
$161.\ x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{14}$ | $162.\ x_{1}^{7}x_{3}^{3}x_{4}^{5}x_{5}^{14}$ | $163.\ x_{1}^{7}x_{2}^{3}x_{4}^{5}x_{5}^{14}$ | $164.\ x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{5}^{14}$ | $165.\ x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{14}$
$166.\ x_{2}^{3}x_{3}^{13}x_{4}^{6}x_{5}^{7}$ | $167.\ x_{2}^{3}x_{3}^{13}x_{4}^{7}x_{5}^{6}$ | $168.\ x_{2}^{7}x_{3}^{3}x_{4}^{13}x_{5}^{6}$ | $169.\ x_{1}^{3}x_{3}^{13}x_{4}^{6}x_{5}^{7}$ | $170.\ x_{1}^{3}x_{3}^{13}x_{4}^{7}x_{5}^{6}$
$171.\ x_{1}^{3}x_{2}^{13}x_{4}^{6}x_{5}^{7}$ | $172.\ x_{1}^{3}x_{2}^{13}x_{4}^{7}x_{5}^{6}$ | $173.\ x_{1}^{3}x_{2}^{13}x_{3}^{6}x_{5}^{7}$ | $174.\ x_{1}^{3}x_{2}^{13}x_{3}^{6}x_{4}^{7}$ | $175.\ x_{1}^{3}x_{2}^{13}x_{3}^{7}x_{5}^{6}$
$176.\ x_{1}^{3}x_{2}^{13}x_{3}^{7}x_{4}^{6}$ | $177.\ x_{1}^{7}x_{3}^{3}x_{4}^{13}x_{5}^{6}$ | $178.\ x_{1}^{7}x_{2}^{3}x_{4}^{13}x_{5}^{6}$ | $179.\ x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{5}^{6}$ | $180.\ x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{4}^{6}$
$181.\ x_{2}^{7}x_{3}^{11}x_{4}^{5}x_{5}^{6}$ | $182.\ x_{1}^{7}x_{3}^{11}x_{4}^{5}x_{5}^{6}$ | $183.\ x_{1}^{7}x_{2}^{11}x_{4}^{5}x_{5}^{6}$ | $184.\ x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{5}^{6}$ | $185.\ x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{4}^{6}$
$186.\ x_{2}^{7}x_{3}^{7}x_{4}^{9}x_{5}^{6}$ | $187.\ x_{1}^{7}x_{3}^{7}x_{4}^{9}x_{5}^{6}$ | $188.\ x_{1}^{7}x_{2}^{7}x_{4}^{9}x_{5}^{6}$ | $189.\ x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{5}^{6}$ | $190.\ x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{6}$
$191.\ x_{2}^{7}x_{3}^{7}x_{4}^{7}x_{5}^{8}$ | $192.\ x_{1}^{7}x_{3}^{7}x_{4}^{7}x_{5}^{8}$ | $193.\ x_{1}^{7}x_{2}^{7}x_{4}^{7}x_{5}^{8}$ | $194.\ x_{1}^{7}x_{2}^{7}x_{3}^{7}x_{5}^{8}$ |  $195.\ x_{1}^{7}x_{2}^{7}x_{3}^{7}x_{4}^{8}$.

For $t\geqslant 4,$

$111.\ x_{2}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $112.\ x_{2}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t+1}-2}x_{5}^{2^{t}-1}$ | $113.\ x_{2}^{3}x_{3}^{2^{t}-1}x_{4}^{2^{t}-3}x_{5}^{2^{t+1}-2}$
---|---|---
$114.\ x_{2}^{2^{t}-1}x_{3}^{3}x_{4}^{2^{t}-3}x_{5}^{2^{t+1}-2}$ | $115.\ x_{1}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $116.\ x_{1}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t+1}-2}x_{5}^{2^{t}-1}$
$117.\ x_{1}^{3}x_{3}^{2^{t}-1}x_{4}^{2^{t}-3}x_{5}^{2^{t+1}-2}$ | $118.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $119.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{4}^{2^{t+1}-2}x_{5}^{2^{t}-1}$
$120.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $121.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-2}$ | $122.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t+1}-2}x_{5}^{2^{t}-1}$
$123.\ x_{1}^{3}x_{2}^{2^{t}-3}x_{3}^{2^{t+1}-2}x_{4}^{2^{t}-1}$ | $124.\ x_{1}^{3}x_{2}^{2^{t}-1}x_{4}^{2^{t}-3}x_{5}^{2^{t+1}-2}$ | $125.\ x_{1}^{3}x_{2}^{2^{t}-1}x_{3}^{2^{t}-3}x_{5}^{2^{t+1}-2}$
$126.\ x_{1}^{3}x_{2}^{2^{t}-1}x_{3}^{2^{t}-3}x_{4}^{2^{t+1}-2}$ | $127.\ x_{1}^{2^{t}-1}x_{3}^{3}x_{4}^{2^{t}-3}x_{5}^{2^{t+1}-2}$ | $128.\ x_{1}^{2^{t}-1}x_{2}^{3}x_{4}^{2^{t}-3}x_{5}^{2^{t+1}-2}$
$129.\ x_{1}^{2^{t}-1}x_{2}^{3}x_{3}^{2^{t}-3}x_{5}^{2^{t+1}-2}$ | $130.\ x_{1}^{2^{t}-1}x_{2}^{3}x_{3}^{2^{t}-3}x_{4}^{2^{t+1}-2}$ | $131.\ x_{2}^{7}x_{3}^{2^{t}-5}x_{4}^{2^{t+1}-3}x_{5}^{2^{t}-2}$
$132.\ x_{1}^{7}x_{3}^{2^{t}-5}x_{4}^{2^{t+1}-3}x_{5}^{2^{t}-2}$ | $133.\ x_{1}^{7}x_{2}^{2^{t}-5}x_{4}^{2^{t+1}-3}x_{5}^{2^{t}-2}$ | $134.\ x_{1}^{7}x_{2}^{2^{t}-5}x_{3}^{2^{t+1}-3}x_{5}^{2^{t}-2}$
$135.\ x_{1}^{7}x_{2}^{2^{t}-5}x_{3}^{2^{t+1}-3}x_{4}^{2^{t}-2}$ | $136.\ x_{2}x_{3}^{2^{t}-1}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $137.\ x_{2}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-2}x_{5}^{2^{t}-1}$
$138.\ x_{2}x_{3}^{2^{t+1}-2}x_{4}^{2^{t}-1}x_{5}^{2^{t}-1}$ | $139.\ x_{2}^{2^{t}-1}x_{3}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $140.\ x_{2}^{2^{t}-1}x_{3}x_{4}^{2^{t+1}-2}x_{5}^{2^{t}-1}$
$141.\ x_{2}^{2^{t}-1}x_{3}^{2^{t}-1}x_{4}x_{5}^{2^{t+1}-2}$ | $142.\ x_{1}x_{3}^{2^{t}-1}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $143.\ x_{1}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-2}x_{5}^{2^{t}-1}$
$144.\ x_{1}x_{3}^{2^{t+1}-2}x_{4}^{2^{t}-1}x_{5}^{2^{t}-1}$ | $145.\ x_{1}x_{2}^{2^{t}-1}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $146.\ x_{1}x_{2}^{2^{t}-1}x_{4}^{2^{t+1}-2}x_{5}^{2^{t}-1}$
$147.\ x_{1}x_{2}^{2^{t}-1}x_{3}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $148.\ x_{1}x_{2}^{2^{t}-1}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-2}$ | $149.\ x_{1}x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-2}x_{5}^{2^{t}-1}$
$150.\ x_{1}x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-2}x_{4}^{2^{t}-1}$ | $151.\ x_{1}x_{2}^{2^{t+1}-2}x_{4}^{2^{t}-1}x_{5}^{2^{t}-1}$ | $152.\ x_{1}x_{2}^{2^{t+1}-2}x_{3}^{2^{t}-1}x_{5}^{2^{t}-1}$
$153.\ x_{1}x_{2}^{2^{t+1}-2}x_{3}^{2^{t}-1}x_{4}^{2^{t}-1}$ | $154.\ x_{1}^{2^{t}-1}x_{3}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $155.\ x_{1}^{2^{t}-1}x_{3}x_{4}^{2^{t+1}-2}x_{5}^{2^{t}-1}$
$156.\ x_{1}^{2^{t}-1}x_{3}^{2^{t}-1}x_{4}x_{5}^{2^{t+1}-2}$ | $157.\ x_{1}^{2^{t}-1}x_{2}x_{4}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $158.\ x_{1}^{2^{t}-1}x_{2}x_{4}^{2^{t+1}-2}x_{5}^{2^{t}-1}$
$159.\ x_{1}^{2^{t}-1}x_{2}x_{3}^{2^{t}-1}x_{5}^{2^{t+1}-2}$ | $160.\ x_{1}^{2^{t}-1}x_{2}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-2}$ | $161.\ x_{1}^{2^{t}-1}x_{2}x_{3}^{2^{t+1}-2}x_{5}^{2^{t}-1}$
$162.\ x_{1}^{2^{t}-1}x_{2}x_{3}^{2^{t+1}-2}x_{4}^{2^{t}-1}$ | $163.\ x_{1}^{2^{t}-1}x_{2}^{2^{t}-1}x_{4}x_{5}^{2^{t+1}-2}$ | $164.\ x_{1}^{2^{t}-1}x_{2}^{2^{t}-1}x_{3}x_{5}^{2^{t+1}-2}$
$165.\ x_{1}^{2^{t}-1}x_{2}^{2^{t}-1}x_{3}x_{4}^{2^{t+1}-2}$ | $166.\ x_{2}^{3}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-3}x_{5}^{2^{t}-2}$ | $167.\ x_{2}^{3}x_{3}^{2^{t+1}-3}x_{4}^{2^{t}-2}x_{5}^{2^{t}-1}$
$168.\ x_{2}^{3}x_{3}^{2^{t+1}-3}x_{4}^{2^{t}-1}x_{5}^{2^{t}-2}$ | $169.\ x_{2}^{2^{t}-1}x_{3}^{3}x_{4}^{2^{t+1}-3}x_{5}^{2^{t}-2}$ | $170.\ x_{1}^{3}x_{3}^{2^{t}-1}x_{4}^{2^{t+1}-3}x_{5}^{2^{t}-2}$
$171.\ x_{1}^{3}x_{3}^{2^{t+1}-3}x_{4}^{2^{t}-2}x_{5}^{2^{t}-1}$ | $172.\ x_{1}^{3}x_{3}^{2^{t+1}-3}x_{4}^{2^{t}-1}x_{5}^{2^{t}-2}$ | $173.\ x_{1}^{3}x_{2}^{2^{t}-1}x_{4}^{2^{t+1}-3}x_{5}^{2^{t}-2}$
$174.\ x_{1}^{3}x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-3}x_{5}^{2^{t}-2}$ | $175.\ x_{1}^{3}x_{2}^{2^{t}-1}x_{3}^{2^{t+1}-3}x_{4}^{2^{t}-2}$ | $176.\ x_{1}^{3}x_{2}^{2^{t+1}-3}x_{4}^{2^{t}-2}x_{5}^{2^{t}-1}$
$177.\ x_{1}^{3}x_{2}^{2^{t+1}-3}x_{4}^{2^{t}-1}x_{5}^{2^{t}-2}$ | $178.\ x_{1}^{3}x_{2}^{2^{t+1}-3}x_{3}^{2^{t}-2}x_{5}^{2^{t}-1}$ | $179.\ x_{1}^{3}x_{2}^{2^{t+1}-3}x_{3}^{2^{t}-2}x_{4}^{2^{t}-1}$
$180.\ x_{1}^{3}x_{2}^{2^{t+1}-3}x_{3}^{2^{t}-1}x_{5}^{2^{t}-2}$ | $181.\ x_{1}^{3}x_{2}^{2^{t+1}-3}x_{3}^{2^{t}-1}x_{4}^{2^{t}-2}$ | $182.\ x_{1}^{2^{t}-1}x_{3}^{3}x_{4}^{2^{t+1}-3}x_{5}^{2^{t}-2}$
$183.\ x_{1}^{2^{t}-1}x_{2}^{3}x_{4}^{2^{t+1}-3}x_{5}^{2^{t}-2}$ | $184.\ x_{1}^{2^{t}-1}x_{2}^{3}x_{3}^{2^{t+1}-3}x_{5}^{2^{t}-2}$ | $185.\ x_{1}^{2^{t}-1}x_{2}^{3}x_{3}^{2^{t+1}-3}x_{4}^{2^{t}-2}$
$186.\ x_{2}^{7}x_{3}^{2^{t}-5}x_{4}^{2^{t}-3}x_{5}^{2^{t+1}-2}$ | $187.\ x_{1}^{7}x_{3}^{2^{t}-5}x_{4}^{2^{t}-3}x_{5}^{2^{t+1}-2}$ | $188.\ x_{1}^{7}x_{2}^{2^{t}-5}x_{4}^{2^{t}-3}x_{5}^{2^{t+1}-2}$
$189.\ x_{1}^{7}x_{2}^{2^{t}-5}x_{3}^{2^{t}-3}x_{5}^{2^{t+1}-2}$ | $190.\ x_{1}^{7}x_{2}^{2^{t}-5}x_{3}^{2^{t}-3}x_{4}^{2^{t+1}-2}$ | $191.\ x_{2}^{7}x_{3}^{2^{t+1}-5}x_{4}^{2^{t}-3}x_{5}^{2^{t}-2}$
$192.\ x_{1}^{7}x_{3}^{2^{t+1}-5}x_{4}^{2^{t}-3}x_{5}^{2^{t}-2}$ | $193.\ x_{1}^{7}x_{2}^{2^{t+1}-5}x_{4}^{2^{t}-3}x_{5}^{2^{t}-2}$ | $194.\ x_{1}^{7}x_{2}^{2^{t+1}-5}x_{3}^{2^{t}-3}x_{5}^{2^{t}-2}$
$195.\ x_{1}^{7}x_{2}^{2^{t+1}-5}x_{3}^{2^{t}-3}x_{4}^{2^{t}-2}$. |  |

###  5.2 The admissible monomials of degree $3(2^{t}-1)+2^{t}$ in $P_{5}^{+}$

As it is known,

$\mathscr{B}_{5}(3(2^{t}-1)+2^{t})=\mathscr{B}_{5}^{0}(3(2^{t}-1)+2^{t})\bigcup\varphi(\mathscr{B}_{5}(2^{t+1}-4))\bigcup
B,$

where $B=(\mathscr{B}_{5}^{+}(3(2^{t}-1)+2^{t})\bigcap{\rm
Ker}((\widetilde{Sq}_{*}^{0})_{(5,3(2^{t}-1)+2^{t})}))=\mathscr{B}^{+}_{5}(\omega_{(5,t)}),$
and
$\varphi:P_{5}\to P_{5},\ \varphi(y)=x_{1}x_{2}x_{3}x_{4}x_{5}y^{2},\ \forall
y\in P_{5}.$ We have
$|\mathscr{B}_{5}^{0}(3(2^{t}-1)+2^{t})|=\left\\{\begin{array}[]{ll}45&\text{if}\
t=1,\\\ 145&\text{if}\ t=2,\\\ 195&\text{if}\ t\geqslant 3,\end{array}\right.$
and $|\mathscr{B}_{5}(2^{t+1}-4)|=\left\\{\begin{array}[]{ll}1&\text{if}\
t=1,\\\ 45&\text{if}\ t=2,\\\ 190&\text{if}\ t=3,\\\ 480&\text{if}\ t=4,\\\
650&\text{if}\ t=5,\\\ 651&\text{if}\ t\geqslant 6.\end{array}\right.$

Set $v_{t}=|\mathscr{B}^{+}_{5}(\omega_{(5,t)})|$ and
$\mathscr{B}^{+}_{5}(\omega_{(5,t)})=\\{b_{3(2^{t}-1)+2^{t},k}:\ 1\leqslant
k\leqslant v_{t}\\}.$ Then, $v_{1}=1,\ v_{2}=60,\ v_{3}=260,\ v_{t}=270$ for
$t\geqslant 4.$

For $t=1,$ according to Sum [31, 35],
$\mathscr{B}^{+}_{5}(\omega_{(5,1)})=\big{\\{}b_{5,1}=x_{1}x_{2}x_{3}x_{4}x_{5}\big{\\}}.$

For $t\geqslant 2,$ the admissible monomials $b_{3(2^{t}-1)+2^{t},k},\
1\leqslant k\leqslant v_{t},$ are determined as follows:

For $t=2,$

$1.\ x_{1}x_{2}x_{3}^{2}x_{4}^{2}x_{5}^{7}$ | $2.\ x_{1}x_{2}x_{3}^{2}x_{4}^{7}x_{5}^{2}$ | $3.\ x_{1}x_{2}x_{3}^{7}x_{4}^{2}x_{5}^{2}$ | $4.\ x_{1}x_{2}^{2}x_{3}x_{4}^{2}x_{5}^{7}$
---|---|---|---
$5.\ x_{1}x_{2}^{2}x_{3}x_{4}^{7}x_{5}^{2}$ | $6.\ x_{1}x_{2}^{2}x_{3}^{7}x_{4}x_{5}^{2}$ | $7.\ x_{1}x_{2}^{7}x_{3}x_{4}^{2}x_{5}^{2}$ | $8.\ x_{1}x_{2}^{7}x_{3}^{2}x_{4}x_{5}^{2}$
$9.\ x_{1}^{7}x_{2}x_{3}x_{4}^{2}x_{5}^{2}$ | $10.\ x_{1}^{7}x_{2}x_{3}^{2}x_{4}x_{5}^{2}$ | $11.\ x_{1}x_{2}x_{3}^{2}x_{4}^{3}x_{5}^{6}$ | $12.\ x_{1}x_{2}x_{3}^{2}x_{4}^{6}x_{5}^{3}$
$13.\ x_{1}x_{2}x_{3}^{3}x_{4}^{2}x_{5}^{6}$ | $14.\ x_{1}x_{2}x_{3}^{3}x_{4}^{6}x_{5}^{2}$ | $15.\ x_{1}x_{2}x_{3}^{6}x_{4}^{2}x_{5}^{3}$ | $16.\ x_{1}x_{2}x_{3}^{6}x_{4}^{3}x_{5}^{2}$
$17.\ x_{1}x_{2}^{2}x_{3}x_{4}^{3}x_{5}^{6}$ | $18.\ x_{1}x_{2}^{2}x_{3}x_{4}^{6}x_{5}^{3}$ | $19.\ x_{1}x_{2}^{2}x_{3}^{3}x_{4}x_{5}^{6}$ | $20.\ x_{1}x_{2}^{3}x_{3}x_{4}^{2}x_{5}^{6}$
$21.\ x_{1}x_{2}^{3}x_{3}x_{4}^{6}x_{5}^{2}$ | $22.\ x_{1}x_{2}^{3}x_{3}^{2}x_{4}x_{5}^{6}$ | $23.\ x_{1}x_{2}^{3}x_{3}^{6}x_{4}x_{5}^{2}$ | $24.\ x_{1}x_{2}^{6}x_{3}x_{4}^{2}x_{5}^{3}$
$25.\ x_{1}x_{2}^{6}x_{3}x_{4}^{3}x_{5}^{2}$ | $26.\ x_{1}x_{2}^{6}x_{3}^{3}x_{4}x_{5}^{2}$ | $27.\ x_{1}^{3}x_{2}x_{3}x_{4}^{2}x_{5}^{6}$ | $28.\ x_{1}^{3}x_{2}x_{3}x_{4}^{6}x_{5}^{2}$
$29.\ x_{1}^{3}x_{2}x_{3}^{2}x_{4}x_{5}^{6}$ | $30.\ x_{1}^{3}x_{2}x_{3}^{6}x_{4}x_{5}^{2}$ | $31.\ x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{5}x_{5}^{2}$ | $32.\ x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{2}x_{5}^{3}$
$33.\ x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{3}x_{5}^{2}$ | $34.\ x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{5}x_{5}^{2}$ | $35.\ x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{2}$ | $36.\ x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{5}x_{5}^{2}$
$37.\ x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{2}x_{5}^{2}$ | $38.\ x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{2}x_{5}^{2}$ | $39.\ x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}x_{5}^{2}$ | $40.\ x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{3}x_{5}^{4}$
$41.\ x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{4}x_{5}^{3}$ | $42.\ x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{3}x_{5}^{3}$ | $43.\ x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{3}x_{5}^{4}$ | $44.\ x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{4}x_{5}^{3}$
$45.\ x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{2}x_{5}^{4}$ | $46.\ x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{2}$ | $47.\ x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{2}x_{5}^{3}$ | $48.\ x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{3}x_{5}^{2}$
$49.\ x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{3}x_{5}^{4}$ | $50.\ x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{4}x_{5}^{3}$ | $51.\ x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{2}x_{5}^{4}$ | $52.\ x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{4}x_{5}^{2}$
$53.\ x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{2}x_{5}^{3}$ | $54.\ x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{3}x_{5}^{2}$ | $55.\ x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{2}x_{5}^{4}$ | $56.\ x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{4}x_{5}^{2}$
$57.\ x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}x_{5}^{2}$ | $58.\ x_{1}^{3}x_{2}^{4}x_{3}x_{4}^{2}x_{5}^{3}$ | $59.\ x_{1}^{3}x_{2}^{4}x_{3}x_{4}^{3}x_{5}^{2}$ | $60.\ x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}x_{5}^{2}.$

For $t=3,$

$1.\ x_{1}x_{2}x_{3}^{6}x_{4}^{6}x_{5}^{15}$ | $2.\ x_{1}x_{2}x_{3}^{6}x_{4}^{15}x_{5}^{6}$ | $3.\ x_{1}x_{2}x_{3}^{15}x_{4}^{6}x_{5}^{6}$ | $4.\ x_{1}x_{2}^{15}x_{3}x_{4}^{6}x_{5}^{6}$ |
---|---|---|---|---
$5.\ x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{4}x_{5}^{15}$ | $6.\ x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{15}x_{5}^{4}$ | $7.\ x_{1}^{3}x_{2}^{3}x_{3}^{15}x_{4}^{4}x_{5}^{4}$ | $8.\ x_{1}^{3}x_{2}^{15}x_{3}^{3}x_{4}^{4}x_{5}^{4}$ |
$9.\ x_{1}^{15}x_{2}x_{3}x_{4}^{6}x_{5}^{6}$ | $10.\ x_{1}^{15}x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{4}$ | $11.\ x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{6}x_{5}^{15}$ | $12.\ x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{15}x_{5}^{6}$ |
$13.\ x_{1}x_{2}^{2}x_{3}^{15}x_{4}^{5}x_{5}^{6}$ | $14.\ x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{6}x_{5}^{15}$ | $15.\ x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{15}x_{5}^{6}$ | $16.\ x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{4}x_{5}^{15}$ |
$17.\ x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{15}x_{5}^{4}$ | $18.\ x_{1}x_{2}^{3}x_{3}^{15}x_{4}^{4}x_{5}^{6}$ | $19.\ x_{1}x_{2}^{3}x_{3}^{15}x_{4}^{6}x_{5}^{4}$ | $20.\ x_{1}x_{2}^{6}x_{3}x_{4}^{6}x_{5}^{15}$ |
$21.\ x_{1}x_{2}^{6}x_{3}x_{4}^{15}x_{5}^{6}$ | $22.\ x_{1}x_{2}^{6}x_{3}^{15}x_{4}x_{5}^{6}$ | $23.\ x_{1}x_{2}^{15}x_{3}^{2}x_{4}^{5}x_{5}^{6}$ | $24.\ x_{1}x_{2}^{15}x_{3}^{3}x_{4}^{4}x_{5}^{6}$ |
$25.\ x_{1}x_{2}^{15}x_{3}^{3}x_{4}^{6}x_{5}^{4}$ | $26.\ x_{1}x_{2}^{15}x_{3}^{6}x_{4}x_{5}^{6}$ | $27.\ x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{6}x_{5}^{15}$ | $28.\ x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{15}x_{5}^{6}$ |
$29.\ x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{4}x_{5}^{15}$ | $30.\ x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{15}x_{5}^{4}$ | $31.\ x_{1}^{3}x_{2}x_{3}^{15}x_{4}^{4}x_{5}^{6}$ | $32.\ x_{1}^{3}x_{2}x_{3}^{15}x_{4}^{6}x_{5}^{4}$ |
$33.\ x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{4}x_{5}^{15}$ | $34.\ x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{15}x_{5}^{4}$ | $35.\ x_{1}^{3}x_{2}^{5}x_{3}^{15}x_{4}^{2}x_{5}^{4}$ | $36.\ x_{1}^{3}x_{2}^{15}x_{3}x_{4}^{4}x_{5}^{6}$ |
$37.\ x_{1}^{3}x_{2}^{15}x_{3}x_{4}^{6}x_{5}^{4}$ | $38.\ x_{1}^{3}x_{2}^{15}x_{3}^{5}x_{4}^{2}x_{5}^{4}$ | $39.\ x_{1}^{15}x_{2}x_{3}^{2}x_{4}^{5}x_{5}^{6}$ | $40.\ x_{1}^{15}x_{2}x_{3}^{3}x_{4}^{4}x_{5}^{6}$ |
$41.\ x_{1}^{15}x_{2}x_{3}^{3}x_{4}^{6}x_{5}^{4}$ | $42.\ x_{1}^{15}x_{2}x_{3}^{6}x_{4}x_{5}^{6}$ | $43.\ x_{1}^{15}x_{2}^{3}x_{3}x_{4}^{4}x_{5}^{6}$ | $44.\ x_{1}^{15}x_{2}^{3}x_{3}x_{4}^{6}x_{5}^{4}$ |
$45.\ x_{1}^{15}x_{2}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{4}$ | $46.\ x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{7}x_{5}^{15}$ | $47.\ x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{15}x_{5}^{7}$ | $48.\ x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{4}x_{5}^{15}$ |
$49.\ x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{15}x_{5}^{4}$ | $50.\ x_{1}x_{2}^{2}x_{3}^{15}x_{4}^{4}x_{5}^{7}$ | $51.\ x_{1}x_{2}^{2}x_{3}^{15}x_{4}^{7}x_{5}^{4}$ | $52.\ x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{4}x_{5}^{15}$ |
$53.\ x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{15}x_{5}^{4}$ | $54.\ x_{1}x_{2}^{7}x_{3}^{15}x_{4}^{2}x_{5}^{4}$ | $55.\ x_{1}x_{2}^{15}x_{3}^{2}x_{4}^{4}x_{5}^{7}$ | $56.\ x_{1}x_{2}^{15}x_{3}^{2}x_{4}^{7}x_{5}^{4}$ |
$57.\ x_{1}x_{2}^{15}x_{3}^{7}x_{4}^{2}x_{5}^{4}$ | $58.\ x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{4}x_{5}^{15}$ | $59.\ x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{15}x_{5}^{4}$ | $60.\ x_{1}^{7}x_{2}x_{3}^{15}x_{4}^{2}x_{5}^{4}$ |
$61.\ x_{1}^{7}x_{2}^{15}x_{3}x_{4}^{2}x_{5}^{4}$ | $62.\ x_{1}^{15}x_{2}x_{3}^{2}x_{4}^{4}x_{5}^{7}$ | $63.\ x_{1}^{15}x_{2}x_{3}^{2}x_{4}^{7}x_{5}^{4}$ | $64.\ x_{1}^{15}x_{2}x_{3}^{7}x_{4}^{2}x_{5}^{4}$ |
$65.\ x_{1}^{15}x_{2}^{7}x_{3}x_{4}^{2}x_{5}^{4}$ | $66.\ x_{1}x_{2}x_{3}^{6}x_{4}^{7}x_{5}^{14}$ | $67.\ x_{1}x_{2}x_{3}^{6}x_{4}^{14}x_{5}^{7}$ | $68.\ x_{1}x_{2}x_{3}^{7}x_{4}^{6}x_{5}^{14}$ |
$69.\ x_{1}x_{2}x_{3}^{7}x_{4}^{14}x_{5}^{6}$ | $70.\ x_{1}x_{2}x_{3}^{14}x_{4}^{6}x_{5}^{7}$ | $71.\ x_{1}x_{2}x_{3}^{14}x_{4}^{7}x_{5}^{6}$ | $72.\ x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{7}x_{5}^{14}$ |
$73.\ x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{14}x_{5}^{7}$ | $74.\ x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{5}x_{5}^{14}$ | $75.\ x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{7}x_{5}^{12}$ | $76.\ x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{12}x_{5}^{7}$ |
$77.\ x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{13}x_{5}^{6}$ | $78.\ x_{1}x_{2}^{2}x_{3}^{12}x_{4}^{7}x_{5}^{7}$ | $79.\ x_{1}x_{2}^{2}x_{3}^{13}x_{4}^{6}x_{5}^{7}$ | $80.\ x_{1}x_{2}^{2}x_{3}^{13}x_{4}^{7}x_{5}^{6}$ |
$81.\ x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{7}x_{5}^{14}$ | $82.\ x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{14}x_{5}^{7}$ |  $83.\ x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{14}$, | $84.\ x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{14}x_{5}^{6}$ |
$85.\ x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{5}x_{5}^{14}$ | $86.\ x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{7}x_{5}^{12}$ | $87.\ x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{12}x_{5}^{7}$ | $88.\ x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{13}x_{5}^{6}$ |
$89.\ x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{4}x_{5}^{14}$ | $90.\ x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{6}x_{5}^{12}$ | $91.\ x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{12}x_{5}^{6}$ | $92.\ x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{14}x_{5}^{4}$ |
$93.\ x_{1}x_{2}^{3}x_{3}^{12}x_{4}^{6}x_{5}^{7}$ | $94.\ x_{1}x_{2}^{3}x_{3}^{12}x_{4}^{7}x_{5}^{6}$ | $95.\ x_{1}x_{2}^{3}x_{3}^{13}x_{4}^{6}x_{5}^{6}$ | $96.\ x_{1}x_{2}^{3}x_{3}^{14}x_{4}^{4}x_{5}^{7}$ |
$97.\ x_{1}x_{2}^{3}x_{3}^{14}x_{4}^{5}x_{5}^{6}$ | $98.\ x_{1}x_{2}^{3}x_{3}^{14}x_{4}^{7}x_{5}^{4}$ | $99.\ x_{1}x_{2}^{6}x_{3}x_{4}^{7}x_{5}^{14}$ | $100.\ x_{1}x_{2}^{6}x_{3}x_{4}^{14}x_{5}^{7}$ |
$101.\ x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{5}x_{5}^{14}$ | $102.\ x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{13}x_{5}^{6}$ | $103.\ x_{1}x_{2}^{6}x_{3}^{7}x_{4}x_{5}^{14}$ | $104.\ x_{1}x_{2}^{6}x_{3}^{7}x_{4}^{7}x_{5}^{8}$ |
$105.\ x_{1}x_{2}^{6}x_{3}^{7}x_{4}^{9}x_{5}^{6}$ | $106.\ x_{1}x_{2}^{6}x_{3}^{11}x_{4}^{5}x_{5}^{6}$ | $107.\ x_{1}x_{2}^{7}x_{3}x_{4}^{6}x_{5}^{14}$ | $108.\ x_{1}x_{2}^{7}x_{3}x_{4}^{14}x_{5}^{6}$ |
$109.\ x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{5}x_{5}^{14}$ | $110.\ x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{7}x_{5}^{12}$ | $111.\ x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{12}x_{5}^{7}$ | $112.\ x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{13}x_{5}^{6}$ |
$113.\ x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{4}x_{5}^{14}$ | $114.\ x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{6}x_{5}^{12}$ | $115.\ x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{12}x_{5}^{6}$ | $116.\ x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{14}x_{5}^{4}$ |

$117.\ x_{1}x_{2}^{7}x_{3}^{6}x_{4}x_{5}^{14}$ | $118.\ x_{1}x_{2}^{7}x_{3}^{6}x_{4}^{7}x_{5}^{8}$ | $119.\ x_{1}x_{2}^{7}x_{3}^{6}x_{4}^{9}x_{5}^{6}$ | $120.\ x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{2}x_{5}^{12}$ |
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$121.\ x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{6}x_{5}^{8}$ | $122.\ x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{8}x_{5}^{6}$ | $123.\ x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{10}x_{5}^{4}$ | $124.\ x_{1}x_{2}^{7}x_{3}^{10}x_{4}^{5}x_{5}^{6}$ |
$125.\ x_{1}x_{2}^{7}x_{3}^{11}x_{4}^{4}x_{5}^{6}$ | $126.\ x_{1}x_{2}^{7}x_{3}^{11}x_{4}^{6}x_{5}^{4}$ | $127.\ x_{1}x_{2}^{7}x_{3}^{14}x_{4}x_{5}^{6}$ | $128.\ x_{1}x_{2}^{14}x_{3}x_{4}^{6}x_{5}^{7}$ |
$129.\ x_{1}x_{2}^{14}x_{3}x_{4}^{7}x_{5}^{6}$ | $130.\ x_{1}x_{2}^{14}x_{3}^{3}x_{4}^{5}x_{5}^{6}$ | $131.\ x_{1}x_{2}^{14}x_{3}^{7}x_{4}x_{5}^{6}$ | $132.\ x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{7}x_{5}^{14}$ |
$133.\ x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{14}x_{5}^{7}$ | $134.\ x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{6}x_{5}^{14}$ | $135.\ x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{14}x_{5}^{6}$ | $136.\ x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{5}x_{5}^{14}$ |
$137.\ x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{7}x_{5}^{12}$ | $138.\ x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{12}x_{5}^{7}$ | $139.\ x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{13}x_{5}^{6}$ | $140.\ x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{4}x_{5}^{14}$ |
$141.\ x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{6}x_{5}^{12}$ | $142.\ x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{12}x_{5}^{6}$ | $143.\ x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{14}x_{5}^{4}$ | $144.\ x_{1}^{3}x_{2}x_{3}^{12}x_{4}^{6}x_{5}^{7}$ |