# On the Betti numbers and Rees algebras of ideals with linear powers

Lisa Nicklasson
Max-Planck Institute for Mathematics in the Sciences
<EMAIL_ADDRESS>
###### Abstract

An ideal $I\subset\mathbb{k}[x_{1},\ldots,x_{n}]$ is said to have linear
powers if $I^{k}$ has a linear minimal free resolution, for all integers
$k>0$. In this paper we study the Betti numbers of $I^{k}$, for ideals $I$
with linear powers. We provide linear relations on the Betti numbers, which
holds for all ideals with linear powers. This is especially useful for ideals
of low dimension. The Betti numbers are computed explicitly, as polynomials in
$k$, for the ideal generated by all square free monomials of degree $d$, for
$d=2,3$ or $n-1$, and the product of all ideals generated by $s$ variables,
for $s=n-1$ or $n-2$. We also study the generators of the Rees ideal, for
ideals with linear powers. Especially, we are interested in ideals for which
the Rees ideal is generated by quadratic elements. This problem is related to
a conjecture on matroids by White.

##  1 Introduction

Let $S=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a polynomial ring over a field
$\mathbb{k}$. For a homogeneous ideal $I\subset S$, recall that the numbers
$\beta_{1},\ldots,\beta_{n}$ in a minimal free resolution

$0\to S^{\beta_{n}}\to\dots\to S^{\beta_{1}}\to S\to S/I\to 0$

are called the _Betti numbers of $S/I$_. We will denote the $i$-th Betti
number of $S/I$ by $\beta_{i}(S/I)$. In [13] the following famous result is
proved.

######  Theorem 1.1.

Let $I\subset S$ be a homogeneous ideal. The Betti numbers
$\beta_{i}(S/I^{k})$, seen as functions of $k$, are polynomials in $k$, for
$k$ large enough.

Given some nice class of ideals, one may ask if it is possible to give
explicit expressions for these polynomials. In this paper we will attempt to
answer this question for ideals with _linear powers_.

######  Definition 1.2.

A homogeneous ideal $I\subseteq S$ is said to have _linear resolution_ if all
the maps $\partial_{i}$ with $i>1$ in a minimal free resolution

$0{\to}S^{\beta_{n}}\overset{\partial_{n}}{\to}S^{\beta_{n-1}}\to\dots\to
S^{\beta_{1}}\overset{\partial_{1}}{\to}S\to S/I\to 0$

of $S/I$ are represented by matrices whose non-zero entries have degree one.
Moreover, the ideal $I$ has _linear powers_ if $I^{k}$ has linear resolution,
for all $k$.

Two examples of ideals with linear powers, that will be of special interest in
this paper, are the following two classes.

A: _The ideals generated by all square-free monomials of degree $d$._

B: _The products of all ideals generated by $s$-sized subsets of the
variables. _

In Section 2 we will see how the Betti numbers of ideals with linear powers
can be computed using the Herzog-Kühl equations. Typically, this is a suitable
approach for ideals of low dimension. Application of this result, together
with a method involving Rees algebras, gives explicit expressions for
$\beta_{i}(S/I^{k})$, as polynomials in $k$, for the ideals in A with $d=2,3,$
and $n-1$, and the ideals in B with $s=n-1$ and $n-2$.

Both A and B are subclasses of a larger class of ideals with linear powers,
namely the _polymatroidal ideals_.

######  Definition 1.3.

Let $I$ be a monomial ideal, and let $\mathcal{G}(I)$ be the minimal
generating set of monomials. The ideal $I$ is called _polymatroidal_ if all
the monomials in $\mathcal{G}(I)$ have the same degree and the following
condition is satisfied. Let $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ and
$x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}$ be two monomials in $\mathcal{G}(I)$, such
that $a_{i}>b_{i}$ for some $i$. Then there is a $j$ such that $a_{j}<b_{j}$
and $(x_{j}/x_{i})x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\in\mathcal{G}(I)$.

It was proved in [12] that polymatroidal ideals have linear resolution.
Combined with the result from [4], that products of polymatroidal ideals are
polymatroidal, it follows that polymatroidal ideals have linear powers.
Another class of ideals with linear powers, which is also found in [4], is the
products of ideals generated by linear forms.

A third class of ideals with linear powers is a certain type of edge ideals of
graphs. We recall the definition of an edge ideal. Let $G$ be a graph with the
vertex set $V=\\{x_{1},\ldots,x_{n}\\}$. The _edge ideal_ of $G$, denoted
$I(G)$, is the monomial ideal generated by all $x_{i}x_{j}$ such that
$\\{x_{i},x_{j}\\}$ is an edge in $G$. The _complementary graph_ of $G$ is the
graph with the same vertex set, and whose edges are all $\\{x_{i},x_{j}\\}$
that are _not_ edges of $G$. A graph is called _chordal_ if every cycle of
length at least four has a chord. It was proved in [7] that an edge ideal
$I(G)$ has a linear resolution if and only if the complementary graph of $G$
is chordal. In fact, these ideals have linear powers. This follows from the
result in [10], which states that monomial ideals generated in degree two,
with linear resolution, have linear powers.

Recall that the _Rees algebra_ of an ideal $I\subset S$ is defined as

$R(I)=\bigoplus_{j\geq 0}I^{j}t^{j}\subset S[t].$

If $I=(f_{1},\ldots,f_{m})$ we have $R(I)=S[f_{1}t,\ldots,f_{m}t]$. Let
$T=S[y_{1},\ldots,y_{m}]$, with the bigrading $\deg x_{i}=(1,0)$ and $\deg
y_{i}=(0,1)$, and define a homomorphism $\phi:T\to S[t]$ by
$\phi(x_{i})=x_{i}$ and $\phi(y_{i})=f_{i}t$. Then the image of $\phi$ is
$R(I)$, and hence $R(I)\cong T/J$ where $J=\ker\phi$. The ideal $J$ is called
the _Rees ideal_ of $I$. If the $f_{i}$’s are all of the same degree, then $J$
is a homogeneous ideal. If the $f_{i}$’s are monomials, then $J$ is a binomial
ideal.

It was asked in [1] if the Rees ideal of a polymatroidal ideal is generated in
the degrees $(0,2)$ and $(1,1)$. This question is a generalization of White’s
conjecture, [15], on matroids. For further discussion on White’s conjecture,
and its connection to polymatroidal ideals, see [1] and [8]. Inspired by this
question, we devote Section 3 to the study of ideals with linear powers, not
necessarily polymatroidal, with the property that their Rees ideals are
generated in the degrees $(0,2)$ and $(1,1)$.

##  2 The Betti numbers of ideals with linear powers

###  2.1 General result

Let $S=\mathbb{k}[x_{1},\ldots,x_{n}]$, and let $I\subset S$ be a homogeneous
ideal with linear powers, generated in degree $d$. The Betti numbers of $S/I$
satisfy certain linear equations known as the _Herzog-Kühl_ -equations. These
equations were first studied in [11]. We will review how to obtain the Herzog-
Kühl equations, for $I^{k}$. The ideal $I^{k}$ is generated in degree $dk$,
and has the minimal graded free resolution

$0\\!\to S(-(dk+n-1))^{\beta_{n}^{(k)}}\\!\\!\\!\to\dots\to
S(-(dk+1))^{\beta_{2}^{(k)}}\\!\\!\\!\to S(-dk)^{\beta_{1}^{(k)}}\\!\\!\\!\to
S\to S/I^{k}\to 0\ .$

The Hilbert series of $S/I^{k}$ is given by

$\frac{1+\sum_{i=1}^{n}(-1)^{i}\beta_{i}^{(k)}t^{dk+i-1}}{(1-t)^{n}}=\frac{p(t)}{(1-t)^{\delta}}\
,$

where $p(t)$ is a polynomial, and $\delta=\dim I$. We rewrite this as

$1+\sum_{i=1}^{n}(-1)^{i}\beta_{i}^{(k)}t^{dk+i-1}=(1-t)^{n-\delta}p(t)\ ,$

and we see that $t=1$ is a zero of the right hand side, and of its
$n-\delta-1$ first derivatives. This gives us the $n-\delta$ Herzog-Kühl
equations

$\left\\{\begin{array}[]{rl}\sum_{i=1}^{n}(-1)^{i}\beta_{i}^{(k)}&=-1\\\\[4.30554pt]
\sum_{i=1}^{n}(-1)^{i}(dk+i-1)\beta_{i}^{(k)}&=0\\\\[4.30554pt]
\sum_{i=1}^{n}(-1)^{i}(dk+i-1)(dk+i-2)\beta_{i}^{(k)}&=0\\\\[4.30554pt]
\vdots&\\\
\sum_{i=1}^{n}(-1)^{i}(dk+i-1)\cdots(dk+i-(n-\delta-1))\beta_{i}^{(k)}&=0\
.\\\\[4.30554pt] \end{array}\right.$ (1)

This system of equations can actually be written on a very nice form, as we
will see in Theorem 2.1 below. We will use the convention $\binom{j}{i}=0$
when $i>j$, for binomial coefficients. Also, we will use the notation
$[c_{ij}]$ for the matrix with the element $c_{ij}$ on row $i$, column $j$.
Similarly, we use the notation $[c_{i}]$ for column vectors.

######  Theorem 2.1.

Let $S=\mathbb{k}[x_{1},\ldots,x_{n}]$. For a homogeneous ideal $I\subseteq S$
with linear powers, generated in degree $d$, define the column vector
${\beta^{(k)}}=(\beta_{i}(S/I^{k}))_{1\leq i\leq n}$. Then

$\left[(-1)^{i+j}\binom{j}{i}\right]_{\begin{subarray}{c}0\leq i<n-\dim I\\\
0\leq j<n\end{subarray}}{\beta^{(k)}}=\left[\binom{dk+i-1}{i}\right]_{0\leq
i<n-\dim I\ \textstyle{.}}$

###### Proof.

We start by writing the system of equations (1) in matrix form, as

$\left[\begin{matrix}-1&1&\cdots&(-1)^{n}\\\\[3.87498pt]
-dk&dk+1&\cdots&(-1)^{n}(dk+n-1)\\\\[3.87498pt]
-dk(dk-1)&\\!\\!\\!(dk\\!+\\!1)dk&&(-1)^{n}(dk\\!+\\!n\\!-\\!1)(dk\\!+\\!n\\!-\\!2)\\\
\vdots&&&\vdots\\\
\\!-dk(dk\\!-\\!1)\cdots(dk\\!-\\!n\\!+\\!\delta\\!+\\!2)&\dots&\dots&(-1)^{n}(dk\\!+\\!n\\!-\\!1)\cdots(dk\\!+\\!\delta-1)\end{matrix}\right.\\!\\!\left|\begin{matrix}-1\\\\[3.87498pt]
0\\\\[3.87498pt] 0\\\ \vdots\\\ 0\end{matrix}\right]_{\ \textstyle{,}}$

where $\delta=\dim I$. Let us index the rows by $i=0,1,\ldots,n-\delta-1$.
When we subtract a $(dk-i)$-multiple of row $i$ from row $i+1$, for
$i=n-\delta-2,n-\delta-3,\ldots,1,0$ we get

$\left[\begin{matrix}-1&1&-1&\cdots&(-1)^{n}\\\\[3.87498pt]
0&1&-2&\cdots&(-1)^{n}(n-1)\\\\[3.87498pt]
0&dk+1&-2(dk+2)&&(-1)^{n}(n-1)(dk+n-1)\\\\[3.87498pt]
0&(dk+1)dk&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-2(dk\\!+\\!2)(dk\\!+\\!1)\\!\\!\\!\\!\\!&&\\!\\!(-1)^{n}(n\\!-\\!1)(dk\\!+\\!n\\!-\\!1)(dk\\!+\\!n\\!-\\!2)\\\
\vdots&&&&\vdots\\\
0&\\!\\!\\!\\!\\!\\!\\!(dk\\!+\\!1)dk\cdots(dk\\!-\\!n\\!+\\!\delta\\!+\\!4)&\dots&\dots&\\!\\!\\!\\!\\!(-1)^{n}(n\\!-\\!1)(dk\\!+\\!n\\!-\\!1)\cdots(dk\\!+\\!\delta)\end{matrix}\right.\\!\\!\left|\begin{matrix}-1\\\\[3.87498pt]
dk\\\\[3.87498pt] 0\\\\[3.87498pt] 0\\\ \vdots\\\ 0\end{matrix}\right]_{\
\textstyle{,}}$

Next, we subtract a $(dk-i+2)$-multiple of row $i$ from row $i+1$, for
$i=n-d-1,\ldots,1$, which gives

$\left[\begin{matrix}-1&1&-1&\cdots&(-1)^{n}\\\\[3.87498pt]
0&1&-2&\cdots&(-1)^{n}(n-1)\\\\[3.87498pt]
0&0&-2&&(-1)^{n}(n-1)(n-2)\\\\[3.87498pt]
0&0&-2(dk+2)&&\\!\\!(-1)^{n}(n\\!-\\!1)(n-2)(dk\\!+\\!n\\!-\\!1)\\\
\vdots&\vdots&&&\vdots\\\
0&0&-2(dk+2)\cdots(dk\\!-\\!n\\!+\\!\delta\\!+\\!6)&\dots&\\!\\!\\!\\!\\!(-1)^{n}(n\\!-\\!1)(n\\!-\\!2)(dk\\!+\\!n\\!-\\!1)\cdots(dk\\!+\\!\delta\\!-\\!1)\end{matrix}\right.\\!\\!\left|\begin{matrix}-1\\\\[3.87498pt]
dk\\\\[3.87498pt] -dk(dk+1)\\\\[3.87498pt] 0\\\ \vdots\\\
0\end{matrix}\right]_{\ \textstyle{.}}$

Continuing in the same fashion eventually gives

$\left[\begin{matrix}\\!-1&1&-1&1&\dots&(-1)^{n-\delta}&\dots&(-1)^{n}\\\\[3.44444pt]
0&1&-2&3&\dots&(-1)^{n-\delta}(n-\delta-1)&\dots&(-1)^{n}(n-1)\\\\[3.44444pt]
0&0&-2&3\cdot
2&&(-1)^{n\\!-\delta}(n\\!-\\!\delta\\!-\\!1)(n\\!-\\!\delta\\!-\\!2)\\!\\!\\!\\!&&\\!\\!\\!\\!(-1)^{n}(n\\!-\\!1)(n\\!-\\!2)\\\
0&0&0&3\cdot 2&&\vdots&&\vdots\\\ 0&0&0&0&&&&\\\\[-3.44444pt]
\vdots&\vdots&\vdots&\vdots&\ddots\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!&(-1)^{n-\delta}(n-\delta-1)!&&\vdots\\\
0&0&0&0&&(-1)^{n-\delta}(n-\delta-1)!&&(-1)^{n}(n\\!-\\!1)\cdots(\delta\\!+\\!1)\end{matrix}\right|\left.\begin{matrix}-1\\\\[3.44444pt]
dk\\\\[3.44444pt] -dk(dk+1)\\\\[3.44444pt] dk(dk+1)(dk+2)\\\ \vdots\\\ \\\
\\!(-1)^{n\\!-\\!d+1}\\!dk\cdots\\!(dk\\!+\\!n\\!-\\!\delta)\\!\end{matrix}\right]$

Notice that we now have $(-1)^{j+1}j!/(j-i)!$ in position $(i,j)$ in the
coefficient matrix. Multiplying each row by $(-1)^{i+1}/i!$ completes the
proof. ∎

We can see that Theorem 2.1 is most useful for ideals of low dimension, since
the number of equations is $n-\dim I$. If we can compute $m$ of the Betti
numbers $\beta_{i}(S/I^{k})$, for some $m\geq\dim I$, we have enough
information to solve the system of equations, and we get explicit expressions
for $\beta_{i}(S/I^{k})$, as polynomials in $k$. For any ideal $J\subseteq S$,
we know that $\beta_{1}(S/J)$ is always the number of minimal generators of
$J$. We can also compute $\beta_{n}(S/J)$ in the following way. Recall that
the Betti numbers satisfy
$\beta_{i}(S/J)=\dim_{\mathbb{k}}\operatorname{Tor}_{i}^{S}(\mathbb{k},S/J)$,
see e. g. [2, Proposition 1.3.1]. Using the properties of the Tor functor we
have an isomorphism of $\mathbb{k}$-spaces

$\operatorname{Tor}_{n}^{S}(\mathbb{k},S/J)=\operatorname{Tor}_{n}^{S}(S/(x_{1},\ldots,x_{n}),S/J)\cong\operatorname{Soc}(S/J),$

where

$\operatorname{Soc}(S/J)=\\{f\in S/J\ |\ fx_{i}=0,\ \mbox{for all}\
i=1,\ldots,n\\}\ $

is the _socle_ of $S/J$. It follows that
$\beta_{n}(S/J)=\dim_{\mathbb{k}}\operatorname{Soc}(S/J)$.

######  Theorem 2.2.

Let $S=\mathbb{k}[x_{1},\ldots,x_{n}]$, and let $I\subset S$ be a homogeneous
ideal with linear powers, generated in degree $d$. Let
$\beta_{i}^{(k)}=\beta_{i}(S/I^{k})$. If $\dim I=1$, then

$\begin{bmatrix}\beta_{2}^{(k)}\\\ \vdots\\\ \vdots\\\
\beta_{n}^{(k)}\end{bmatrix}=\left[\binom{i}{j}\right]_{\begin{subarray}{c}0\leq
i\leq n-2\\\ \begin{subarray}{c}0\end{subarray}\leq j\leq
n-2\end{subarray}}\left[\beta_{1}^{(k)}-\binom{dk+i}{i}\right]_{0\leq i\leq
n-2{\ \textstyle{.}}}$

If $\dim I=2$, then

$\begin{bmatrix}\beta_{3}^{(k)}\\\ \vdots\\\ \vdots\\\
\beta_{n}^{(k)}\end{bmatrix}=\left[\binom{i}{j}\right]_{\begin{subarray}{c}0\leq
i\leq n-3\\\ \begin{subarray}{c}0\end{subarray}\leq j\leq
n-3\end{subarray}}\left[\beta_{1}^{(k)}-(-1)^{n+i}\binom{n-2}{i}\beta_{n}^{(k)}-\binom{dk+i}{i}\right]_{0\leq
i\leq n-3{\ \textstyle{.}}}$

###### Proof.

Subtracting $\beta_{1}^{(k)}$ from both sides of the first equation in Theorem
2.1 results in the system

$\begin{array}[]{l}\begin{matrix}\ \ \ \beta_{2}^{(k)}&\ \beta_{3}^{(k)}&\
\beta_{4}^{(k)}&\ \beta_{5}^{(k)}&\ \dots&\ \ \beta_{n}^{(k)}\end{matrix}\\\
\left[\begin{matrix}-1&1&-1&1&\dots&(-1)^{n-1}\\\\[4.30554pt]
1&-\binom{2}{1}&\binom{3}{1}&-\binom{4}{1}&\dots&(-1)^{n}\binom{n}{1}\\\\[4.30554pt]
0&1&-\binom{3}{2}&\binom{4}{2}&\dots&(-1)^{n+1}\binom{n}{2}\\\\[4.30554pt]
0&0&1&-\binom{4}{3}&&\vdots\\\ \vdots&\vdots&\vdots&\vdots&&\\\
0&0&&&\end{matrix}\right.\left|\begin{matrix}1-\beta_{1}^{(k)}\\\\[4.30554pt]
dk\\\\[4.30554pt] \binom{dk+1}{2}\\\\[4.30554pt] \binom{dk+2}{3}\\\ \vdots\\\
\binom{dk+n-\delta}{n-\delta-1}\\\ \end{matrix}\right]_{\
\textstyle{.}}\end{array}$

Adding the first row to the second, then the second row to the third, and so
on, gives

$\left[\begin{matrix}-1&1&-1&1&\dots&(-1)^{n-1}\\\\[4.30554pt]
0&-1&\binom{2}{1}&-\binom{3}{1}&\dots&(-1)^{n}\binom{n-1}{1}\\\\[4.30554pt]
0&0&-1&\binom{3}{2}&\dots&(-1)^{n+1}\binom{n-1}{2}\\\\[4.30554pt]
0&0&0&-1&&\vdots\\\ \vdots&\vdots&\vdots&\vdots&&\\\
0&0&&&\end{matrix}\right.\left|\begin{matrix}1-\beta_{1}^{(k)}\\\\[4.30554pt]
dk+1-\beta_{1}^{(k)}\\\\[4.30554pt]
\binom{dk+2}{2}-\beta_{1}^{(k)}\\\\[4.30554pt]
\binom{dk+3}{3}-\beta_{1}^{(k)}\\\ \vdots\\\
\binom{dk+n-\delta-1}{n-\delta-1}-\beta_{1}^{(k)}\\\ \end{matrix}\right]_{\
\textstyle{,}}$

where the relation $\binom{j+1}{i}-\binom{j}{i-1}=\binom{j}{i}$ has been used.
Multiplying each row by $-1$ results in the system

$\left[(-1)^{i+j}\binom{j}{i}\right]_{\begin{subarray}{c}0\leq i<n-\dim I\\\
0\leq j\leq n-2\end{subarray}}\left[\beta_{i}^{(k)}\right]_{2\leq i\leq
n}=\left[\beta_{1}^{(k)}-\binom{dk+i}{i}\right]_{0\leq i<n-\dim I{\
\textstyle{.}}}$

The inverse of the square matrix $\big{[}(-1)^{i+j}\binom{j}{i}\big{]}$ with
$0\leq i,j\leq s$ is $\big{[}\binom{j}{i}\big{]}$. For a proof of this fact,
see e. g. [3]. This proves the case $\dim I=1$. Moving all multiples of
$\beta_{n}^{(k)}$ to the right hand side of the above equation gives

$\displaystyle\left[(-1)^{i+j}\binom{j}{i}\right]_{\begin{subarray}{c}0\leq
i<n-\dim I\\\ 0\leq j\leq n-3\end{subarray}}$
$\displaystyle\left[\beta_{i}^{(k)}\right]_{2\leq i\leq n-1}\\!$
$\displaystyle=\left[\beta_{1}^{(k)}\\!-(-1)^{n+i}\binom{n-2}{i}\beta_{n}^{(k)}-\binom{dk+i}{i}\right]_{0\leq
i<n-\dim I{\ \textstyle{.}}}$

If $\dim I=2$, we now have a square coefficient matrix, and multiplying by its
inverse completes the proof. ∎

###  2.2 Application to square-free monomial ideals

Let $I$ be the ideal in $S=\mathbb{k}[x_{1},\ldots,x_{n}]$, generated by all
square-free monomials of degree $d$, for some fixed integer $d$, and
$\beta_{i}^{(k)}=\beta_{i}(S/I^{k})$. Such ideals are polymatroidal, and
therefore have linear powers. We have $\dim I=d-1$. The $k$-th power $I^{k}$
is generated by all monomials $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ of degree
$dk$, such that $a_{i}\leq k$, and the number of such monomials can be
computed using the principle of inclusion-exclusion. A monomial
$x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ of degree $dk$ with
$a_{i_{1}},\ldots,a_{i_{m}}>k$, for some indices $i_{1},\ldots,i_{m}$, can be
considered as a monomial of degree $dk-m(k+1)$, multiplied by
$(x_{i_{1}}\cdots x_{i_{m}})^{k+1}$. It follows that the number of monomials
$x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ with $a_{i}>k$, for at least $m$ of the
$a_{i}$’s is

$\binom{n}{m}\binom{n+dk-m(k+1)-1}{n-1}_{\ \textstyle{,}}$

and hence

$\beta_{1}^{(k)}=\sum_{\begin{subarray}{c}m\geq 0\\\ m(k+1)\leq
dk\end{subarray}}\\!\\!(-1)^{m}\binom{n}{m}\binom{n+dk-m(k+1)-1}{n-1}_{\
\textstyle{.}}$ (2)

To compute $\operatorname{Soc}(S/I^{k})$ it is sufficient to determine all
monomials $f$ such that $f\notin I^{k}$ but $fx_{i}\in I^{k}$, for all $i$.
Such a monomial $f$ must be of degree at least $dk-1$. Suppose
$f=x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}$ is a monomial of degree $dk$ or higher,
with this property. We can factorize $f$ as

$f=gx_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\ \mbox{where}\ g=\prod_{b_{i}\geq
k}x_{i}^{b_{i}-k}\ \mbox{and}\ a_{i}=\begin{cases}b_{i}&\mbox{for}\ b_{i}\leq
k,\\\ k&\mbox{otherwise.}\end{cases}$

Since $f\notin I^{k}$ we have $\sum a_{i}<dk$. Then, for any $i$ such that
$x_{i}|g$, we have $fx_{i}\notin I^{k}$. Hence, all elements of
$\operatorname{Soc}(S/I^{k})$ are of degree $dk-1$. More precisely

$\operatorname{Soc}(S/I^{k})=\operatorname{span}\\{x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}\ |\ a_{i}<k,\sum_{i=1}^{n}a_{i}=dk-1\\},$

and we get

$\displaystyle\beta_{n}^{(k)}=\dim_{\mathbb{k}}\operatorname{Soc}(S/I^{k})$
$\displaystyle=\sum_{\begin{subarray}{c}m\geq 0\\\ mk\leq
dk-1\end{subarray}}\\!\\!(-1)^{m}\binom{n}{m}\binom{n+dk-1-mk-1}{n-1}$ (3)
$\displaystyle=\sum_{m=0}^{d-1}(-1)^{m}\binom{n}{m}\binom{n+(d-m)k-2}{n-1}_{\
\textstyle{.}}$ (4)

If $d=2,$ or $3$, we can now use Theorem 2.2 to compute the remaining Betti
numbers.

######  Corollary 2.3.

Let $S=\mathbb{k}[x_{1},\ldots,x_{n}]$, and let $I\subseteq S$ be the ideal
generated by all square-free monomials of degree two. Let
$\beta_{i}^{(k)}=\beta_{i}(S/I^{k})$. Then

$\beta_{1}^{(k)}=\binom{n+2k-1}{n-1}-n\binom{n+k-2}{n-1}_{\ \textstyle{,}}$

and

$\begin{bmatrix}\beta_{2}^{(k)}\\\ \vdots\\\ \vdots\\\
\beta_{n}^{(k)}\end{bmatrix}=\left[\binom{j}{i}\right]_{\begin{subarray}{c}0\leq
i\leq n-2\\\ 0\leq j\leq
n-2\end{subarray}}\left[\beta_{1}^{(k)}-\binom{2k+i}{i}\right]_{0\leq i\leq
n-2{\ \textstyle{.}}}$

###### Proof.

The expression for $\beta_{1}^{(k)}$ follows from substituting $d=2$ in (2).
Since $\dim I=1$, Theorem 2.2 can be used to compute the remaining Betti
numbers. ∎

######  Example 2.4.

Let $S=\mathbb{k}[x_{1},x_{2},x_{3},x_{4}]$, and let $I\subseteq S$ be the
ideal generated by all square-free monomials of degree two. By Corollary 2.3,
the Betti numbers of $S/I^{k}$ can now be expressed as the following
polynomials in $k$

$\beta_{1}^{(k)}=\binom{3+2k}{3}-4\binom{k+2}{3}=\frac{1}{3}(2k^{3}+6k^{2}+7k+3),$

and

$\displaystyle\begin{bmatrix}\beta_{2}^{(k)}\\\ \beta_{3}^{(k)}\\\
\beta_{4}^{(k)}\\\ \end{bmatrix}$ $\displaystyle=\begin{bmatrix}1&1&1\\\
0&1&2\\\ 0&0&1\end{bmatrix}\begin{bmatrix}\beta_{1}^{(k)}-1\\\
\beta_{1}^{(k)}-(2k+1)\\\ \beta_{1}^{(k)}-(2k+1)(k+1)\\\ \end{bmatrix}$
$\displaystyle=\frac{1}{3}\begin{bmatrix}1&1&1\\\ 0&1&2\\\
0&0&1\end{bmatrix}\begin{bmatrix}2k^{3}+6k^{2}+7k\\\ 2k^{3}+6k^{2}+k\\\
2k^{3}-2k\\\ \end{bmatrix}=\begin{bmatrix}2k^{3}+4k^{2}+k\\\
2k^{3}+2k^{2}-k\\\ \frac{2}{3}(k^{3}-k)\end{bmatrix}_{\ \textstyle{.}}$

######  Corollary 2.5.

Let $S=\mathbb{k}[x_{1},\ldots,x_{n}]$, and let $I\subseteq S$ be the ideal
generated by all square-free monomials of degree three. Let
$\beta_{i}^{(k)}=\beta_{i}(S/I^{k})$. Then

$\beta_{1}^{(k)}=\binom{n+3k-1}{n-1}-n\binom{n+2k-2}{n-1}+\binom{n}{2}\binom{n+k-3}{n-1}_{\
\textstyle{,}}$
$\beta_{n}^{(k)}=\binom{n+3k-2}{n-1}-n\binom{n+2k-2}{n-1}+\binom{n}{2}\binom{n+k-2}{n-1}$

and

$\begin{bmatrix}\beta_{2}^{(k)}\\\ \vdots\\\ \vdots\\\
\beta_{n-1}^{(k)}\end{bmatrix}=\left[\binom{j}{i}\right]_{\begin{subarray}{c}0\leq
i\leq n-3\\\ 0\leq j\leq
n-3\end{subarray}}\left[\beta_{1}^{(k)}-(-1)^{n+i}\beta_{n}^{(k)}\binom{n-2}{i}-\binom{3k+i}{i}\right]_{0\leq
i\leq n-3{\ \textstyle{.}}}$

###### Proof.

The expressions for $\beta_{1}^{(k)}$ and $\beta_{n}^{(k)}$ is obtained by
substituting $d=3$ in (2) and (3). Since $\dim I=2$, Theorem 2.2 can be used
to compute the remaining Betti numbers. ∎

######  Example 2.6.

Let $S=\mathbb{k}[x_{1},\ldots,x_{5}]$, and let $I\subseteq S$ be the ideal
generated by all square-free monomials of degree three. By Corollary 2.5, the
Betti numbers of $S/I^{k}$ can now be expressed as the following polynomials
in $k$

$\displaystyle\beta_{1}^{(k)}$
$\displaystyle=\binom{4+3k}{4}-5\binom{3+2k}{4}+\binom{5}{2}\binom{2+k}{4}$
$\displaystyle=\frac{1}{24}(11k^{4}+50k^{3}+85k^{2}+70k+24),$
$\displaystyle\beta_{5}^{(k)}$
$\displaystyle=\binom{3+3k}{4}-5\binom{3+2k}{4}+\binom{5}{2}\binom{3+k}{4}$
$\displaystyle=\frac{1}{24}(11k^{4}-18k^{3}-11k^{2}+18k),$

and

$\displaystyle\begin{bmatrix}\beta_{2}^{(k)}\\\ \beta_{3}^{(k)}\\\
\beta_{4}^{(k)}\\\ \end{bmatrix}$ $\displaystyle=\begin{bmatrix}1&1&1\\\
0&1&2\\\ 0&0&1\end{bmatrix}\begin{bmatrix}\beta_{1}^{(k)}+\beta_{5}^{(k)}-1\\\
\beta_{1}^{(k)}-3\beta_{5}^{(k)}-(3k+1)\\\
\beta_{1}^{(k)}+3\beta_{5}^{(k)}-\frac{(3k+2)(3k+1)}{2}\\\
\end{bmatrix}=\begin{bmatrix}\frac{11}{6}k^{4}+\frac{11}{2}k^{3}+\frac{17}{3}k^{2}+2k\\\\[3.01385pt]
\frac{11}{4}k^{4}+4k^{3}+\frac{1}{4}k^{2}-k\\\\[3.01385pt]
\frac{11}{6}k^{4}-\frac{1}{6}k^{3}-\frac{7}{3}k^{2}+\frac{2}{3}k\end{bmatrix}_{\
\textstyle{.}}$

###  2.3 Application to the product of ideals generated by variables

In this section let $I$ be the product of all ideals generated by $s$-sized
subsets of the variables, i. e.

$I=\prod_{1\leq i_{1}<\dots<i_{s}\leq n}(x_{i_{1}},\ldots,x_{i_{s}}),$

for some $s\leq n$. Ideals generated by variables are polymatroidal. Since
products of polymatroidal ideals are polymatroidal, $I$ is polymatroidal and
hence has linear powers. The generators are monomials of degree
$\binom{n}{s}$, and $\dim I=n-s$. As in the previous section, we will consider
the two cases $\dim I=2$ and $3$, which here means $s=n-1$ and $n-2$.

Let us start with the case $s=n-1$. Then $I$ is the product of all ideals
$(x_{i_{1}},\ldots,x_{i_{n-1}})$, and it can easily be seen that $I$ is
generated by all monomials of degree $n$, except the pure powers
$x_{1}^{n},x_{2}^{n},\ldots,x_{n}^{n}$. It follows that $I^{k}$ is generated
by all monomials $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ of degree $nk$ with
$a_{i}\leq k(n-1)$ for all $i$. With this information we can now compute
$\beta_{1}(S/I^{k})$ and then apply Theorem 2.2 to get the other Betti
numbers.

######  Corollary 2.7.

Let $S=\mathbb{k}[x_{1},\ldots,x_{n}]$,

$I=\prod_{1\leq i_{1}<\dots<i_{n-1}\leq
n}(x_{i_{1}},\ldots,x_{i_{n-1}})\subset S,$

and let $\beta_{i}^{(k)}=\beta_{i}(S/I^{k})$. Then

$\beta_{1}^{(k)}=\binom{n(k+1)-1}{n-1}-n\binom{n+k-2}{n-1}_{\ \textstyle{,}}$

and

$\begin{bmatrix}\beta_{2}^{(k)}\\\ \vdots\\\ \vdots\\\
\beta_{n}^{(k)}\end{bmatrix}=\left[\binom{j}{i}\right]_{\begin{subarray}{c}0\leq
i\leq n-2\\\ 0\leq j\leq
n-2\end{subarray}}\left[\beta_{1}^{(k)}-\binom{2k+i}{i}\right]_{0\leq i\leq
n-2{\ \textstyle{.}}}$

###### Proof.

The number of monomials of degree $nk$ is $\binom{n+nk-1}{n-1}$. The monomials
of degree $nk$ that have $a_{i}>k(n-1)$, for some $i$, can be written as a
monomial of degree $k-1$ multiplied by $x_{i}^{k(n-1)+1}$, and the number of
such monomials is $n\binom{n+k-2}{n-1}$. Since only one variable can have
degree greater than $k(n-1)$, without making the total degree too high, we can
now conclude that

$\beta_{1}^{(k)}=\binom{n+nk-1}{n-1}-n\binom{n+k-2}{n-1}_{\ \textstyle{.}}$

Since $\dim I=1$, the second part follows from Theorem 2.2. ∎

Let us now assume $n\geq 3$, and consider the case $s=n-2$.

######  Lemma 2.8.

The ideal

$\prod_{1\leq i_{1}<\dots<i_{n-2}\leq n}(x_{i_{1}},\ldots,x_{i_{n-2}})$

is generated by all monomials $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ of degree
$\binom{n}{2}$ such that $a_{i}\leq\binom{n-1}{2}$, for $i=1,\ldots,n$, where
at least three $a_{i}$’s are positive.

###### Proof.

Let us denote the ideal by $I$. We can see that $a_{i}\leq\binom{n-1}{2}$ for
any $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\in I$, since there are only
$\binom{n-1}{2}$ different ideals $(x_{i_{1}},\ldots,x_{i_{n-2}})$ containing
a specific variable $x_{i}$. Also, a monomial $x_{1}^{a_{1}}x_{2}^{a_{2}}$ can
not be contained in $I$, since the ideal $(x_{3},\ldots,x_{n})$ is one of the
ideals in the product defining $I$. The same argument holds for any other
monomial in two variables. Hence any $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\in I$
must have at least three positive $a_{i}$’s. We must now prove that all
monomials satisfying these conditions are contained in $I$.

Let $f=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ be a monomial of degree
$\binom{n}{2}$ such that $a_{i}\leq\binom{n-1}{2}$, for all $i$, and at least
three $a_{i}$’s are positive. We want to prove that $f\in I_{n-2}$. This will
be proved by induction over $\max_{i}(a_{i})$, starting from the highest
possible value $\max(a_{i})=\binom{n-1}{2}$. Without loss of generality, we
may, in the base case, assume that $a_{1}=\binom{n-1}{2}$. Notice that
$I=J_{1}J_{2}$ for

$J_{1}=\prod_{2\leq i_{1}<\dots<i_{n-2}\leq
n}\\!\\!(x_{i_{1}},\ldots,x_{i_{n-2}})\ \ \mbox{and}\ \ J_{2}=\prod_{2\leq
i_{1}<\dots<i_{n-3}\leq n}\\!\\!(x_{1},x_{i_{1}},\ldots,x_{i_{n-3}})\ .$

The ideal $J_{1}$ is generated by all monomials of degree $n-1$ in the
variables $x_{2},\ldots,x_{n}$, except the pure powers. In particular
$x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}$ is one of the generators. It is clear that
$x_{1}^{\binom{n-1}{2}}=x_{1}^{a_{1}}$ is one of the generators in $J_{2}$,
and it follows that $f=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ is one of the
generators in $I$.

Next, suppose that $f$ is such that $\max_{i}(a_{i})=m<\binom{n-1}{2}$, and
assume that the statement is true for all monomials $x_{1}^{b_{1}}\cdots
x_{n}^{b_{n}}$ with $\max_{i}(b_{i})>m$. Without loss of generality, we may
assume that $a_{1}=m$. First, notice that there is some $i$ such that
$\frac{x_{1}}{x_{i}}f$ is a monomial satisfying our conditions, i. e.
containing at least three variables. Indeed, if such an $i$ would not exist,
then $f$ would be on the form $f=x_{1}^{m}x_{i}x_{j}$. But this would imply

$m=\binom{n}{2}-2\geq\binom{n}{2}-\binom{n-1}{1}=\binom{n-1}{2}_{\
\textstyle{,}}$

contradicting $m<\binom{n-1}{2}$. Again without loss of generality, we assume
that

$g=\frac{x_{1}}{x_{2}}f=x_{1}^{m+1}x_{2}^{a_{2}-1}x_{3}^{a_{3}}\cdots
x_{n}^{a_{n}}$

is a monomial in at least three variables. By the induction hypothesis $g\in
I$. Then we also have $x_{1}^{a_{2}-1}x_{2}^{m+1}x_{3}^{a_{3}}\cdots
x_{n}^{a_{n}}\in I,$ by symmetry. Since $I$ is polymatroidal and
$m+1>a_{2}-1$, we can conclude that $f=\frac{x_{2}}{x_{1}}g\in I$. ∎

It follows that $I^{k}$ is generated by all monomials $x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}$ of degree $k\binom{n}{2}$ such that $a_{i}\leq k\binom{n-1}{2}$
and at least three $a_{i}$’s are positive. In a similar way as for the square-
free monomials ideals, we can see that the socle elements must have degree
$k\binom{n}{2}-1$. It follows that the socle is spanned by the monomials
$x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ of degree $k\binom{n}{2}-1$ such that
$a_{i}<k\binom{n-1}{2}$ and at least three $a_{i}$’s positive. We can now use
Theorem 2.2 to compute the Betti numbers of the product of all ideals
generated by $n-2$ variables.

######  Corollary 2.9.

Let $S=\mathbb{k}[x_{1},\ldots,x_{n}]$, and

$I=\prod_{1\leq i_{1}<\dots<i_{n-2}\leq
n}(x_{i_{1}},\ldots,x_{i_{n-2}})\subset S.$

Let $\beta_{i}^{(k)}=\beta_{i}(S/I^{k})$. Then

$\beta_{1}^{(k)}=\binom{n+k\binom{n}{2}-1}{n-1}-n\binom{n+k(n-1)-2}{n-1}-\binom{n}{2}\Big{(}1+k(n-1)\Big{(}\frac{n}{2}-1\Big{)}\Big{)},$
$\beta_{n}^{(k)}=\binom{n+k\binom{n}{2}-2}{n-1}-n\binom{n+k(n-1)-2}{n-1}-\binom{n}{2}\Big{(}1+k(n-1)\Big{(}\frac{n}{2}-1\Big{)}\Big{)}$

and

$\begin{bmatrix}\beta_{2}^{(k)}\\\ \vdots\\\ \vdots\\\
\beta_{n-1}^{(k)}\end{bmatrix}=\left[\binom{j}{i}\right]_{\begin{subarray}{c}0\leq
i\leq n-3\\\ 0\leq j\leq
n-3\end{subarray}}\left[\beta_{1}^{(k)}-(-1)^{n+i}\beta_{n}^{(k)}\binom{n-2}{i}-\binom{k\binom{n}{2}+i}{i}\right]_{0\leq
i\leq n-3}$

###### Proof.

To find $\beta_{1}^{(k)}$ we shall compute the number of monomials
$x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ of degree $k\binom{n}{2}$ such that
$a_{i}\leq k\binom{n-1}{2}$, with at least three positive $a_{i}$’s. A
monomial $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ of degree $k\binom{n}{2}$ can
have at most one $a_{i}>k\binom{n-1}{2}$, or the total degree will be too
high. It follows that the number of monomials $x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}$ of degree $k\binom{n}{2}$ such that $a_{i}\leq k\binom{n-1}{2}$
is

$\binom{n+k\binom{n}{2}-1}{n-1}-n\binom{n+k(n-1)-2}{n-1}_{\ \textstyle{.}}$

From this we must subtract the number of monomials

$x_{i}^{a_{i}}x_{j}^{k\binom{n}{2}-a_{i}}\ \mbox{such that}\ a_{i}\leq
k\binom{n-1}{2},\ k\binom{n}{2}-a_{i}\leq k\binom{n-1}{2}_{\ \textstyle{.}}$

The condition on $a_{i}$ simplifies to

$k(n-1)\leq a_{i}\leq k\binom{n-1}{2}_{\ \textstyle{,}}$

so for each pair of variables $x_{i},x_{j}$ there are

$k\binom{n-1}{2}-k(n-1)+1=1+k(n-1)\Big{(}\frac{n}{2}-1\Big{)}$

such monomials. This proves

$\beta_{1}^{(k)}=\binom{n+k\binom{n}{2}-1}{n-1}-n\binom{n+k(n-1)-2}{n-1}-\binom{n}{2}\Big{(}1+k(n-1)\Big{(}\frac{n}{2}-1\Big{)}\Big{)}_{\
\textstyle{.}}$

To find $\beta_{n}^{k}$ we shall compute the number of monomials
$x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ of degree $k\binom{n}{2}-1$ such that
$a_{i}<k\binom{n-1}{2}$ and at least three $a_{i}$’s positive. The same
argument as above results in

$\beta_{n}^{(k)}=\binom{n+k\binom{n}{2}-2}{n-1}-n\binom{n+k(n-1)-2}{n-1}-\binom{n}{2}\Big{(}1+k(n-1)\Big{(}\frac{n}{2}-1\Big{)}\Big{)}.$

The expression for the remaining Betti numbers follows directly from Theorem
2.2. ∎

##  3 The Rees algebra of monomial ideals with linear powers

The object of this section is to study the generators of the Rees ideal, for
certain monomial ideals with linear powers.

For a bigraded module $M$, we let $M_{(i,j)}$ denote the graded component of
elements of degree $(i,j)$. We also let $M_{(*,j)}=\bigoplus_{i\geq
0}M_{(i,j)}$, and we say that the elements of $M_{(*,j)}$ are of degree
$(*,j)$, and analogously for $M_{(i,*)}$.

###  3.1 The degrees of the generators of the Rees ideal

We start by the following observation on the Rees ideal of ideals with linear
resolution.

######  Lemma 3.1.

Let $I$ be an ideal such that $I^{k}$ has a linear resolution, and let $J$ be
the Rees ideal of $I$. Then the elements of degree $(s,k)$, with $s\geq 1$, in
$J$ are generated by the elements of degree $(1,k)$ in $J$.

###### Proof.

Suppose $I=(f_{1},\ldots,f_{m})$, and let $h\in J$ be homogeneous of degree
$(s,k)$. Then $h=\sum_{i=0}^{N}g_{i}h_{i}$, where $g_{i}\in S$, and the
$h_{i}$’s are the monomials of degree $k$ in the variables
$y_{1},\ldots,y_{m}$. The monomials $h_{i}$ are mapped to a (not necessarily
minimal) generating set of $I^{k}$ under $\phi$. By definition of the Rees
ideal, $\phi(h)=0$, which implies $\sum_{i=1}^{m}g_{i}\phi(h_{i})=0$. But
since $I^{k}$ has a linear resolution, all relations on the generators are
generated by relations in degree one. It follows that $h$ is generated by
elements of degree $(1,k)$. ∎

An ideal $I$ is said to be of _fiber type_ if its Rees ideal is generated in
degrees $(0,*)$ and $(*,1)$. This terminology was introduced in [9], where
they prove that polymatroidal ideals are of fiber type. By [14, Theorem 3.1]
edge ideals of graphs are also of fiber type.

######  Lemma 3.2.

Let $I$ be an ideal of fiber type with linear resolution. Then the Rees ideal
of $I$ is generated in degrees $(0,*)$ and $(1,1)$.

###### Proof.

By Lemma 3.1 the elements if degree $(*,1)$ in the Rees ideal are generated by
elements of degree $(1,1)$. Since $I$ is of fiber type it follows that the
Rees ideal is generated in degrees $(0,*)$ and $(1,1)$. ∎

We are especially interested in monomial ideals with linear resolution where
the elements of degree $(0,*)$ in the Rees ideal are generated in degree
$(0,2)$. Notice that the $(0,*)$-part of the Rees ideal is the same as the
toric ideal defined by the monomial generators. More precisely, if $I$ is
generated by monomials $f_{1},\ldots,f_{m}$ define a map
$\xi:\mathbb{k}[y_{1},\ldots,y_{m}]\to S$ by $\xi(y_{i})=f_{i}$. Then the
$(0,*)$-part of the Rees ideal of $I$ is given by $\ker\xi$. It was proved in
[8] that polymatroidal ideals with the so called _strong exchange property_
have toric ideal generated in degree 2.

######  Definition 3.3.

A polymatroidal ideal $I$ is said to have the _strong exchange property (SEP)_
if it satisfies the following condition. Let $\mathcal{G}(I)$ be the
generating set of monomials. If $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ and
$x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}$ are elements of $\mathcal{G}(I)$, such
that $a_{i}>b_{i}$ and $a_{j}<b_{j}$, then $(x_{j}/x_{i})x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}\in\mathcal{G}(I)$.

An example of polymatroidal ideals with the SEP are the ideals generated by
all square-free monomials in some degree, that we saw in Section 2.2. The
generators of the Rees ideal reflects the ”exchange relations”.

######  Example 3.4.

We return to algebra $S=\mathbb{k}[x_{1},x_{2},x_{3},x_{4}]$ and the square-
free monomial ideal

$I=(x_{1}x_{2},x_{1}x_{3},x_{1}x_{4},x_{2}x_{3},x_{2}x_{4},x_{3}x_{4})\subset
S$

from Example 2.4. Let $T=S[y_{12},y_{13},y_{14},y_{23},y_{24},y_{34}]$, and
define $\phi:T\to S[t]$ by $x_{i}\mapsto x_{i}$ and $y_{ij}\mapsto
x_{i}x_{j}t$. Then the Rees ideal $J=\ker\phi$ is generated by the binomials

$\displaystyle x_{3}y_{12}-x_{2}y_{13},\ x_{1}y_{23}-x_{3}y_{12},\
x_{3}y_{14}-x_{4}y_{13},\ x_{2}y_{14}-x_{4}y_{12},$ $\displaystyle
x_{3}y_{24}-x_{4}y_{23},\ x_{1}y_{24}-x_{4}y_{12},\ x_{2}y_{34}-x_{4}y_{23},\
x_{1}y_{34}-x_{4}y_{13}$

of degree $(1,1)$ and

$y_{12}y_{34}-y_{13}y_{24},\ y_{12}y_{34}-y_{14}y_{23}$

of degree $(0,2)$.

Another class of polymatroidal ideals with toric ideal generated in degree 2
are the _transversal_ ideals, see [5, Theorem 3.5]. A polymatroidal ideal is
called transversal if it is the product of ideals generated by variables. So
in particular, the ideals studied in Section 2.3 are transversal. Next, we
will see a class of edge ideals with toric ideal generated in degree 2.

Let $G=(V,E)$ be a bipartite graph, with the partition
$V=\\{x_{1},\ldots,x_{n}\\}\cup\\{y_{1},\ldots,y_{m}\\}$ of the vertices. Let
$e_{ij}$ denote the edge $\\{x_{i},y_{j}\\}$. The graph $G$ is called a
_Ferrers graph_ if, after possibly re-indexing of the vertices,

$e_{ij}\in E\ \ \implies\ \ e_{rs}\in E\ \mbox{for all}\ 1\leq r\leq i,\
\mbox{and}\ 1\leq s\leq j.$

The name comes from the fact that the graph can be represented by a Ferrers
diagram. The edge ideal of a Ferrers graph is called a _Ferrers ideal_. It is
proved in [6] that the edge ideal of a bipartite graph has a linear resolution
if and only if it is a Ferrers ideal.

######  Lemma 3.5.

Let $J$ be the Rees ideal of a Ferrers ideal. The elements of degree $(0,*)$
in $J$ are generated by the elements of degree $(0,2)$ in $J$.

###### Proof.

The elements of degree $(0,s)$ in $J\subseteq T$ are linear combinations of
elements like

$f=e_{i_{1}j_{1}}\cdots e_{i_{s}j_{s}}-e_{i_{1}k_{1}}\cdots e_{i_{s}k_{s}}\
\mbox{where}\ \\{j_{1},\ldots,j_{s}\\}=\\{k_{1},\ldots k_{s}\\}.$

Let $L$ be the ideal generated by all elements on the form
$e_{i_{1}j_{1}}e_{i_{2}j_{2}}-e_{i_{1}j_{2}}e_{i_{2}j_{1}}$. We want to prove
that $f=0$ in $T/L$. We may assume that $i_{1}\geq i_{2}\dots\geq i_{s}$, and
$j_{1}\geq k_{1}$. Since $\\{j_{1},\ldots,j_{s}\\}=\\{k_{1},\ldots k_{s}\\}$
there is an index $r$ such that $k_{r}=j_{1}$. It follows that
$e_{i_{r}k_{1}}\in E$, since $e_{i_{1}j_{1}}\in E$, and $i_{r}\leq i_{1}$ and
$k_{1}\leq j_{1}$. Then
$e_{i_{1}k_{1}}e_{i_{r}k_{r}}=e_{i_{i}j_{1}}e_{i_{r}k_{1}}$ in $T/L$, and

$f=e_{i_{1}j_{1}}\cdots e_{i_{s}j_{s}}-e_{i_{1}k_{1}}\cdots
e_{i_{s}k_{s}}=e_{i_{1}j_{1}}(e_{i_{2}j_{2}}\cdots
e_{i_{s}j_{s}}-e_{i_{2}\hat{k}_{2}}\cdots e_{i_{s}\hat{k}_{s}}),$

where $\hat{k}_{i}=k_{i}$ for $i\neq r$, and $\hat{k}_{r}=k_{1}$. The result
now follows by induction over $s$. ∎

To summarize, we now have the following theorem.

######  Theorem 3.6.

If $I$ is a polymatroidal ideal with the SEP, a transversal polymatroidal
ideal, or a Ferrers ideal, then the Rees ideal of $I$ is generated in degrees
$(0,2)$ and $(1,1)$.

As pointed out in the introduction, there is reason the believe that Theorem
3.6 holds for all polymatroidal ideals. Theorem 3.6 does _not_ hold for all
edge ideals with linear resolution, as the below example shows. For more
details on the Rees algebra of an edge ideal see [14].

######  Example 3.7.

Let $G$ be the graph in Figure 2. As we can see in Figure 2, the complementary
graph is chordal, so the edge ideal

$I(G)=(x_{1}x_{2},x_{1}x_{3},x_{2}x_{3},x_{2}x_{4},x_{2}x_{5},x_{3}x_{5},x_{3}x_{6},x_{4}x_{5},x_{5}x_{6})$

has linear powers. However, the Rees ideal is generated in the degrees
$(1,1),(0,2)$ and $(0,3)$. The generator of degree $(0,3)$ corresponds to the
relation

$x_{1}x_{2}\cdot x_{3}x_{6}\cdot x_{4}x_{5}=x_{1}x_{3}\cdot x_{2}x_{4}\cdot
x_{5}x_{6}\ .$

Notice also that $I(G)$ is not polymatroidal. To realize this, consider the
two monomials $x_{1}x_{2}$ and $x_{5}x_{6}$, that are in the generating set of
$I(G)$. The monomial $x_{1}x_{2}$ has higher $x_{2}$-degree than $x_{5}x_{6}$,
but neither $(x_{5}/x_{2})x_{1}x_{2}=x_{1}x_{5}$, nor
$(x_{6}/x_{2})x_{1}x_{2}=x_{1}x_{6}$ belongs to $I(G)$.

Figure 1: The graph $G$

Figure 2: The complementary graph of $G$

###  3.2 The Betti numbers of the ideal generated by all square-free monomials
of degree $n-1$

In this last section we will return to the square-free monomial ideals. Let
$S=\mathbb{k}[x_{1},\ldots,x_{n}]$, and let $I\subset S$ be the ideal
generated by all square-free monomials of degree $n-1$. We will use the Rees
algebra $R(I)$ to compute the Betti numbers of $S/I^{k}$.

Let $f_{i}=x_{1}\cdots x_{n}/x_{i}$, i. e. the product of all the variables
except $x_{i}$. Then $I=(f_{1},\ldots,f_{n})$. As before, we let
$T=S[y_{1},\ldots,y_{n}]$, and $\phi:T\to S[t]$ be defined by $x_{i}\mapsto
x_{i}$ and $y_{i}\mapsto f_{i}t$. Since $I$ is a polymatroidal ideal with the
SEP, the Rees ideal $J=\ker\phi$ is generated in the degrees $(0,2)$ and
$(1,1)$, by Theorem 3.6. In fact, there are no non-trivial relations in degree
$(0,2)$, and we can see that $J$ is generated by all elements on the form
$x_{i}y_{i}-x_{j}y_{j}$. A minimal generating set for $J$ is then

$x_{1}y_{1}-x_{2}y_{2},x_{1}y_{1}-x_{3}y_{3},\ldots,x_{1}y_{1}-x_{n}y_{n}.$

These elements constitutes a regular sequence of length $n-1$ in $T$, and we
can then use the Koszul complex to find the bigraded minimal resolution

$\displaystyle\begin{split}0\to T(-n+1&,-n+1)^{\binom{n-1}{n-1}}\to\ldots\\\
\ldots\to&T(-2,-2)^{\binom{n-1}{2}}\to T(-1,-1)^{\binom{n-1}{1}}\to T(0,0)\to
T/J\to 0\end{split}$ (5)

of $T/J\cong R(I)$. Now notice that $\phi$ maps $T_{(*,k)}$ to $I^{k}$, and
the kernel is $J_{(*,k)}$. Hence $[T/J]_{(*,k)}\cong I^{k}$ as $S$-modules.
Also,

$T(-a,-b)_{(*,k)}=\bigoplus_{\sum
u_{i}=k-b}\\!\\!\\!\\!S(-a)y_{1}^{u_{1}}\cdots
y_{n}^{u_{n}}\cong{S(-a)^{\binom{n+k-b-1}{n-1}}}_{\ \textstyle{.}}$

Hence the degree $(*,k)$ part of (5) gives us the minimal resolution

$0\to S(-(n-1))^{\binom{k}{n-1}}\to\dots\to
S(-i)^{\binom{n+k-i-1}{n-1}\binom{n-1}{i}}\to\dots\to
S(0)^{\binom{n+k-1}{n-1}}\to I^{k}\to 0$ (6)

of $I^{k}$.

######  Theorem 3.8.

Let $S=\mathbb{k}[x_{1},\ldots,x_{n}]$, and let $I\subset S$ be the ideal
generated by all square-free monomials of degree $n-1$. Then the Betti numbers
of $S/I^{k}$ are given by

$\beta_{i}(S/I^{k})=\binom{n+k-i}{n-1}\binom{n-1}{i-1}_{\textstyle{.}}$

###### Proof.

By (6) we have the minimal resolution

$0\to S^{\binom{k}{n-1}}\to\dots\to
S^{\binom{n+k-i-1}{n-1}\binom{n-1}{i}}\to\dots\to S^{\binom{n+k-1}{n-1}}\to
S\to S/I^{k}\to 0$

of $S/I^{k}$. ∎

##  4 Further questions and problems

Lastly, we post some problems that arise from the topics of this paper. We
will refer to the classes A and B introduced in Section 1.

  1. 1.

In Section 2 we computed the Betti numbers for the ideals in A with $d=2,3$
and $n-1$, and the ideals in B with $s=n-1$, and $n-2$. Can we cover more
instances of these two classes, by combining Theorem 2.1 with some other
method? Are there other classes for which we can use Theorem 2.1 to compute
some of the Betti numbers?

  2. 2.

The main question that we came across in Section 3 is whether all
polymatroidal ideals have their Rees ideals generated in the degrees $(0,2)$
and $(1,1)$. One may also ask if the Rees algebras of the ideals in Theorem
3.6 are Koszul, as the defining ideals are generated in total degree two.

  3. 3.

Is there a nice description of all monomial ideals with linear powers, which
have their Rees ideals generated in degrees $(0,2)$ and $(1,1)$? For example,
one can try to classify all edge ideals with this property.

## Acknowledgements

The author would like to thank Ralf Fröberg, Christian Gottlieb, and Samuel
Lundqvist for the rewarding discussions about the topics of this paper.

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