# Complete Quadrics: Schubert calculus for Gaussian models and semidefinite
programming

Laurent Manivel  Institut de Mathématiques de Toulouse ; UMR 5219, Université
de Toulouse & CNRS, F-31062 Toulouse Cedex 9, France<EMAIL_ADDRESS>,
Mateusz Michałek  University of Konstanz, Germany, Fachbereich Mathematik und
Statistik, Fach D 197 D-78457 Konstanz, Germany  mateusz.michalek@uni-
konstanz.de ,  Leonid Monin  University of Bristol, School of Mathematics, BS8
1TW, Bristol, UK<EMAIL_ADDRESS>,  Tim Seynnaeve  University of
Bern, Mathematical Institute, Sidlerstrasse 5, 3012 Bern, Switzerland
<EMAIL_ADDRESS>and  Martin Vodička  Max Planck Institute for
Mathematics in the Sciences, 04103 Leipzig, Germany<EMAIL_ADDRESS>
###### Abstract.

We establish connections between: the maximum likelihood degree (ML-degree)
for linear concentration models, the algebraic degree of semidefinite
programming (SDP), and Schubert calculus for complete quadrics. We prove a
conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We
also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit
formula for the degree of SDP. The interactions between the three fields shed
new light on the asymptotic behaviour of enumerative invariants for the
variety of complete quadrics. We also extend these results to spaces of
general matrices and of skew-symmetric matrices.

###### 2020 Mathematics Subject Classification:

primary: 62R01, 14M17, 14N10 secondary: 05E05, 14C17, 14E05, 14M15, 14Q15,
60G15

##  1\. Introduction

### Maximum likelihood degree and quadrics

Although this paper is mainly about enumerative geometry and symmetric
functions, the main motivations come from algebraic statistics and
multivariate Gaussian models. These are generalizations of the well-known
Gaussian distribution to higher dimensions. In the one-dimensional case, in
order to determine a Gaussian distribution on ${\mathbb{R}}$, one needs to
specify its mean $\mu\in{\mathbb{R}}$ and its variance
$\sigma\in{\mathbb{R}}_{>0}$. In the $n$-dimensional case, the mean is a
vector $\mu\in{\mathbb{R}}^{n}$, and the second parameter is a positive-
definite $n\times n$ covariance matrix $\Sigma$. The corresponding Gaussian
distribution on ${\mathbb{R}}^{n}$ is given by

$f_{\mu,\Sigma}(x):=\frac{1}{\sqrt{(2\pi)^{n}\mathop{\rm
det}\nolimits\Sigma}}e^{-\frac{1}{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu)},$

where T denotes the transpose. Equivalently to determining it by $\mu$ and
$\Sigma$, one may represent the distribution by $\mu$ and the _concentration
matrix_ $K:=\Sigma^{-1}$, which is also positive definite. Our primary
interest lies in _linear concentration models_ , i.e. statistical models which
assume that $K$ belongs to a fixed $d$-dimensional space $\mathcal{L}$ of
$n\times n$ symmetric matrices. These were introduced by Anderson half a
century ago [1]. In particular, this means that $\Sigma$ should belong to the
set $\mathcal{L}^{-1}$ of inverses of matrices from $\mathcal{L}$.

In statistics, typically one gathers data as sample vectors
$x_{1},\dots,x_{s}\in{\mathbb{R}}^{n}$. This allows to estimate the mean $\mu$
as the mean of the $x_{i}$’s. Furthermore, each $x_{i}$ provides a matrix
$\Sigma_{i}:=(x_{i}-\mu)(x_{i}-\mu)^{T}$. Next, one considers the _sample
covariance matrix_ $S$, that is the mean of the $\Sigma_{i}$’s. Note that in
most situations, it is not true that $S\in\mathcal{L}^{-1}$. The aim is then
to find $\Sigma$ that best explains the observations. From the point of view
of statistics, it is natural to maximize the likelihood function

$f_{\mu,\Sigma}(x_{1})\cdots f_{\mu,\Sigma}(x_{s}),$

that is, to find a positive definite matrix $\Sigma\in\mathcal{L}^{-1}$ for
which the above value is maximal. Classical theorems in statistics assert that
the solution to this optimization problem is essentially geometric [4, Theorem
3.6, Theorem 5.5], [24, Theorem 4.4]. Namely, under mild genericity
assumptions, the optimal $\Sigma$ is the unique positive definite matrix in
$\mathcal{L}^{-1}$ that maps to the same point as $S$ under projection from
$\mathcal{L}^{\perp}$.

This is one of the main reasons why the complex variety that is the Zariski
closure of $\mathcal{L}^{-1}$ (which abusing notation we also denote by
$\mathcal{L}^{-1}$) and the rational map $\pi$ defined as the projection from
$\mathcal{L}^{\perp}$ are intensively studied in algebraic statistics. Note
that for generic $\mathcal{L}$, and after projectivization, $\pi$ becomes a
finite map. The following is the central definition of the article.

######  Definition 1.1 (ML-degree).

The _ML-degree_ of a linear concentration model represented by a space
$\mathcal{L}$ is the degree of the projection from the space
$\mathcal{L}^{\perp}$ restricted to the variety $\mathcal{L}^{-1}$.

The ML-degree is the basic measure of the complexity of the model. When
$\mathcal{L}$ is a generic space, the ML-degree only depends on the size $n$
of the symmetric matrices and on the (affine) dimension $d$ of $\mathcal{L}$.
By a theorem of Teissier [39, 40] (cf. [22, Corollary 2.6]) or Sturmfels and
Uhler [38, Theorem 1], the ML-degree equals the degree of the variety
$\mathcal{L}^{-1}$. Following Sturmfels and Uhler [38] we denote it by
$\phi(n,d)$. We refer algebraists interested in statistics to [10] for more
information about the subject.

######  Definition 1.2.

For $n\in{\mathbb{Z}}_{>0}$ and $1\leq d\leq\binom{n+1}{2}$, we define
$\phi(n,d)$ to be the degree of the variety $\mathcal{L}^{-1}$, where
$\mathcal{L}$ is a general $d$-dimensional linear subspace of
$S^{2}{\mathbb{C}}^{n}$.

Thus, our main result concerns a very basic algebro-geometric object: the
degree of the variety obtained by inverting all symmetric matrices in a
general linear space. In Section 4, we confirm the following conjecture of
Sturmfels and Uhler [38, p. 611], [22, Conjecture 2.8]. This theorem is
actually a corollary of a similar polynomiality result for the _algebraic
degree of semidefinite programming_ , which will be discussed in the next
section.

######  Theorem 1.3.

For any fixed positive integer $d$, the ML-degree $\phi(n,d)$ is polynomial in
$n$.

Astonishingly, it appears that the numbers $\phi(n,d)$ were studied for the
last 150 years! In 1879 Schubert presented his fundamental results on quadrics
satisfying various tangency conditions [31]. His contributions shaped the
field of enumerative geometry, inspiring many mathematicians for centuries to
come. A nondegenerate quadric being given, the set of its tangent hyperplanes
(its projective dual, in modern language) is nothing else than the inverse
quadric. This implies that $\phi(n,d)$ is also the solution to the following
enumerative problem:

> What is the number of nondegenerate quadrics in $n$ variables, passing
> through ${{n+1}\choose 2}-d$ general points and tangent to $d-1$ general
> hyperplanes?

In modern language, such problems can be solved by performing computations in
the cohomology ring of the _variety of complete quadrics_. This is now a
classical topic with many beautiful results [32, 33, 42, 43, 8, 9, 17, 7, 41,
20]. In particular, the cohomology ring has been described by generators and
relations, and algorithms have been devised that allow to compute any given
intersection number. But this only applies for $n$ fixed. Algebraic statistics
suggested to change the perspective and to fix $d$ instead of $n$. This
explains, in a way, why the polynomiality property of $\phi(n,d)$ is only
proved now.

### Semidefinite programming and projective duality

The second domain of mathematics that inspired our research is semidefinite
programming (SDP), a very important and effective subject in optimization
theory. The goal is to study linear optimization problems over spectrahedra.
This subject is a direct generalization of linear programming, that is
optimization of linear functions over polyhedra. For a short introduction to
the topic we refer to [23, Chapter 12].

The coordinates of the optimal solution for an SDP problem, defined over
rational numbers, are algebraic numbers. Their algebraic degree is governed by
_the algebraic degree of semidefinite programming_. For more information we
refer to the fundamental article [25]. To stress the importance of this degree
let us just quote this paper:

> _”The algebraic degree of semidefinite programming addresses the
> computational complexity at a fundamental level. To solve the semidefinite
> programming exactly essentially reduces to solve a class of univariate
> polynomial equations whose degrees are the algebraic degree._ ”

Let us provide a precise definition of the algebraic degree of SDP, in the
language of algebraic geometry, without referring to optimization. (However,
the fact that this definition is correct is actually a nontrivial result [25,
Theorem 13].)

######  Definition 1.4.

For $0<m<\binom{n+1}{2}$ and $0<r<n$, let $\mathcal{L}\subset
S^{2}{\mathbb{C}}^{n}$ be a general linear space of symmetric matrices, of
(affine) dimension $m+1$, and let $SD_{m}^{r,n}\subset\mathbb{P}(\mathcal{L})$
denote the projectivization of the cone of matrices of rank at most $r$ in
$\mathcal{L}$. The _algebraic degree of semidefinite programming_
$\delta(m,n,r)$ is the degree of the projective dual $(SD_{m}^{r,n})^{*}$ of
S$D_{m}^{r,n}$ if this dual is a hypersurface, and zero otherwise.

Projective duality is a very classical topic, to which a huge literature has
been devoted. Computing the degree of a dual variety is well-known to be very
hard, especially when the variety in question is singular, which is often the
case for our $SD_{m}^{r,n}$. Nevertheless, Ranestad and Graf von Bothmer [14]
suggested to use conormal varieties, and managed to obtain an algebraic
expression for $\delta(m,n,r)$ in terms of what we call the Lascoux
coefficients. These are integer coefficients that govern the Segre classes of
the symmetric square of a given vector bundle; algebraically, they are defined
by the formal identity

$\prod_{1\leq i\leq j\leq
s}\frac{1}{1-(x_{i}+x_{j})}=\sum_{I}\psi_{I}s_{\lambda(I)}(x_{1},\ldots,x_{s}),$

where the sum is over the increasing sets $I=(i_{1}<i_{2}<\cdots<i_{s})$ of
nonnegative integers, $\lambda(I)=(i_{s}-s+1,\ldots,i_{2}-1,i_{1})$ is the
associated partition, and $s_{\lambda(I)}(x_{1},\ldots,x_{s})$ the
corresponding Schur function in the variables $x_{1},\ldots,x_{s}$. These
coefficients were introduced and studied in [17, 26], whose influence on our
work cannot be underestimated. Graf von Bothmer and Ranestad found a formula
for $\delta(m,n,r)$ in terms of the Lascoux coefficients (see 3.7). Diving
into the combinatorics of those coefficients, in Section 4 we prove the
following polynomiality result:

######  Theorem 1.5.

For any fixed $m,s>0$, the function $\delta(m,n,n-s)$ is a polynomial in $n$.

Moreover, in Section 5 we confirm [25, Conjecture 21], providing another
explicit formula for $\delta(m,n,r)$, and another proof of the above theorem.

######  Theorem 1.6.

(NRS, Conjecture 21) Let $m,n,s$ be positive integers. Then

$\delta(m,n,n-s)=\sum_{\sum I\leq m-s}(-1)^{m-s-\sum
I}\psi_{I}b_{I}(n)\binom{m-1}{m-s-\sum I}$

where the sum goes trough all sets of nonnegative integers of cardinality $s$.

In this formula, $\Sigma I=i_{1}+\cdots+i_{s}$, and $b_{I}(n)$ is a polynomial
function of $n$ defined in Section 5. Actually, $b_{I}(n)$ is obtained by
evaluating a Q-Schur polynomial on $n$ identical variables; by the work of
Stembridge [35], it counts certain shifted tableaux of shape determined by
$I$, numbered by integers not greater than $n$. This also proves the
polynomiality of the ML-degree, since elementary relations in the cohomology
ring of the variety of complete quadrics imply the fundamental identity (see
3.6):

$\phi(n,d)=\frac{1}{n}\sum_{\binom{s+1}{2}\leq d}s\delta(d,n,n-s).$

Our approach also applies to linear spaces of general square matrices or of
skew-symmetric matrices, and allows to obtain closed formulas for the dual
degrees of their determinantal loci. In particular, the analogues of the Nie-
Ranestad-Sturmfels conjecture hold true. This means in particular that these
dual degrees can be computed by the class formula, essentially as if those
determinantal loci were always smooth (which is certainly not the case in
general!). Since the dual degree is well-known to be extremely sensitive to
singularities, it would be very interesting to have a conceptual explanation
of this phenomenon.

### Computations

The proofs of the Theorems 1.3, 1.5, 1.6 give rise to explicit polynomial
formulas for $\delta(m,n,n-s)$ and $\phi(n,d)$, which can be evaluated using
software. We illustrate this on a Sage worksheet, available on

https://mathrepo.mis.mpg.de/.

So far the exact formula for $\phi(n,d)$ was only known for $d\leq 5$ [6, 36,
38, 22]. We compute it explicitly for $d\leq 50$, confirming in particular
[22, Conjecture 5.1].

## Acknowledgements

We would like to thank Andrzej Weber and Jarosław Wiśniewski for many
interesting discussions. We are grateful to Piotr Pragacz for important
remarks about the article.

##  2\. Notation and preliminaries

###  2.A. Partitions, Schur polynomials

######  Notation 2.1.

For a set of nonnegative integers $I=\\{i_{1},\dots,i_{r}\\}$, we assume
$i_{1}<\cdots<i_{r-1}<i_{r}$ and we define the corresponding partition

$\lambda(I):=(i_{r}-(r-1),i_{r-1}-(r-2),\dots,i_{2}-1,i_{1}).$

Analogously, for a partition $\lambda=(\lambda_{1},\dots,\lambda_{r})$, which
means $\lambda_{1}\geq\cdots\geq\lambda_{r}\geq 0$ (zeroes are allowed), we
define the corresponding set

$I(\lambda):=\\{\lambda_{r},\lambda_{r-1}+1,\dots,\lambda_{2}+r-2,\lambda_{1}+r-1\\}.$

The length of a partition $\lambda$, (i.e. the number of nonzero entries) will
be denoted by $\mathop{\rm length}\nolimits(\lambda)$. By $|\lambda|$ we
denote the size of the partition $\sum_{i=1}^{r}\lambda_{r}$. By
$\tilde{\lambda}$ we denote the partition conjugate to $\lambda$, e.g.
$\widetilde{(3,1)}=(2,1,1)$.

We will abbreviate $\\{0,\dots,n-1\\}$ to $[n]$. Let $\sum
I:=i_{1}+\dots+i_{r}$ denote the sum of elements of $I$ and $\\#I=r$ its
cardinality. For two sets $I=\\{i_{1},\dots,i_{r}\\}$ and
$J=\\{j_{1},\dots,j_{r}\\}$ we say that $I\leq J$ if $i_{k}\leq j_{k}$ for all
$1\leq k\leq r$.

######  Notation 2.2.

For a partition $\lambda$ we denote by $s_{\lambda}$ the corresponding Schur
polynomial.

######  Definition 2.3.

Let $I,J$ be two sets of nonnegative integers of cardinality $r$. We define
the numbers $s_{I,J}$ to be the unique integers which satisfy the polynomial
equation

$s_{\lambda(I)}(x_{1}+1,\dots,x_{r}+1)=\sum_{J\leq
I}s_{I,J}s_{\lambda(J)}(x_{1},\dots,x_{r}).$

Note that since $s_{\lambda(I)}$ is a homogeneous polynomial, we also have the
identity

$s_{\lambda(I)}(x_{1}-1,\dots,x_{r}-1)=\sum_{J\leq I}(-1)^{\sum I-\sum
J}s_{I,J}s_{\lambda(J)}(x_{1},\dots,x_{r}).$

As a consequence, the triangular matrices $(s_{I,J})_{I,J}$ and $((-1)^{\sum
I-\sum J}s_{I,J})_{I,J}$ are inverses of each other.

######  Lemma 2.4.

Let $I=\\{i_{1},\dots,i_{r}\\}$ and $J=\\{j_{1},\dots,j_{r}\\}$ be two sets of
nonnegative integers. Let $M_{I,J}=(m_{kl})$ be the $r\times r$ matrix with
$m_{kl}=\binom{i_{k}}{j_{l}}$. Then

  * a)

$s_{I,J}=\mathop{\rm det}\nolimits(M_{I,J})$

  * b)

$s_{(d)}(x_{1}+1,\dots,x_{r}+1)=\sum_{i=0}^{d}\binom{d+r-1}{d-i}s_{(i)}(x_{1},\dots,x_{r})$

###### Proof.

Part a) is proved in [19, Section I.3, example 10]. In particular, it implies

$s_{[r+d],[r+i]}=\binom{d+r-1}{r+i-1}=\binom{d+r-1}{d-i}.$

From this, the equation in part b) becomes the defining equation for
$s_{I,J}$.

###  2.B. Lascoux coefficients

######  Definition 2.5.

We define the _Lascoux coefficients_ $\psi_{I}$ by the following formula:

$s_{(d)}(\\{x_{i}+x_{j}\mid 1\leq i\leq j\leq
r\\})=\sum_{\tiny\begin{matrix}\lambda(I)\vdash d\\\
\\#I=r\end{matrix}}\psi_{I}s_{\lambda(I)}(x_{1},\ldots,x_{r}),$

Here $s_{(d)}$ is the complete symmetric polynomial of degree $d$, in the
$\binom{r+1}{2}$ variables $x_{i}+x_{j}$. Hence, the coefficients $\psi_{I}$
appear in the expansion in the Schur basis, of the complete symmetric
polynomial evaluated at sums of variables.

Equivalently, the Lascoux coefficients appear in the expansion of the $d$-th
Segre class of the second symmetric power of any rank $r$ vector bundle in
terms of its Schur classes. In particular, for the universal bundle
$\mathcal{U}$ over the Grassmannian $G(r,n)$ for $n\geq r+d$,

$Seg_{d}(S^{2}\mathcal{U})=\sum_{\tiny\begin{matrix}\lambda(I)\vdash d\\\
\\#I=r\end{matrix}}{\psi_{I}\sigma_{\lambda(I)}},$

where $\sigma_{\lambda}$ denote the Schubert classes in the Chow ring of the
Grassmannian. (For $r\leq n<r+d$ the identity is still true, but some of the
Schubert classes $\sigma_{\lambda(I)}$ will be zero.)

######  Example 2.6.

Let us consider $r=2$ and $n=4$, i.e. the Grassmannian $G(2,4)$. The rank two
universal vector bundle $\mathcal{U}$ has two Chern roots $x_{1},x_{2}$.
Recall that the cohomology ring of $G(2,4)$ is six-dimensional with basis
corresponding to Young diagrams contained in the $2\times 2$ square. We have
formal equalities:

$x_{1}+x_{2}=-\yng(1),\quad x_{1}\cdot x_{2}=\yng(1,1).$

The Chern roots of $S^{2}\mathcal{U}$ are $2x_{1},x_{1}+x_{2},2x_{2}$.
Computing the elementary symmetric polynomials in those we obtain the three
respective Chern classes:

$-3\,\yng(1),\quad 2\,\yng(2)+6\,\yng(1,1),\quad-4\,\yng(2,1).$

By inverting the Chern polynomial we obtain the Segre classes:

$3\,\yng(1),\quad\mathbf{7}\,\yng(2)+\emph{3}\,\yng(1,1),\quad
10\,\yng(2,1),\quad 10\,\yng(2,2).$

Their coefficients are the Lascoux coefficients, namely:

$\psi_{0,2}=3,\quad\psi_{0,3}=\mathbf{7},\quad\psi_{1,2}=\emph{3},\quad\psi_{1,3}=10,\quad\psi_{2,3}=10.$

We use boldface and emphasis above and below to indicate the same numbers. We
may also compute them by expanding complete symmetric polynomials, where now
$x_{1},x_{2}$ are simply formal variables.

$s_{(2)}(2x_{1},x_{1}+x_{2},2x_{2})=7x_{1}^{2}+7x_{2}^{2}+10x_{1}x_{2}=$
$=7(x_{1}^{2}+x_{1}x_{2}+x_{2}^{2})+3x_{1}x_{2}=\mathbf{7}s_{(2,0)}(x_{1},x_{2})+\emph{3}s_{(1,1)}(x_{1},x_{2}).$

We note that Lascoux coefficients appear in many publications with different
notations. In particular one needs to be careful with the shift:
$\psi_{\\{j_{1},\ldots,j_{r}\\}}$ as defined above equals
$\psi_{\\{j_{1}+1,\ldots,j_{r}+1\\}}$ in [14]. On the other hand our notation
is consistent with [16, 25].

The lemma below gives a closed formula for the Lascoux coefficients, in terms
of Pfaffians.

######  Lemma 2.7 ([26, Prop. 7.12]).

Let $I=\\{i_{1},\dots,i_{r}\\}$ be a set of nonnegative integers. If $r=1,2$
then $\psi_{I}$ is given by $\psi_{\\{i\\}}=2^{i}$ and
$\psi_{\\{i,j\\}}=\sum_{k=i+1}^{j}\binom{i+j}{k}$ respectively. For $r>2$,
$\psi_{I}$ can be computed as

$\psi_{I}=\mathop{\rm Pf}\nolimits(\psi_{\\{i_{k},i_{l}\\}})_{0<k<l\leq
n}\text{ for even }r,$ $\psi_{I}=\mathop{\rm
Pf}\nolimits(\psi_{\\{i_{k},i_{l}\\}})_{0\leq k<l\leq n}\text{ for odd }r,$

where $\psi_{\\{i_{0},i_{k}\\}}:=\psi_{\\{i_{k}\\}}.$

###  2.C. SDP-degree and ML-degree

Recall the Definitions 1.2 and 1.4 of the ML-degree $\phi(n,d)$ and the SDP-
degree $\delta(m,n,r)$.

######  Remark 2.8.

For our polynomiality results in Section 4, it will be useful to extend the
definitions of $\phi$ and $\delta$ as follows:

  * •

For $d>\binom{n+1}{2}$, we put $\phi(n,d)=0$.

  * •

For $m=0$ and $r<n$, we define $\delta(0,n,r)=0$.

  * •

For $m\geq\binom{n+1}{2}$ or $s\geq n$, we put $\delta(m,n,n-s)=0$, with one
exception: in the case $m=\binom{n+1}{2}$ and $s=n$, we define
$\delta(\binom{n+1}{2},n,0)=1$. See also 3.9.

Now $\phi(n,d)$ is defined for all $n,d>0$, and $\delta(m,n,n-s)$ is defined
for all $m,n,s>0$.

For later reference, we recall the description of $\delta(m,n,r)$ in terms of
the bidegrees of a conormal variety:

######  Theorem 2.9 ([25, Theorem 10]).

Let $Z_{r}\subseteq{\mathbb{P}}(S^{2}V)\times{\mathbb{P}}(S^{2}V^{*})$ be the
conormal variety to the variety $SD^{r}\subseteq{\mathbb{P}}(S^{2}V)$ of
matrices of rank at most $r$. Explicitely, $Z_{r}$ consisit of pairs of
symmetric matrices $(X,Y)$, up to scalars, with $\mathop{\rm rk}\nolimits
X\leq r$, $\mathop{\rm rk}\nolimits Y\leq n-r$, and $X\cdot Y=0$. Then the
multidegree of $Z_{r}$ is given by

$[Z_{r}]=\sum_{m=0}^{\binom{n+1}{2}}\delta(m,n,r)H_{1}^{m}H_{n-1}^{\binom{n+1}{2}-m}.$

Here $H_{1}$ and $H_{n-1}$ denote the pull-backs of the hyperplane classes
from ${\mathbb{P}}(S^{2}V)$ and ${\mathbb{P}}(S^{2}V^{*})$.

######  Remark 2.10.

From this description, we immediately get the following duality relation (see
also [25, Proposition 9])

$\delta(m,n,n-s)=\delta(\binom{n+1}{2}-m,n,s).$

##  3\. ML-degrees via complete quadrics

###  3.A. Spaces of complete quadrics

Let $V$ be a $n$-dimensional vector space over ${\mathbb{C}}$. The space of
complete quadrics $CQ(V)$ is a particular compactification of the space of
smooth quadrics in ${\mathbb{P}}(V)$, or equivalently, of the space of
invertible symmetric matrices (up to scalar)
${\mathbb{P}}(S^{2}(V))^{\circ}\subset{\mathbb{P}}(S^{2}(V))$. The space of
complete quadrics $CQ(V)$ has several equivalent descriptions, below we will
describe some of them. For more information we refer the reader to [17, 41,
20].

For $A\in S^{2}(V)$ let $\bigwedge^{k}A\in S^{2}(\bigwedge^{k}V)$ be the
corresponding form on $\bigwedge^{k}V$. If we view $A$ as a symmetric matrix,
then $\bigwedge^{k}A$ is the matrix of $k\times k$ minors of $A$. In
particular, $\bigwedge^{n-1}A$ is the inverse of $A$ up to scaling (by the
determinant).

######  Definition 3.1.

The space of complete quadrics $CQ(V)$ is the closure of
$\phi({\mathbb{P}}(S^{2}(V))^{\circ})$, where

$\phi:{\mathbb{P}}(S^{2}(V))^{\circ}\to{\mathbb{P}}(S^{2}(V))\times{\mathbb{P}}\left(S^{2}\left(V\wedge
V\right)\right)\times\ldots\times{\mathbb{P}}\left(S^{2}\left(\bigwedge^{n-1}V\right)\right)$

is given by

$[A]\mapsto\left([A],[\bigwedge^{2}A],\ldots,[\bigwedge^{n-1}A]\right).$

To simplify the notation we will also denote $CQ(V)$ by $CQ_{n}$.

The natural projection to the $j$-th factor induces a regular map

$\pi_{j}:CQ(V)\to{\mathbb{P}}\left(S^{2}\left(\bigwedge^{j}V\right)\right).$

In particular, the map $\pi_{1}:CQ(V)\to{\mathbb{P}}(S^{2}(V))$, which is an
isomorphism on $\phi({\mathbb{P}}(S^{2}(V))^{\circ})$, can be described as a
sequence of blow-downs. This provides the second description of the space of
complete quadrics.

######  Definition 3.2.

The space of complete quadrics $CQ(V)$ is the successive blow-up of
${\mathbb{P}}(S^{2}(V))$:

$CQ(V)=Bl_{\widetilde{D}^{n-1}}Bl_{\widetilde{D}^{n-2}}\ldots
Bl_{D^{1}}{\mathbb{P}}(S^{2}(V)),$

where $\widetilde{D}^{i}$ is the proper transform of the space of symmetric
matrices of rank at most $i$ under the previous blow-ups.

######  Theorem 3.3 ([43, Theorem 6.3]).

Definitions 3.1 and 3.2 of the space of complete quadrics are equivalent.

The space of complete quadrics admits other descriptions, we would like to
mention two of them. The first one describes the space of complete quadrics as
an equivariant compactification of the space of invertible symmetric matrices,
which is a spherical homogeneous space:

${\mathbb{P}}(S^{2}(V))^{\circ}\simeq\mathop{\rm
SL}\nolimits_{n}/N(\mathop{\rm SO}\nolimits_{n}),$

where $N(\mathop{\rm SO}\nolimits_{n})$ is the normalizer of $\mathop{\rm
SO}\nolimits_{n}$. The second description realises the space of complete
quadrics as a subvariety of the Kontsevich moduli space of stable maps to the
Lagrangian Grassmannian, see for details [41, 22].

The space of complete quadrics has two natural series of special classes of
divisors. The first series $S_{1},\ldots,S_{n-1}$ is given by the classes of
the (strict transforms) of the exceptional divisors $E_{1},\ldots,E_{n-1}$ of
the successive blow-ups in Definition 3.2 (which are precisely the
$\mathop{\rm SL}\nolimits_{n}$-invariant prime divisors on $CQ(V)$). The
second series $L_{1},\ldots,L_{n-1}$ is obtained by pulling back the
hyperplane classes by the projections $\pi_{1},\ldots,\pi_{n-1}$.

######  Proposition 3.4.

The classes $L_{1},\ldots,L_{n-1}$ form a basis of $\mathop{\rm
Pic}\nolimits(CQ(V))$, in which the classes $S_{1},\ldots,S_{n-1}$ are given
by the relations

$S_{i}=-L_{i-1}+2L_{i}-L_{i+1},$

with $L_{0}=L_{n}:=0$.

###### Proof.

These relations were already known to Schubert [30]. For a modern treatment,
see for example [20, Proposition 3.6 and Theorem 3.13].

The inverse relations are given by the $(n-1)\times(n-1)$ matrix $M$ given by:

$M_{i,j}=\min(i,j)-\frac{ij}{n},$

in particular we have:

$nL_{1}=(n-1)S_{1}+(n-2)S_{2}+\ldots+S_{n-1},$ (3.1)
$nL_{n-1}=S_{1}+2S_{2}+\ldots+(n-1)S_{n-1}.$

###  3.B. Intersection theory

We are ready to relate the computation of the $M\\!L$-degree and of the
algebraic degree of semidefinite programming to the intersection theory of
$CQ(V)$.

######  Proposition 3.5.

The $M\\!L$-degree and the algebraic degree of semidefinite programming can be
computed as the following intersection numbers on $CQ(V)$:

$\phi(n,d)=\int_{CQ_{n}}L_{1}^{\binom{n+1}{2}-d}L_{n-1}^{d-1},$
$\delta(m,n,r)=\int_{E_{r}}L_{1}^{\binom{n+1}{2}-m-1}L_{n-1}^{m-1}=\int_{CQ_{n}}S_{r}L_{1}^{\binom{n+1}{2}-m-1}L_{n-1}^{m-1}.$

###### Proof.

Since the morphisms $\pi_{1}$ and $\pi_{n-1}$ resolve the inversion map
${\mathbb{P}}(S^{2}V)\dashrightarrow{\mathbb{P}}(S^{2}V^{*})$, we can compute
the degree of $\mathcal{L}^{-1}$, for
$\mathcal{L}\subseteq{\mathbb{P}}(S^{2}V^{*})$ a general $d-1$-dimensional
linear subspace, as
$\pi_{1}^{*}(H_{1}^{\binom{n+1}{2}-d})\pi_{n-1}^{*}(H_{n-1}^{d-1})$, where
$H_{1}$ and $H_{n-1}$ are hyperplane classes in ${\mathbb{P}}(S^{2}V)$ and
${\mathbb{P}}(S^{2}V^{*})$ respectively. Indeed the factor
$H_{1}^{\binom{n+1}{2}-d}$ imposes $\binom{n+1}{2}-d$ linear conditions to
matrices in ${\mathbb{P}}(S^{2}V)$, defining a linear subspace
${\mathbb{P}}(\mathcal{L})$ of projective dimension $d-1$, while the factor
$H_{n-1}^{d-1}$ imposes $d-1$ linear conditions on
${\mathbb{P}}(\mathcal{L}^{-1})$, hence computing the degree $\phi(n,d)$.

For $\delta(m,n,r)$, the main observation is that $E_{r}$ is birationally
isomorphic to the conormal variety $Z_{r}$. Indeed, the inductive properties
of the spaces of complete quadrics imply that $E_{r}$ can be described as a
space of relative complete quadrics; more precisely

$E_{r}=CQ_{r}(\mathcal{U})\times_{G(r,n)}CQ_{n-r}(\mathcal{Q}^{*}).$

The equality above may be derived from the quotient construction of the
variety of complete quadrics, cf. [22], [41]. In particular $E_{r}$ is
birationally isomorphic to
$Y_{r}={\mathbb{P}}(S^{2}\mathcal{U})\times_{G(r,n)}{\mathbb{P}}(S^{2}\mathcal{Q}^{*})$.
As observed for example in [14], this is also a smooth model for the conormal
variety $Z_{r}$. As the divisors $L_{i}$ are base point free, by 2.9

$\delta(m,n,r)=\int_{Z_{r}}L_{1}^{\binom{n+1}{2}-m-1}L_{n-1}^{m-1}=\int_{Y_{r}}L_{1}^{\binom{n+1}{2}-m-1}L_{n-1}^{m-1}$

can be computed on $E_{r}$, and this implies our claim.

Since $S_{r}$ projects in ${\mathbb{P}}(S^{2}(V))$ to the locus of matrices of
rank at most $r$, we must have $S_{r}L_{1}^{\binom{n+1}{2}-m-1}=0$ when $m$ is
smaller that the codimension of this locus. Similarly $S_{r}L_{n-1}^{m-1}=0$
when $m$ is big enough. One can deduce that the following _Pataki
inequalities_ are necessary and sufficient conditions for $\delta(m,n,r)$ to
be nonzero [25, Proposition 5 and Theorem 7]:

(3.2) ${{n-r+1}\choose{2}}\leq m\leq{{n+1}\choose{2}}-{{r+1}\choose{2}}.$

We can then use (3.1) to write the ML-degree in terms of the SDP-degree:

######  Corollary 3.6.

For any $n,d>0$, the following fundamental relation does hold:

$\phi(n,d)=\frac{1}{n}\sum_{1\leq\binom{s+1}{2}\leq d}{s\delta(d,n,n-s)}.$

###### Proof.

First, for $d>\binom{n+1}{2}$, we have both sides equal to $0$, and for
$d=\binom{n+1}{2}$ both sides equal $1$ (see 2.8), so the relation holds. For
$1\leq d<\binom{n+1}{2}$, we can write

$\displaystyle\phi(n,d)$
$\displaystyle=\int_{CQ_{n}}L_{1}^{\binom{n+1}{2}-d}L_{n-1}^{d-1}$ (3.3)
$\displaystyle=\frac{1}{n}\int_{CQ_{n}}L_{1}^{\binom{n+1}{2}-d-1}L_{n-1}^{d-1}\sum_{s=1}^{n-1}{sS_{n-s}}$
$\displaystyle=\frac{1}{n}\sum_{s=1}^{n-1}{s\delta(d,n,n-s)}=\frac{1}{n}\sum_{1\leq\binom{s+1}{2}\leq
d}{s\delta(d,n,n-s)}.$

The last equality follows from the Pataki inequalities, since
$\delta(d,n,n-s)=0$ whenever $\binom{s+1}{2}>d$.

All our computations can now be reduced to the intersection theory of the
Grassmannians.

######  Theorem 3.7 ([14, Theorem 1.1]).

For $0<m<\binom{n+1}{2}$ and $0<r<n$,

$\delta(m,n,r)=\sum_{\begin{subarray}{c}I\subset[n]\\\ \\#I=n-r\\\
\sum{I}=m-n+r\end{subarray}}{\psi_{I}\psi_{[n]\setminus I}}$

###### Idea of proof.

As already mentioned, $\delta(m,n,r)$ can be computed as an intersection
number on $Y_{r}$, which is a fiber bundle over the Grassmannian $G(r,n)$. By
push-forward, one obtains

$\delta(m,n,r)=\int_{G(r,n)}\mathop{\rm
Seg}\nolimits_{\left(\binom{n+1}{2}-m-\binom{r+1}{2}\right)}(S^{2}\mathcal{U})\mathop{\rm
Seg}\nolimits_{\left(m-\binom{n-r+1}{2}\right)}(S^{2}\mathcal{Q}^{*}).$

We then obtain the theorem by expanding these Segre classes, and using the
fundamental duality properties of Schubert classes.

######  Remark 3.8.

Recall that our definition of $\psi_{I}$ is shifted w.r.t. [14], which
explains why our formula looks slightly different.

######  Remark 3.9.

One can easily verify that the above formula is still true for the extended
definition of $\delta$ from 2.8. The only nontrivial case is
$\delta\left(\binom{n+1}{2},n,0\right)=\psi_{[n]}\psi_{[n]\setminus[n]}=1$.

###  3.C. Representation theory

In this subsection we will establish a formula which expresses the ML-degree
as a linear combination of dimensions of irreducible representations of
$\mathop{\rm SL}\nolimits_{n}$. Our construction is based on the following
folklore lemma.

######  Lemma 3.10.

Let $X$ be a smooth complete $N$-dimensional algebraic variety, and
$D_{1},D_{2}$ two divisors on $X$. Then the following identity holds:

$\int_{X}D_{1}^{i}D_{2}^{N-i}=\chi\left((1-\mathcal{O}(-D_{1}))^{i}(1-\mathcal{O}(-D_{2}))^{N-i}\right),$

where $\chi$ denotes the holomorphic Euler characteristic.

###### Proof.

By the additivity and multiplicativity properties of the Chern character we
have

$\mathop{\rm ch}\nolimits(1-{\mathcal{O}}(-D_{i}))=\sum_{k\geq
1}(-1)^{k+1}\frac{D_{i}^{k}}{k!},$

and therefore

$\mathop{\rm
ch}\nolimits\\!\left((1-\mathcal{O}(-D_{1}))^{i}(1-\mathcal{O}(-D_{2}))^{N-i}\right)=\\!\left(\sum_{k\geq
1}(-1)^{k+1}\frac{D_{1}^{k}}{k!}\right)^{i}\\!\\!\\!\left(\sum_{k\geq
1}(-1)^{k+1}\frac{D_{2}^{k}}{k!}\right)^{N-i}\\!\\!\\!\\!\\!\\!=\int_{X}D_{1}^{i}D_{2}^{N-i}.$

Finally, by the Riemann-Roch theorem we get

$\chi\left((1-\mathcal{O}(-D_{1}))^{i}(1-\mathcal{O}(-D_{2}))^{N-i}\right)=\int_{X}D_{1}^{i}D_{2}^{N-i}\mathop{\rm
td}\nolimits(X)=\int_{X}D_{1}^{i}D_{2}^{N-i}.$

We are going to apply Lemma 3.10 to the computation of the ML-degree

$\phi(n,d)=\int_{CQ_{n}}L_{1}^{N+1-d}L_{n-1}^{d-1},$

where $N=\binom{n+1}{2}-1$ denotes the dimension of the variety of complete
quadrics $CQ_{n}$. We will denote by $\Lambda$ the character lattice of
$\mathop{\rm SL}\nolimits_{n}$, and for a fundamental $\lambda\in\Lambda$, we
will denote by $V_{\lambda}$ the corresponding irreducible representation of
$\mathop{\rm SL}\nolimits_{n}$. We will also denote by
$\alpha_{1},\ldots,\alpha_{n-1}$ and $\omega_{1},\ldots,\omega_{n-1}$ the
simple roots and fundamental weights of $\mathop{\rm SL}\nolimits_{n-1}$,
respectively.

For a character $\lambda\in\Lambda$, let us define a $\mathop{\rm
SL}\nolimits_{n}$-representation $W_{\lambda}$ by

$W_{\lambda}:=\bigoplus_{\nu\in\Lambda}V_{2\nu}^{*},$

where the sum is taken over dominant weights of the form
$\nu=\lambda-\sum_{i=1}^{n-1}k_{i}\alpha_{i}$, with
$k_{i}\in{\mathbb{Z}}_{\geq 0}$. In particular, $W_{\lambda}=0$ if $\lambda$
cannot be represented as a sum $\nu+\sum_{i=1}^{n-1}k_{i}\alpha_{i}$, with
$\nu$ a dominant weight and $k_{i}\in{\mathbb{Z}}_{\geq 0}$. For
$\lambda=(i-1)\omega_{1}+(j-1)\omega_{2}-\sum_{l=1}^{n-1}\omega_{l}$, we will
denote the representation $W_{\lambda}$ by $W_{i,j}^{n}$.

######  Theorem 3.11.

With the notation as above, the following identity holds:

$\phi(n,d)=1+\sum_{\begin{subarray}{c}0\leq i\leq N-d+1\\\ 0\leq j\leq d-1\\\
i+j>0\end{subarray}}(-1)^{i+j+N}\binom{N+1-d}{i}\binom{d-1}{j}\dim(W^{n}_{i,j}).$

###### Proof.

By Lemma 3.10 we have:

$\phi(n,d)=\chi((1-{\mathcal{L}}_{1}^{-1})^{N+1-d}(1-{\mathcal{L}}_{n-1}^{-1})^{d-1})=\\\
\sum_{\begin{subarray}{c}0\leq i\leq N+1-d\\\ 0\leq j\leq
d-1\end{subarray}}(-1)^{i+j}\binom{N+1-d}{i}\binom{d-1}{j}\chi({\mathcal{L}}_{1}^{-i}\otimes{\mathcal{L}}_{n-1}^{-j}).$

Now, both ${\mathcal{L}}_{1}$ and ${\mathcal{L}}_{n-1}$ are globally
generated, and since they are pull-backs of ample line bundles by birational
morphisms their Iitaka dimensions
$\kappa({\mathcal{L}}_{1})=\kappa({\mathcal{L}}_{n-1})=N$. By [3, Theorem
2.2], the cohomology of their negative powers must therefore vanish in degree
lower than the dimension, so that for $i\geq 0,j\geq 0$ and $i+j>0$

$\chi({\mathcal{L}}_{1}^{-i}\otimes{\mathcal{L}}_{n-1}^{-j})=(-1)^{N}h^{N}(CQ_{n},{\mathcal{L}}_{1}^{-i}\otimes{\mathcal{L}}_{n-1}^{-j})=(-1)^{N}h^{0}(CQ_{n},K_{CQ_{n}}\otimes{\mathcal{L}}_{1}^{i}\otimes{\mathcal{L}}_{n-1}^{j}).$

The canonical divisor $K_{CQ_{n}}$ of the space of complete quadrics is given
by ([18, Corollary 3])

$K_{CQ_{n}}=-L_{1}-L_{n-1}-\sum_{l=1}^{n-1}L_{l}.$

In particular, $CQ_{n}$ is Fano [20] and $\chi(CQ_{n},{\mathcal{O}})=1$.
Finally, the spaces of global sections of line bundles on complete quadrics
have been computed, as $\mathop{\rm SL}\nolimits_{n}$-representations, by De
Concini and Procesi. It is direct corollary of [8, Theorem 8.3] that

$H^{0}(CQ_{n},{\mathcal{L}}_{1}^{a_{1}}\otimes\ldots\otimes{\mathcal{L}}_{n-1}^{a_{n-1}})=W_{a_{1}\omega_{1}+\ldots+a_{n-1}\omega_{n-1}}.$

In particular,
$H^{0}(CQ_{n},K_{CQ_{n}}\otimes{\mathcal{L}}_{1}^{i}\otimes{\mathcal{L}}_{n-1}^{j})=W^{n}_{ij}$,
so our result follows.

######  Example 3.12.

Let us compute $\phi(3,1),\phi(3,2)$ and $\phi(3,3)$ with the help of Theorem
3.11. First we notice that:

$W^{3}_{i,0}=W^{3}_{0,i}=0\text{ for }i\leq 5,\quad
W^{3}_{i,1}=W^{3}_{1,i}=0\text{ for }i\leq 3.$

Therefore we get:

$\phi(3,1)=1+\sum_{\begin{subarray}{c}1\leq i\leq
5\end{subarray}}(-1)^{i+1}\binom{5}{i}\dim\left(W^{3}_{i,0}\right)=1;$
$\phi(3,2)=1+\sum_{\begin{subarray}{c}0\leq i\leq 4\\\ 0\leq j\leq 1\\\
i+j>0\end{subarray}}(-1)^{i+j+1}\binom{4}{i}\dim\left(W^{3}_{i,j}\right)=1+\dim(W^{3}_{4,1})=1+\dim
V_{0}=2;$ $\phi(3,3)=1+\sum_{\begin{subarray}{c}0\leq i\leq 3\\\ 0\leq j\leq
2\\\
i+j>0\end{subarray}}(-1)^{i+j+1}\binom{3}{i}\binom{2}{j}\dim\left(W^{3}_{i,j}\right)=1-3\cdot\dim(W^{3}_{2,2})+\dim(W^{3}_{3,2}),$

where $W^{3}_{2,2}=V_{0}$ and $W^{3}_{3,2}=V_{2\omega_{1}}^{*}$. By Weyl’s
dimension formula:

$\dim V_{i\omega_{1}+j\omega_{2}}=\frac{(i+1)(j+1)(i+j+2)}{2},$

we get $\phi(3,3)=1-3\cdot 1+6=4$.

######  Remark 3.13.

Our representation theoretic approach gives a closed formula for the ML-
degree. However, already for $n=4$, the computation analogous to Example 3.12
is quite involved. One reason why this computation is more complicated than
other formulas is that we obtain the answer to our intersection problem as a
virtual representation, not only its dimension. One could say that this gives
more information than we ask for.

##  4\. Polynomiality of the ML-degree

In this section and the next one, we present three proofs of the following
polynomiality result for the algebraic degree of semidefinite programming:

######  Theorem 4.1.

For any fixed $m,s>0$, the function $\delta(m,n,n-s)$ is a polynomial in $n$.
Moreover this polynomial vanishes at $n=0$.

As an immediate corollary, we obtain one of the main results of this paper:
the polynomiality of the ML-degree for linear concentration models. This
property was first conjectured by Sturmfels and Uhler [38] and confirmed in
small, special cases in [6, 36, 22].

######  Theorem 4.2.

For any fixed $d>0$, the function $\phi(n,d)$ is a polynomial for $n>0$.

###### Proof.

For all $n,d>0$, by Corollary 3.6, we have:

(4.1) $\phi(n,d)=\frac{1}{n}\sum_{1\leq\binom{s+1}{2}\leq
d}{s\delta(d,n,n-s)}.$

By 4.1 every term in the right hand side of (4.1) is a polynomial divisible by
$n$, hence the theorem follows.

Each of our proofs of 4.1 has its advantages. The first one is quite
elementary, being based on algebraic recursive formulas, which also have a
geometric meaning. It provides very efficient methods for explicit
computations. The second proof is more technical, however it allows to derive
the leading coefficients of the polynomials we study. The last one is simply a
corollary of the conjecture of Nie, Ranestad and Sturmfels that we prove in
Section 5.

Our first two proofs of 4.1 are based on the following theorem.

######  Theorem 4.3.

Let $I=\\{i_{1},\ldots,i_{r}\\}$ be a set of strictly increasing nonnegative
integers. For $n\geq 0$ the function:

$LP_{I}(n):=\begin{cases}\psi_{[n]\setminus I}&\text{ if }I\subseteq[n],\\\
0&\text{ otherwise}.\end{cases}$

is a polynomial.

Before we prove Theorem 4.3 let us note that it immediately implies Theorem
4.1. Indeed, by 3.7, we have

$\delta(m,n,n-s)=\sum_{\begin{subarray}{c}I\subset[n]\\\ \\#I=s\\\
\sum{I}=m-s\end{subarray}}{\psi_{I}\psi_{[n]\setminus
I}}=\sum_{\begin{subarray}{c}\\#I=s\\\
\sum{I}=m-s\end{subarray}}{\psi_{I}LP_{I}(n)}$

By Theorem 4.3, each of the summands is a polynomial in $n$ that vanishes for
$n=0$. Thus $\delta(m,n,n-s)$ is also a polynomial in $n$, which proves
Theorem 4.1, and hence Theorem 4.2.

In the remainder of this section, we will present two proofs of Theorem 4.3.
But let us first give a few examples.

######  Example 4.4.

By induction, one can check the following formulas for $LP_{I}$, when $I$ has
cardinality one or two:

$LP_{(i)}(n)=\binom{n}{j+1},\qquad LP_{(0,j)}(n)=j\binom{n+1}{j+2},$

and more generally, for $i<j$,

$LP_{(i,j)}(n)=\frac{(j-i)[n+1]_{j+2}}{(i+1)!(j+1)!(i+j+2)!}\sum_{d=0}^{i}(-1)^{d}a_{i,d}(i+j+1-d)![n]_{i-d},$

where $a_{i,d}=\prod_{k=0}^{d-1}(i-k)(i-k+1)$ and
$[n]_{d}=n(n-1)\cdots(n-d+1)$.

###  4.A. First proof

The following recursive relations are central for our first proof.

######  Lemma 4.5.

  1. (1)

For $j_{1}>0$ we have:

(4.2)
$\psi_{\\{j_{1},\ldots,j_{r}\\}}=(r+1)\psi_{\\{0,j_{1},\ldots,j_{r}\\}}-2\sum_{\ell=1}^{r}{\psi_{\\{0,j_{1},\ldots,j_{\ell}-1,\ldots,j_{r}\\}}},$

where the summation is over all $\ell$ for which $j_{\ell}-1>j_{\ell-1}$ and
we set $j_{0}:=0$.

  2. (2)

For $j_{1}=0$ we have:

(4.3) $\psi_{\\{j_{1},j_{2},\ldots,j_{r}\\}}=\sum_{j_{\ell}\leq
j^{\prime}_{\ell}<j_{\ell+1}}{\psi_{\\{j^{\prime}_{1},\ldots,j^{\prime}_{r-1}\\}}}.$

###### Proof.

The first formula is: [26, p. 446], [16, (A.15.7)] and [28, p. 163-166].

To prove the second formula, recall that $s_{(d)}$ is the complete homogeneous
symmetric polynomial of degree $d$, and that we have:

$s_{(d)}(\\{x_{i}+x_{j}\mid 1\leq i\leq j\leq
r\\})=\sum_{\tiny\begin{matrix}\lambda(I)\vdash d\\\
\\#I=r\end{matrix}}\psi_{I}s_{\lambda(I)}(x_{1},\ldots,x_{r}).$

Substituting $x_{r}=0$ we obtain:

$\sum_{i=0}^{d}s_{(i)}(\\{x_{i}+x_{j}\mid 1\leq i\leq j\leq
r-1\\})s_{(d-i)}(x_{1},\dots,x_{r-1})=$ $s_{(d)}(\\{x_{i}+x_{j}\mid 1\leq
i\leq j\leq
r-1\\},x_{1},\dots,x_{r-1})=\sum_{\tiny\begin{matrix}\lambda(I)\vdash d\\\
\mathop{\rm length}\nolimits(\lambda(I))\leq
r-1\end{matrix}}\psi_{I}s_{\lambda(I)}(x_{1},\ldots,x_{r-1}).$

We note that $\mathop{\rm length}\nolimits(\lambda(I))\leq r-1$ if and only if
$0\in I$. On the other hand we may apply Pieri’s rule to

$\sum_{i=0}^{d}s_{(i)}(\\{x_{i}+x_{j}\mid 1\leq i\leq j\leq
r-1\\})s_{(d-i)}(x_{1},\dots,x_{r-1})=$
$\sum_{i=0}^{d}\left(\sum_{\tiny\begin{matrix}\lambda(I)\vdash i\\\
\\#I=r-1\end{matrix}}\psi_{I}s_{\lambda(I)}(x_{1},\ldots,x_{r-1})\right)s_{(d-i)}(x_{1},\dots,x_{r-1}).$

Comparing the coefficients of Schur polynomials in both expressions gives the
formula.

###### First proof of Theorem 4.3.

We proceed by induction first on $\\#I$, then on $\sum{I}:=\sum_{i_{j}\in
I}i_{j}$. The base case is $I=\emptyset$, when $\psi_{\\{0,\ldots,n-1\\}}=1$.

For the induction step, fix $I$, and assume the theorem has been proven for
all $I^{\prime}$ with $\\#I^{\prime}<\\#I$, and for all $I^{\prime}$ with
$\\#I^{\prime}=\\#I$ and $\sum{I^{\prime}}<\sum{I}$. We consider two cases:

Case 1. $i_{1}=0$. We claim that for every $n\geq 0$,

$LP_{I}(n)=(n-r+1)LP_{I\setminus\\{0\\}}(n)-2\sum_{\ell:i_{\ell+1}>i_{\ell}+1}{LP_{I\setminus\\{0,i_{\ell}\\}\sqcup\\{i_{\ell}+1\\}}(n)},$

where for summation we formally assume $i_{r+1}=+\infty$. Indeed: if $n\leq
i_{r}$ then both sides are $0$, and if $n>i_{r}$ then the equation is
precisely Lemma 4.5 (1).

Case 2. $i_{1}>0$. We claim that for every $n\geq 0$,

$LP_{I}(n)-LP_{I}(n-1)=\sum_{J}{LP_{J}(n-1)},$

where the sum is over all $J\neq I$ of the form
$\\{i_{1}-\epsilon_{1},\ldots,i_{r}-\epsilon_{r}\\}$ with
$\epsilon_{\ell}\in\\{0,1\\}$. Again, if $n\leq i_{r}$ then both sides are
$0$, and if $n>i_{r}$ then the equation is precisely Lemma 4.5 (2).

In both cases, it follows that $LP_{I}$ is a polynomial.

###  4.B. Second proof

Our second proof is based on an explicit interpretation of $\psi_{I}$ as a sum
of minors in the Pascal triangle. We denote by $E$ the Pascal triangle matrix,
i.e. $E_{ij}=\binom{i}{j}$. We will always consider only finite submatrices of
$E$ so despite the fact that it is an infinite matrix there will be no
computations with infinite matrices.

######  Notation 4.6.

For an $n\times n$ matrix $A$ and sets $I,J\subset[n]$ we denote by $A_{I,J}$
the $\\#I\times\\#J$ matrix which is obtained from $A$ by taking rows indexed
by $I$ and columns indexed by $J$. Here we index rows and columns from 0. In
the case $I=J$ we write simply $A_{I,I}=A_{I}$.

For sets $K,C\subset{\mathbb{N}}$ with $\\#K=\\#C$ we denote $V(K,C)$ the
Vandermonde matrix with entries $V(K,C)_{ij}=k_{i+1}^{c_{j+1}}$. We also set
$V(K):=V(K,[\\#K])$, i.e. $V(K)_{ij}=k_{i+1}^{j}$.

For two sets $A,B\subset{\mathbb{N}}$ we denote by $\varepsilon^{A,B}$ the
sign of the permutation of $A\cup B$ determined by $A,B$ if they are disjoint.
If they are not, we define $\varepsilon^{A,B}=0$.

We begin with a characterization of $\psi_{I}$ as a sum of the minors of the
matrix $E$ which follows from [16, Proposition 2.8].

######  Proposition 4.7.

The following equality holds:

$\psi_{I}=\sum_{J\leq I}\mathop{\rm det}\nolimits(E_{I,J}).$

In what follows we will need the following lemma that may be easily proved by
induction.

######  Lemma 4.8.

Let $a,b$ be nonnegative integers.

  * a)

If $a>b$ then $\sum_{i=0}^{a}(-1)^{i}{a\choose i}i^{b}=0$.

  * b)

If $a=b$ then $\sum_{i=0}^{a}(-1)^{a-i}{a\choose i}i^{b}=a!$.

To compute special minors of the matrix $E$ we use the following lemma.

######  Lemma 4.9.

Let $I=\\{i_{1},\dots,i_{r}\\}\subset[n]$ be a set of nonnegative integers.
Then

$\mathop{\rm det}\nolimits E_{[n]\setminus[r],[n]\setminus
I}=\frac{\prod_{1\leq j<k\leq n-r}{(i_{k}-i_{j})}}{(r-1)!(r-2)!\dots
2!1!}=\frac{\mathop{\rm det}\nolimits(V(I))}{(r-1)!(r-2)!\dots 2!1!}$

###### Proof.

We fix $r$ and proceed by induction on $n$. The case $i_{r}<n-1$ is trivial.
In the case $i_{r}=n-1$ we express the determinant via Laplace expansion on
the $n$-th row, use the induction hypothesis and Lemma 4.8 to conclude.

Now we are able to present our second proof of Theorem 4.3.

###### Second proof of Theorem 4.3.

Let $\\#I=r$ and $m:=i_{r}+1$. First, assume $n\geq m$. We use the formula
from Proposition 4.7. We express the determinants $E_{[n]\setminus I,J}$ using
the Laplace expansion along the first $m-r$ rows, we choose the columns
indexed by set $L$. For the rest we use the Lemma 4.9. To simplify notation we
let $K:=[n]\setminus J$.

$\displaystyle\psi_{[n]\setminus I}$ $\displaystyle=\sum_{J\leq[n]\setminus
I}\mathop{\rm det}\nolimits(E_{[n]\setminus I,J})$
$\displaystyle=\sum_{J\leq[n]\setminus I}\sum_{\begin{subarray}{c}L\subseteq
J\\\ \\#L=m-r\end{subarray}}\varepsilon^{L,J\setminus L}\mathop{\rm
det}\nolimits(E_{[m]\setminus I,L})\mathop{\rm
det}\nolimits(E_{[n]\setminus[m],J\setminus L})$
$\displaystyle=\sum_{\begin{subarray}{c}\\#L=m-r\\\ L\leq[m]\setminus
I\end{subarray}}\mathop{\rm det}\nolimits(E_{[m]\setminus
I,L})\sum_{\begin{subarray}{c}\\#K=r\\\ K\cap L=\emptyset\\\
K\subset[n]\end{subarray}}\varepsilon^{L,[n]\setminus(K\cup L)}\mathop{\rm
det}\nolimits(E_{[n]\setminus[m],[n]\setminus(K\cup L)})$
$\displaystyle=\sum_{\begin{subarray}{c}\\#L=m-r\\\ L\leq[m]\setminus
I\end{subarray}}\mathop{\rm det}\nolimits(E_{[m]\setminus
I,L})\sum_{\begin{subarray}{c}\\#K=r\\\
K\subset[n]\end{subarray}}\varepsilon^{L,K}\varepsilon^{L,[n]\setminus
L}\frac{\mathop{\rm det}\nolimits(V(L\cup K))}{(m-1)!(m-2)!\dots 2!1!}$
$\displaystyle=\sum_{\begin{subarray}{c}\\#L=m-r\\\ L\leq[m]\setminus
I\end{subarray}}\frac{\varepsilon^{L,[n]\setminus L}\mathop{\rm
det}\nolimits(E_{[m]\setminus I,L})}{(m-1)!(m-2)!\dots
2!1!}\sum_{\begin{subarray}{c}\\#K=r\\\ K\subset[n]\end{subarray}}\mathop{\rm
det}\nolimits(V^{*}(L\cup K))$

where $V^{*}(L\cup K)$ is the matrix $V(L\cup K)$ where we first put the rows
indexed by $L$. Note that we may drop the assumption $L\leq[m]\setminus I$,
since otherwise $\mathop{\rm det}\nolimits(E_{[m]\setminus I,L})=0$.
Similarly, we can extend our sum and drop the condition $L\cap K=\emptyset$
since we add only zero terms. If we fix $L$ and denote the elements of $K$ by
$k_{1}<\dots<k_{r}$, then $\mathop{\rm det}\nolimits(V^{*}(L\cup K))$ is
clearly a polynomial in $k_{1},\dots,k_{r}$. Then

$\sum_{\begin{subarray}{c}\\#K=r\\\ K\subset[n]\end{subarray}}\mathop{\rm
det}\nolimits(V^{*}(L\cup K))=\sum_{0\leq k_{1}<\dots<k_{r}<n}\mathop{\rm
det}\nolimits(V^{*}(L\cup K))$

is a polynomial in $n$ for the fixed $L$. Moreover, the sum through $L$ does
not depend on $n$ and therefore also $\psi_{[n]\setminus I}$ is a polynomial
in $n$. Our computations are correct only for $n\geq m$. However, the last
expression makes sense and is a polynomial for all $n\geq 0$. Clearly, it is
equal $0$ for $n<m$. This proves the theorem.

With this approach we can even compute the leading coefficient of $LP_{I}$.
For this we will need two technical lemmas. The proof of the first one is
straightforward, e.g. by induction.

######  Lemma 4.10.

Let $a_{1},\dots,a_{r}$ be nonnegative integers. Then

$\sum_{0\leq k_{1}<\dots<k_{r}<n}k_{1}^{a_{1}}k_{2}^{a_{2}}\dots
k_{r}^{a_{r}}$

is a polynomial in $n$ of degree $\sum_{i=1}^{r}a_{i}+r$, with leading
coefficient $\frac{1}{(a_{1}+1)(a_{1}+a_{2}+2)\dots(a_{1}+\dots+a_{r}+r)}$.

######  Lemma 4.11.

The following identity of rational functions in $r$ variables holds:

$\sum_{\sigma\in
S_{r}}(-1)^{\sigma}\frac{1}{(x_{\sigma(1)})(x_{\sigma(1)}+x_{\sigma(2)})\dots(x_{\sigma(1)}+\dots+x_{\sigma(r)})}=\frac{\prod_{i>j}(x_{i}-x_{j})}{\prod_{i}x_{i}\prod_{i>j}(x_{i}+x_{j})}$

###### Proof.

We proceed by induction on $r$. It is easy to check that for $r=1,2$ the
statement holds. For $r>2$, we split the sum depending on $\sigma(r)$ and
apply the induction hypothesis to the partial sums:

$\displaystyle\sum_{\sigma\in
S_{r}}(-1)^{\sigma}\frac{1}{(x_{\sigma(1)})(x_{\sigma(1)}+x_{\sigma(2)})\dots(x_{\sigma(1)}+\dots+x_{\sigma(r)})}=$
$\displaystyle=\frac{1}{x_{1}+\dots+x_{r}}\sum_{k=1}^{r}\sum_{\begin{subarray}{c}\sigma\in
S_{r}\\\
\sigma(r)=k\end{subarray}}(-1)^{\sigma}\frac{1}{(x_{\sigma(1)})(x_{\sigma(1)}+x_{\sigma(2)})\dots(x_{\sigma(1)}+\dots+x_{\sigma(r-1)})}$
$\displaystyle=\frac{1}{x_{1}+\dots+x_{r}}\sum_{k=1}^{r}(-1)^{r-k}\frac{\prod_{i>j;i,j\neq
k}(x_{i}-x_{j})}{\prod_{i\neq k}x_{i}\prod_{i>j;i,j,\neq k}(x_{i}+x_{j})}$
$\displaystyle=\frac{1}{(x_{1}+\dots+x_{r})\prod_{i}x_{i}\prod_{i>j}(x_{i}+x_{j})}\sum_{k=1}^{r}(-1)^{r-k}x_{k}\prod_{i>j;i,j\neq
k}(x_{i}-x_{j})\prod_{i\neq k}(x_{i}+x_{k})$
$\displaystyle=\frac{1}{(x_{1}+\dots+x_{r})\prod_{i}x_{i}\prod_{i>j}(x_{i}+x_{j})}Q(x_{1},\dots,x_{r}),$

where $Q$ is a homogeneous polynomial of degree ${r\choose 2}+1$. Moreover,
$Q$ is skew-symmetric, that if we exchange values of $x_{i}$ and $x_{j}$ we
just change the sign. Therefore

$Q(x_{1},\dots,x_{r})=\prod_{i>j}(x_{i}-x_{j})R(x_{1},\dots,x_{r})$

for $R$ a symmetric polynomial of degree one. This implies that $R$ is a
multiple of $x_{1}+\cdots+x_{r}$. Finally, it is easy to check that the
coefficient of $x_{r}^{r}x_{r-1}^{r-2}x_{r-2}^{r-3}\dots x_{2}$ in $Q$ is $1$.
Therefore $R=x_{1}+\dots+x_{r}$ and the proof is complete.

######  Theorem 4.12.

The polynomial $LP_{I}$ has degree $\sum I+\\#I$. Its leading coefficient is
equal to

$\frac{\prod_{j>k}(i_{j}-i_{k})}{(i_{1}+1)!\dots(i_{r}+1)!\prod_{j>k}(i_{j}+i_{k}+2)}$

###### Proof.

We continue with the calculation from the second proof of Theorem 4.3. We do
Laplace expansion of Vandermonde by first $m-r$ rows. We get

$\displaystyle\sum_{\begin{subarray}{c}\\#L=m-r\\\ L\leq[m]\setminus
I\end{subarray}}\varepsilon^{L,[m]\setminus L}\mathop{\rm
det}\nolimits(E_{[m]\setminus I,L})\sum_{\begin{subarray}{c}\\#K=r\\\
K\subset[n]\end{subarray}}\mathop{\rm det}\nolimits(V^{*}(L\cup K))=$
$\displaystyle=\sum_{\begin{subarray}{c}\\#L=m-r\\\ L\leq[m]\setminus
I\end{subarray}}\varepsilon^{L,[m]\setminus L}\mathop{\rm
det}\nolimits(E_{[m]\setminus I,L})\sum_{\begin{subarray}{c}\\#C=m-r\\\
C\subset[m]\end{subarray}}\varepsilon^{C,[m]\setminus C}\mathop{\rm
det}\nolimits(V(L,C))\sum_{\begin{subarray}{c}\\#K=r\\\
K\subset[n]\end{subarray}}\mathop{\rm det}\nolimits(V(K,[m]\setminus C))$
$\displaystyle=\sum_{\begin{subarray}{c}\\#C=m-r\\\
C\subset[m]\end{subarray}}\varepsilon^{C,[m]\setminus
C}\sum_{\begin{subarray}{c}\\#L=m-r\\\ L\leq[m]\setminus
I\end{subarray}}\varepsilon^{L,[m]\setminus L}\mathop{\rm
det}\nolimits(E_{[m]\setminus I,L})\mathop{\rm
det}\nolimits(V(L,C))\sum_{\begin{subarray}{c}\\#K=r\\\
K\subset[n]\end{subarray}}\mathop{\rm det}\nolimits(V(K,[m]\setminus C))$
$\displaystyle=\sum_{\begin{subarray}{c}\\#C=m-r\\\
C\subset[m]\end{subarray}}\varepsilon^{C,[m]\setminus C}\mathop{\rm
det}\nolimits\left(\text{diag}(1,-1,\dots,(-1)^{m-1})E_{[m]\setminus
I,[m]}V([m],C)\right)\sum_{\begin{subarray}{c}\\#K=r\\\
K\subset[n]\end{subarray}}\mathop{\rm det}\nolimits(V(K,[m]\setminus C)).$

Consider the matrix $A:=\left(\text{diag}(1,-1,\dots,-1^{m-1})E_{[m]\setminus
I,[m]}V([m],C)\right)$. Let $[m]\setminus I=\\{b_{1},\dots,b_{m-r}\\}$,
$C=\\{c_{1},\dots,c_{m-r}\\}$, where, as always, we assume that the elements
of these sets are ordered increasingly. Notice that $c_{m-r}<b_{m-r}$ implies
that the last row of the matrix $A$ is 0 by Lemma 4.8 and so is $\mathop{\rm
det}\nolimits(A)$. In general, if $c_{i}<b_{i}$, then
$A_{[m-r]\setminus[i-1],[i]}=0$ and we also get $\mathop{\rm det}\nolimits
A=0$.

The necessary condition for $\mathop{\rm det}\nolimits A\neq 0$ is $c_{i}\geq
b_{i}$ for all $1\leq i\leq m-r$. Therefore, we will sum only through such
sets $C$. In the border case when $C=[m]\setminus I$ we get that the matrix
$A$ is upper triangular and by Lemma 4.8 we have $\varepsilon^{C,[m]\setminus
C}\mathop{\rm det}\nolimits A=(b_{1})!\dots(b_{m-r})!$.

The sum $\sum_{\begin{subarray}{c}\\#K=r\\\
K\subset[n]\end{subarray}}\mathop{\rm det}\nolimits(V(K,[m]\setminus C))$ is
clearly a polynomial in $n$ of degree at most $\sum([m]\setminus
C)+r={m\choose 2}+r-\sum C$. Since we are summing only trough $C$ with $\sum
C\geq\sum([m]\setminus I)$ we immediately get that the degree of the
polynomial $P$ is at most $\sum I+r$. Moreover, the only summand which
contributes to the term of degree ${\sum I+r}$ is the one with $C=[m]\setminus
I$. We finish the proof of the theorem by computing this summand. In this case
we get the polynomial

$\widetilde{LP}_{I}(n):=\sum_{\begin{subarray}{c}\\#K=r\\\
K\subset[n]\end{subarray}}\mathop{\rm det}\nolimits(V(K,[m]\setminus
C))=\sum_{\sigma\in S_{r}}\sum_{0\leq
k_{1}<\dots<k_{r}<n}(-1)^{\sigma}k_{1}^{i_{\sigma(1)}}\dots
k_{r}^{i_{\sigma(r)}}.$

By Lemma 4.10 the leading coefficient of $\widetilde{LP}_{I}$ is

$\sum_{\sigma\in\mathbb{S}_{r}}(-1)^{\sigma}\frac{1}{(i_{\sigma(1)}+1)(i_{\sigma(1)}+i_{\sigma(2)}+2)\dots(i_{\sigma(1)}+\dots+i_{\sigma(r)}+r)}$

Now we apply Lemma 4.11 for $x_{j}=i_{j}+1$ to conclude that the leading
coefficient of $\widetilde{LP}_{I}$ is

$\frac{\prod_{j>k}(i_{j}-i_{k})}{\prod_{j}(i_{j}+1)\prod_{j>k}(i_{j}+i_{k}+2)}$

which is obviously non-zero. This shows that the degree of the polynomial
$LP_{I}$ is $\sum I+r$ and its leading coeffient is

$\frac{1}{(m-1)!(m-2)!\dots
1!}\cdot(b_{1}!)\dots(b_{m-r})!\cdot\frac{\prod_{j>k}(i_{j}-i_{k})}{\prod_{j}(i_{j}+1)\prod_{j>k}(i_{j}+i_{k}+2)}=$
$=\frac{\prod_{j>k}(i_{j}-i_{k})}{(i_{1})!\dots(i_{r})!\prod_{j}(i_{j}+1)\prod_{j>k}(i_{j}+i_{k}+2)}.$

######  Corollary 4.13.

The polynomial $\delta(m,n,n-s)$ from 4.1 has degree $m$, and the polynomial
$\phi(n,d)$ from 4.2 has degree $d-1$.

##  5\. The Nie-Ranestad-Sturmfels conjecture

In this section we present a proof of the formula for the degree of
semidefinite programming which was conjectured by Nie, Ranestad and Sturmfels
[25]. The formula was known so far only for special values of the parameters.
To state it we introduce the following coefficients.

######  Definition 5.1 (Coefficients $b_{I}$).

Let $I$ be a set of $s$ nonnegative integers. We define $b_{I}(n)$ by the
following formula:

$b_{I}(n)=Q_{I+\mathbf{1}_{s}}(\underbrace{1/2,\dots,1/2}_{n\;\mathrm{times}}),$

where $I+\mathbf{1}_{s}$ is the set obtained from $I$ by adding one to each of
its elements. The function $Q_{I+\mathbf{1}_{s}}$ is the Schur $Q$-function
[19, Section III.8] and its argument $1/2$ appears $n$ times.

These coefficients may be computed recursively as described in [25, Section
6]. We note that in this reference the authors use the convention that $I$ is
a subset of the set $\\{1,\dots,n\\}$ while in this article
$I\subset[n]=\\{0,\dots,n-1\\}$. This results in the difference in notation
for the coefficient $b_{I}$ exchanging $I$ and $I+\mathbf{1}_{s}$.

The main theorem of this section, confirming the Nie-Ranestad-Sturmfels
conjecture, is the following.

######  Theorem 5.2.

([25], Conjecture 21) Let $m,n,s$ be positive integers. Then

$\delta(m,n,n-s)=\sum_{\sum I\leq m-s}(-1)^{m-s-\sum
I}\psi_{I}b_{I}(n)\binom{m-1}{m-s-\sum I}.$

where the sum goes through all sets of nonnegative integers of cardinality
$s$.

As we already mentioned, 4.1 is an immediate corollary of 5.2, since the
coefficients $b_{I}(n)$ are known to be polynomials. Hence, as soon as we have
proven 5.2, we have a third proof of 4.1.

######  Remark 5.3.

We note that if the Pataki inequality (3.2) $m\geq\binom{s+1}{2}$ is not
satisfied, then both sides of the equality above are trivially zero.

For the rest of the section we fix the numbers $m,n,s$ as in the statement of
the theorem. Theorem 5.2 presents a relation between numbers $b_{I}(n)$ and
$\psi_{I}$, our proof of which will be algebraic, with the coefficients
$s_{I,J}$ from Definition 2.3 playing a prominent role. The following lemma
describes the relations between $b_{I}(n)$ and $s_{I,J}$:

######  Lemma 5.4.

Let $I$ be a set of $s$ nonnegative integers. Then

$b_{I}(n)=\sum_{J\leq I}\left(\frac{1}{2}\right)^{\sum I-\sum
J}s_{I,J}LP_{J}(n),$ $LP_{I}(n)=\sum_{J\leq I}\left(-\frac{1}{2}\right)^{\sum
I-\sum J}s_{I,J}b_{J}(n).$

These two identities are equivalent, by the discussion following Definition
2.3. We present two proofs: one based on simple algebra, and one on more
sophisticated methods from algebraic geometry.

For the first proof, let us recall two statements from linear algebra which
will allow us to prove Pfaffian formulas also for the set complements.

######  Lemma 5.5.

(Jacobi’s Theorem.) Let $A$ be an $n\times n$ matrix, and $A^{C}$ its cofactor
matrix. Then

$\mathop{\rm det}\nolimits(A_{[n]\setminus I,[n]\setminus J})=\mathop{\rm
det}\nolimits(A^{C}_{I,J})\mathop{\rm det}\nolimits(A)^{\\#I-1}$

for all sets $I,J\subset[n]$ with $\\#I=\\#J$.

######  Corollary 5.6.

The cofactor matrix $A^{C}$ of an $n\times n$ skew-symmetric matrix $A$ is
given by

$A^{C}_{ij}=\mathop{\rm Pf}\nolimits(A_{[n]\setminus{\\{i,j\\}}})\mathop{\rm
Pf}\nolimits(A).$

######  Lemma 5.7.

Let $I=\\{i_{1},\dots,i_{r}\\}$ be a set of nonnegative integers. Then

$\psi_{[n]\setminus I}=\mathop{\rm
Pf}\nolimits(\psi_{[n]\setminus{\\{i_{k},i_{l}\\}}})_{0<k<l\leq r}\text{ for
even }\\#I,$ $\psi_{[n]\setminus I}=\mathop{\rm
Pf}\nolimits(\psi_{[n]\setminus{\\{i_{k},i_{l}\\}}})_{0\leq k<l\leq r}\text{
for odd }\\#I,$

where
$\psi_{[n]\setminus{\\{i_{0},i_{k}\\}}}:=\psi_{[n]\setminus{\\{i_{k}\\}}}.$

###### Proof.

Let us consider the case where both $n$ and $\\#I$ are even. Consider the
skew-symmetric matrix $A$ such that $A_{k,l}=\psi_{\\{k,l\\}}$ for $0\leq
k<l<n$. Then using Lemmas 5.5 and 2.7 we get

$\psi_{[n]\setminus I}=\mathop{\rm Pf}\nolimits(A_{[n]\setminus
I})=\mathop{\rm Pf}\nolimits(A^{C}_{I})\mathop{\rm
Pf}\nolimits(A)^{\\#I-1}=\mathop{\rm Pf}\nolimits(A^{C}_{I}),$

since $\mathop{\rm det}\nolimits(A)=\psi_{\\{0,1,\dots,n-1\\}}=1$. Moreover,
by Corollary 5.6, the entries of the cofactor matrix $A^{C}$ are $\mathop{\rm
Pf}\nolimits(A_{[n]\setminus{\\{k,l\\}}})\mathop{\rm
Pf}\nolimits(A)=\psi_{[n]\setminus{\\{k,l\\}}}$ which proves the lemma in this
case.

The proof in the other cases is similar. The only difference is that we
consider a different matrix $A$. If $n$ is odd we take
$A=(\psi_{\\{k,l\\}})_{-1\leq k<l<n}$ and if $n$ is even and $\\#I$ is odd we
take $A=(\psi_{\\{k,l\\}})_{-2\leq k<l<n}$. We interpret $\psi_{\\{-1,k\\}}$
and $\psi_{\\{-2,k\\}}$ as $\psi_{\\{k\\}}$ and we put $\psi_{\\{-1,-2\\}}=1$.
Then we conclude in the same way.

######  Corollary 5.8.

$\\#I\psi_{[n]\setminus{I}}=\begin{cases}2\sum_{1\leq k<l\leq
r}{(-1)^{k+l+1}\psi_{[n]\setminus{\\{i_{k},i_{l}\\}}}\psi_{[n]\setminus{(I\setminus\\{i_{k},i_{l}\\})}}}&\text{
if $\\#I$ is even}\\\ 2\sum_{0\leq k<l\leq
r}{(-1)^{k+l+1}\psi_{[n]\setminus{\\{i_{k},i_{l}\\}}}\psi_{[n]\setminus{(I\setminus\\{i_{k},i_{l}\\})}}}&\text{
if $\\#I$ is odd.}\end{cases}$

where
$\psi_{[n]\setminus{\\{i_{0},i_{k}\\}}}:=\psi_{[n]\setminus{\\{i_{k}\\}}}.$

###### Proof.

For every skew-symmetric $r\times r$ matrix $A$ (with $r$ even) and every
$k=1,\ldots,r$, we have the following recursive formula for the Pfaffian:

$\mathop{\rm Pf}\nolimits(A)=\sum_{l=1}^{k-1}{(-1)^{k+l}a_{k,l}\mathop{\rm
Pf}\nolimits(A_{\hat{k}\hat{l}})}-\sum_{l=k+1}^{r}{(-1)^{k+l}a_{k,l}\mathop{\rm
Pf}\nolimits(A_{\hat{k}\hat{l}})},$

where $A_{\hat{k}\hat{l}}$ is the submatrix obtained by removing the $k$-th
and $l$-th rows and columns. Summing over all $k$ gives the desired equality.

######  Remark 5.9.

If we define $\psi_{[n]\setminus{I}}=0$ for $I=\\{i_{1},\ldots,i_{r}\\}$ a
multiset/partition with at least one repeated entry, the recursion from 5.8
still holds.

######  Remark 5.10.

5.8 can be seen as a recursive relation between the polynomials $LP_{I}(n)$
from 4.3. In particular, we can obtain in this way one more proof of 4.3.

###### First proof of Lemma 5.4.

We will use induction on the length of $I$, which we will denote by $s$. The
base of induction, i.e. the cases $s=1,2$ are left for the reader.

We proceed with the general case $s>2$. We will assume that $s$ is even; the
odd case is analogous. Since $b_{I}=\mathop{\rm
Pf}\nolimits(b_{i_{p},i_{q}})_{1\leq p<q\leq s}$, we have (as in 5.8) the
following recursive relations between the $b_{I}$’s:

$sb_{I}=2\sum_{1\leq p<q\leq
n}{(-1)^{p+q+1}b_{\\{i_{p},i_{q}\\}}b_{I\setminus\\{i_{p},i_{q}\\}}}.$

In order to use induction, we need to show that

$s\sum_{J\leq I}{2^{\sum{J}}s_{I,J}\psi_{[n]\setminus{J}}}=\hskip 113.81102pt$
$2\sum_{1\leq p<q\leq
n}{(-1)^{p+q+1}\left(\sum_{J\leq\\{i_{p},i_{q}\\}}{2^{\sum{J}}s_{\\{i_{p},i_{q}\\},J}\psi_{[n]\setminus{J}}}\right)\left(\sum_{J\leq
I\setminus\\{i_{p},i_{q}\\}}{2^{\sum{J}}s_{I\setminus\\{i_{p},i_{q}\\},J}\psi_{[n]\setminus{J}}}\right)}.$

This follows immediately from the following claim:

######  Claim 5.11.

For every $J\leq I$, where $J$ can have repeated elements,

${s_{I,J}\psi_{[n]\setminus{J}}}=\frac{2}{s}\sum_{1\leq p<q\leq
n}{(-1)^{p+q+1}\left(\sum_{1\leq
s<t<n}{s_{\\{i_{p},i_{q}\\},\\{j_{s},j_{t}\\}}\psi_{[n]\setminus{\\{j_{s},j_{t}\\}}}s_{I\setminus\\{i_{p},i_{q}\\},J\setminus\\{j_{s},j_{t}\\}}\psi_{[n]\setminus{(J\setminus\\{j_{s},j_{t}\\})}}}\right)}.$

Indeed, using Laplace expansion, for any $t,u$ we can write:

$s_{I,J}=\sum_{p<q}{(-1)^{p+q+t+u}s_{\\{i_{p},i_{q}\\},\\{j_{t},j_{u}\\}}s_{I\setminus\\{i_{p},i_{q}\\},J\setminus\\{j_{t},j_{u}\\}}}.$

Hence, the right hand side can be rewritten as

$2s_{I,J}\sum_{1\leq
t<u<n}{(-1)^{t+u+1}\psi_{[n]\setminus{\\{j_{t},j_{u}\\}}}\psi_{[n]\setminus{(J\setminus\\{j_{t},j_{u}\\})}}}.$

It remains to show that

$s\psi_{[n]\setminus{J}}=2\sum_{1\leq
t<u<n}{(-1)^{t+u+1}\psi_{[n]\setminus{\\{j_{t},j_{u}\\}}}\psi_{[n]\setminus{(J\setminus\\{j_{t},j_{u}\\})}}}.$

But this is precisely 5.8, and this concludes the first proof of the formula.

The ideas of the second proof were suggested to us by Andrzej Weber.

###### Second proof of Lemma 5.4.

We start with a projection formula, which is a special case of [13, (4.7)].
The proofs of the formula were first provided by Pragacz [28, 29]. Note that
this formula is stated in terms of Schur $P$-polynomials, while we work with
Schur $Q$-polynomials which accounts for an additional factor of a power of
two.

For a vector bundle $\mathcal{E}$ of rank $n$ over some base $X$, we consider
the relative Grassmannian $G^{s}(\mathcal{E})$ of rank $s$ quotients of
$\mathcal{E}$, with its projection $\pi$ to $X$. We denote by $\mathcal{K}$
and $\mathcal{Q}$ the relative tautological subbundle and quotient bundle of
$\pi^{*}\mathcal{E}$, of respective ranks $r=n-s$ and $s$. Then

(5.1)
$Q_{I+\mathbf{1}_{s}}(\mathcal{E})=\pi_{*}(c_{top}(\mathcal{K}\otimes\mathcal{Q})Q_{I+\mathbf{1}_{s}}(\mathcal{Q})),$

where by $+\mathbf{1}_{s}$ we mean adding $1$ to all $s$ elements of $I$ (cf.
[13, Example 2, p. 50]). Moreover, [13, (4.5)], [27, Proposition 2.2] can be
written as

$Q_{I+\mathbf{1}_{s}}(\mathcal{Q})=2^{s}c_{top}(\wedge^{2}\mathcal{Q})s_{\lambda(I)+\mathbf{1}_{s}}(\mathcal{Q})=c_{top}(S^{2}\mathcal{Q})s_{\lambda(I)}(\mathcal{Q}).$

Since $\pi^{*}\mathcal{E}$ is an extension of $\mathcal{Q}$ by $\mathcal{K}$,
the bundle $\pi^{*}S^{2}\mathcal{E}$ admits a filtration whose successive
quotients are $S^{2}\mathcal{Q}$, $\mathcal{K}\otimes\mathcal{Q}$ and
$S^{2}\mathcal{K}$. Hence the identity

$c(\mathcal{K}\otimes\mathcal{Q})c(S^{2}\mathcal{Q})=s(S^{2}\mathcal{K})\pi^{*}c(S^{2}\mathcal{E}).$

Equation (5.1) can thus be rewritten as

$Q_{I+\mathbf{1}_{s}}(\mathcal{E})=c(S^{2}\mathcal{E})\pi_{*}(s(S^{2}\mathcal{K})s_{\lambda(I)}(\mathcal{Q}))_{|deg=\Sigma
I+s},$

where the last symbols mean we only keep the component of degree $\sum I+s$.

Now suppose that $\mathcal{E}=\mathcal{E}_{0}\otimes L$ for some line bundle
$L$ and a trivial vector bundle $\mathcal{E}_{0}$. Then $G^{s}(\mathcal{E})$
is a trivial bundle over $X$, while $\mathcal{K}=\mathcal{K}_{0}\otimes L$ and
$\mathcal{Q}=\mathcal{Q}_{0}\otimes L$ are obtained by pull-back of the
tautological and quotient bundles $\mathcal{K}_{0}$, $\mathcal{Q}_{0}$ over a
fixed Grassmannian $G^{s}(\mathbf{C}^{n})$ (we omit the pull-backs for
simplicity). By Definition 2.3 (where formally the $x_{i}$’s are the Chern
roots of $\mathcal{Q}$ and we need to homogenize by using $c_{1}(L)$), we
have:

$s_{\lambda(I)}(\mathcal{Q})=\sum_{J\leq
I}s_{I,J}s_{\lambda(J)}(\mathcal{Q}_{0})\delta^{\Sigma I-\Sigma J},$

where $\delta=c_{1}(L)$. Moreover, the Segre classes of
$S^{2}\mathcal{K}_{0}^{*}$ and $S^{2}\mathcal{K}$ are related by the formula

$s(S^{2}\mathcal{K})=\sum_{\ell\geq
0}(1+2\delta)^{-\binom{r+1}{2}-\ell}s_{(\ell)}(S^{2}\mathcal{K}_{0}^{*}).$

Plugging these two formulas into the previous one, we get
$Q_{I+\mathbf{1}_{s}}(\mathcal{E})$ as

$\sum_{J\leq
I}\sum_{L}(1+2\delta)^{\binom{n+1}{2}-\binom{r+1}{2}-|\lambda(L)|}\delta^{\Sigma
I-\Sigma
J}s_{I,J}\psi_{L}\pi_{*}(s_{\lambda(L)}(\mathcal{K}_{0}^{*})s_{\lambda(J)}(\mathcal{Q}_{0}))_{|deg=\Sigma
I+s}.$

Now recall that the Schur classes $s_{\alpha}(\mathcal{K}_{0}^{*})$ and
$s_{\beta}(\mathcal{Q}_{0})$, for partitions $\alpha\subset(s^{r})$ and
$\beta\subset(s^{k})$, that are non zero, give dual bases of Schubert cycles
on the Grassmannian $G^{s}(\mathbf{C}^{n})$. This can be expressed as

$\pi_{*}(s_{\lambda(L)}(\mathcal{K}_{0}^{*})s_{\lambda(J)}(\mathcal{Q}_{0}))=\delta_{L,[n]/J},$

where $\delta_{L,[n]/J}$ is the Kronecker delta. Note that $L=[n]/J$ implies
that $|\lambda(L)|+|\lambda(J)|=sr$. We thus get the formula

$Q_{I+\mathbf{1}_{s}}(\mathcal{E})=\Big{(}\sum_{J\leq I}(1+2\delta)^{s+\Sigma
J}\delta^{\Sigma I-\Sigma J}s_{I,J}\psi_{[n]/J}\Big{)}_{|deg=\Sigma I+s}.$

But since the degree of the polynomial in brackets is exactly $\Sigma I+s$, we
just need to keep its top degree component, that is

$Q_{I+\mathbf{1}_{s}}(\mathcal{E})=\sum_{J\leq I}2^{\Sigma
J+s}s_{I,J}\psi_{[n]/J}\delta^{\Sigma I+s}.$

We conclude by applying formally this formula to the bundle
$\mathcal{E}=\mathcal{O}(1/2)^{\oplus n}$ over the projective space.

######  Lemma 5.12.

Let $J$ be a set of nonnegative integers of length $s$ with $\sum J\leq m-s$.
Then

$\sum_{\begin{subarray}{c}I\geq J\\\ \sum I\leq
m-s\end{subarray}}\psi_{I}\left(-\frac{1}{2}\right)^{\sum I-\sum
J}s_{I,J}\binom{m-1}{m-s-\sum I}=\begin{cases}0&\text{ if }\sum J<m-s\\\
\psi_{J}&\text{ if }\sum J=m-s\par\end{cases}$

###### Proof.

We prove the lemma at the same time for all the $J$’s by multiplying the above
equation by the Schur polynomial $s_{\lambda(J)}(x_{1},\dots,x_{s})$ and
summing up. Since Schur polynomials form a basis of the space of symmetric
polynomials, the statement of the lemma is equivalent to the following
polynomial identity:

$\sum_{\sum J\leq m-s}\sum_{\begin{subarray}{c}I\geq J\\\ \sum I\leq
m-s\end{subarray}}\psi_{I}\left(-\frac{1}{2}\right)^{\sum I-\sum
J}s_{I,J}\binom{m-1}{m-s-\sum I}s_{\lambda(J)}(x_{1},\dots,x_{s})=$ $\hskip
85.35826pt=\sum_{\sum J=m-s}\psi_{J}s_{\lambda(J)}(x_{1},\dots,x_{s}).$

By 2.5, the right hand side is equal to
$s_{(m-s-\binom{s}{2})}(x_{i}+x_{j}|1\leq i\leq j\leq s)$. For the left hand
side we can use Definition 2.3 of the coefficients $s_{I,J}$:

$\displaystyle\sum_{\sum J\leq m-s}\sum_{\begin{subarray}{c}I\geq J\\\ \sum
I\leq m-s\end{subarray}}\psi_{I}\left(-\frac{1}{2}\right)^{\sum I-\sum
J}s_{I,J}\binom{m-1}{m-s-\sum I}s_{\lambda(J)}(x_{1},\dots,x_{s})$
$\displaystyle=$ $\displaystyle\sum_{\sum I\leq
m-s}\psi_{I}\binom{m-1}{m-s-\sum I}\sum_{J\leq
I}\left(-\frac{1}{2}\right)^{\sum I-\sum
J}s_{I,J}s_{\lambda(J)}(x_{1},\dots,x_{s})$ $\displaystyle=$
$\displaystyle\sum_{\sum I\leq m-s}\psi_{I}\binom{m-1}{m-s-\sum
I}s_{\lambda(I)}(x_{1}-1/2,\dots,x_{s}-1/2)$ $\displaystyle=$
$\displaystyle\sum_{i=\binom{s}{2}}^{m-s}\sum_{\sum
I=i}\binom{m-1}{m-s-i}\psi_{I}s_{\lambda(I)}(x_{1}-1/2,\dots,x_{s}-1/2)$
$\displaystyle=$
$\displaystyle\sum_{i=\binom{s}{2}}^{m-s}\binom{m-1}{m-s-i}\sum_{\sum
I=i}\psi_{I}s_{\lambda(I)}(x_{1}-1/2,\dots,x_{s}-1/2)$ $\displaystyle=$
$\displaystyle\sum_{i=\binom{s}{2}}^{m-s}\binom{m-1}{m-s-i}s_{(i-\binom{s}{2})}(x_{i}+x_{j}-1|1\leq
i\leq j\leq s)$ $\displaystyle=$ $\displaystyle
s_{(m-s-\binom{s}{2})}(x_{i}+x_{j}|1\leq i\leq j\leq s).$

In the last equality we applied Lemma 2.4 to the variables $x_{i}+x_{j}-1$.

Now we are able to present the proof of Theorem 5.2:

###### Proof of Theorem 5.2.

We replace $b_{I}(n)$ by the expression from Lemma 5.4, change the order of
summation and use Lemma 5.12 in the last step:

$\displaystyle\sum_{\sum I\leq m-s}(-1)^{m-s-\sum
I}\psi_{I}b_{I}(n)\binom{m-1}{m-s-\sum I}=$ $\displaystyle=\sum_{\sum I\leq
m-s}\sum_{J\leq I}s_{I,J}\psi_{[n]\setminus J}\left(\frac{1}{2}\right)^{\sum
I-\sum J}(-1)^{m-s-\sum I}\psi_{I}\binom{m-1}{m-s-\sum I}$
$\displaystyle=\sum_{\sum J\leq m-s}(-1)^{m-s-\sum J}\psi_{[n]\setminus
J}\sum_{\begin{subarray}{c}I\geq J\\\ \sum I\leq
m-s\end{subarray}}s_{I,J}\left(-\frac{1}{2}\right)^{\sum I-\sum
J}\psi_{I}\binom{m-1}{m-s-\sum I}$ $\displaystyle\hskip 85.35826pt=\sum_{\sum
J=m-s}(-1)^{m-s-\sum J}\psi_{[n]\setminus J}\psi_{J}=\delta(m,n,n-s).$

##  6\. General square matrices

The results from the previous sections have natural analogues if we replace
the space of symmetric matrices (“type C”) with the space of skew-symmetric
matrices (“type D”), or with the space of general matrices (“type A”). This
section will be devoted to the latter case, and the next section to the former
one.

###  6.A. Codegrees of smooth determinantal loci

Let $M_{n}$ denote the space of complex matrices of size $n$, and
$D^{n-r,n}\subset\mathbb{P}(M_{n})$ the locus of matrices of rank at most
$n-r$. Denote by $D^{n-r,n}_{m}$ its intersection with a general
$m$-dimensional projective space. Its dimension is $d=m-r^{2}$ when this is
non negative, otherwise it is empty. The analogues of the Pataki’s
inequalities are given by:

######  Proposition 6.1.

The dual variety of $D^{n-r,n}_{m}$ is a hypersurface if and only if

$r^{2}\leq m\leq n^{2}-(n-r)^{2}.$

As was done in [25] for symmetric matrices, the degree of this dual variety
can be computed by classical means when $D^{n-r,n}_{m}$ is smooth, which is
equivalent to $r^{2}\leq m\leq r^{2}+2r$. The class formula gives, in terms of
topological Euler characteristics,

$\mathop{\rm
deg}\nolimits(D^{n-r,n}_{m})^{*}=(-1)^{d}\Big{(}\chi(D^{n-r,n}_{m})-2\chi(D^{n-r,n}_{m-1})+\chi(D^{n-r,n}_{m-2})\Big{)}.$

Euler characteristics of smooth degeneracy loci have been computed by Pragacz
[26]. For $\varphi:F\rightarrow E$ a morphism of vector bundles of ranks $f,e$
over a variety $X$, the formula given in [13, page 57] is

$\chi(D_{r}(\varphi))=\int_{X}P_{r}(E,F)c(X),$

where $c(X)$ denotes the total Chern class of $X$, while $P_{r}(E,F)$ is a
universal polynomial in the Chern classes of $E$ and $F$. Explicitely,

$P_{r}(E,F)=\sum_{\lambda,\mu}(-1)^{|\lambda|+|\mu|}D_{\lambda,\mu}^{n-r,m-r}s_{(m-r)^{n-r}+\lambda,\tilde{\mu}}(E-F),$

where the sum is over partitions $\lambda$ and $\mu$ of length $n-r$ and $m-r$
respectively, and $\tilde{\mu}$ is the dual partition of $\mu$. Moreover the
coefficients denoted $D_{\lambda,\mu}^{n-r,m-r}$ in [13] encode the Segre
classes of a tensor product of vector bundles. (We will rather use in the
sequel the notations of [16], see Definition 6.7.)

We want to apply this formula to $D^{n-r,n}_{m}$, which we consider as the
degeneracy locus $D_{n-r}(\varphi)$ of the tautological morphism
$\varphi:F=\mathcal{O}(-1)^{\oplus n}\longrightarrow\mathcal{O}^{\oplus n}$
over $X=\mathbb{P}^{m}$. Since
$c(\mathbb{P}^{m})-2hc(\mathbb{P}^{m-1})+h^{2}c(\mathbb{P}^{m-2})=(1+h)^{m-1}$,
where $h$ denotes the hyperplane class, we get the formula

$\mathop{\rm
deg}\nolimits(D^{n-r,n}_{m})^{*}=\sum_{\lambda,\mu}(-1)^{|\lambda|+|\mu|}\binom{m-1}{r^{2}+|\lambda|+|\mu|}D_{\lambda,\mu}^{r,r}s_{(r)^{r}+\lambda,\tilde{\mu}}(\underbrace{1,\ldots,1}_{n\;\mathrm{times}}),$

the sum being taken over partitions $\lambda$ and $\mu$ of length $r$. Note
that the dependence on $n$ for $r$ and $m$ fixed is only in the last term,
more precisely in the number of one’s on which the Schur functions are
evaluated. This dependence is well known to be polynomial in $n$; very
explicitely, for any partition $\nu$,

$s_{\nu}(\underbrace{1,\ldots,1}_{n\;\mathrm{times}})=\dim
S_{\nu}\mathbb{C}^{n}=c_{\nu}(n)/h(\nu),$

where $c_{\nu}$ is the content polynomial and $h(\nu)$ is the product of the
hook lengths of $\nu$ [19]. A priori this formula is only valid in the range
$r^{2}\leq m\leq r^{2}+2r$, when $D^{n-r,n}_{m}$ is smooth. That it should be
true in general would be an analogue of the NRS conjecture in type A. We will
prove below that this statement is correct.

We introduce the following notations, similar to those we used for symmetric
matrices.

######  Definition 6.2.

We define $\delta_{A}(m,n,r)$ to be the degree of the variety
$(D_{m}^{r,n})^{*}$ if it is a hypersurface, and zero otherwise. Here
$D_{m}^{r,n}$ is the variety of $n\times n$ matrices of rank at most $r$,
intersected with a general (projective) $m$ dimensional subspace.
Equivalently, if we let $Z_{r}\subset\mathbb{P}(V^{*}\otimes
V)\times\mathbb{P}(V^{*}\otimes V)$ be the variety of pairs of matrices
$(X,Y)$, up to scalars, with $X\cdot Y=Y\cdot X=0$, $\mathop{\rm rk}\nolimits
X\leq r$, $\mathop{\rm rk}\nolimits Y\leq n-r$, then the multidegree of
$Z_{r}$ is equal to

$[Z_{r}]=\sum_{m}{\delta_{A}(m,n,r)H_{1}^{n^{2}-m}H_{n-1}^{m}},$

where $H_{1}$ and $H_{n-1}$ denote the pull-backs of the hyperplane classes
from $\mathbb{P}(V^{*}\otimes V)$ and $\mathbb{P}(V^{*}\otimes V)$,
respectively.

######  Definition 6.3.

The number $\phi_{A}(n,d)$ is the degree of the variety $\mathcal{L}^{-1}$,
where $\mathcal{L}\subseteq\mathbb{P}(M_{n})$ is a general linear subspace of
dimension $d-1$.

###  6.B. Complete collineations

The correct space to work with is the _space of complete collineations_ [34,
42, 44, 17, 41, 20]. It can actually be defined for rectangular matrices, but
for sake of simplicity we will restrict ourselves to square matrices.

######  Definition 6.4.

Let $V$ and $W$ be two vector spaces of equal dimension $n$. The space
$\mathbb{P}(V^{*}\otimes W)$ represents linear maps from $V$ to $W$; the open
subset of rank $n$ linear maps is denoted by $\mathbb{P}(V^{*}\otimes
W)^{\circ}$. Then the space of complete collineations $CC(V,W)$ is defined as
the closure of the image of the map

$\phi:\mathbb{P}(V^{*}\otimes W)^{\circ}\to\mathbb{P}(V^{*}\otimes
W)\times\mathbb{P}\left(\bigwedge^{2}{V^{*}}\otimes\bigwedge^{2}{W}\right)\times\ldots\times\mathbb{P}\left(\bigwedge^{n-1}{V^{*}}\otimes\bigwedge^{n-1}{W}\right),$

given by

$[A]\mapsto([A],[\wedge^{2}A],\ldots,[\wedge^{n-1}A]).$

As before, in coordinates this map sends a matrix to its minors of various
sizes.

As in the symmetric case, the space of complete collineations can be
constructed by blowing-up ${\mathbb{P}}(M_{n})$ along the subvariety of rank
one matrices, then the strict transform of the subvariety of matrices of rank
at most two, and so on. As such, it admits a first series
$S_{1},\ldots,S_{n-1}$ of special classes of divisors: the classes of (the
strict transforms of) the exceptional divisors $E_{1},\ldots,E_{n-1}$ of these
successive blow-ups. A second natural series $L_{1},\ldots,L_{n-1}$ of classes
of divisors can be obtained by pulling back the hyperplane classes under the
projections
$\pi_{i}:CC(V,W)\to\mathbb{P}\left(\bigwedge^{i}{V^{*}}\otimes\bigwedge^{i}{W}\right)$.

The analogue of 3.4 holds:

######  Proposition 6.5.

$L_{1},\ldots,L_{n-1}$ form a basis of $\mathop{\rm Pic}\nolimits(CC(V,W))$,
in which the $S_{i}$’s are given by the formulas

$S_{i}=-L_{i-1}+2L_{i}-L_{i+1},$

with the convention that $L_{0}=L_{n}:=0$.

###### Proof.

Follows from [20, Proposition 3.6, Theorem 3.13].

######  Proposition 6.6.

The numbers $\phi_{A}$ and $\delta_{A}$ can be computed as intersection
products of the variety of complete collineations:

$\phi_{A}(n,d)=\int_{CC_{n}}L_{1}^{n^{2}-d}L_{n-1}^{d-1},$
$\delta_{A}(m,n,r)=\int_{CC_{n}}S_{r}L_{1}^{n^{2}-m-1}L_{n-1}^{m-1}=\int_{E_{r}}L_{1}^{n^{2}-m-1}L_{n-1}^{m-1}.$

This implies the analogue of section 3.B:

(6.1) $\phi_{A}(n,d)=\frac{1}{n}\sum_{r=1}^{n-1}{r\delta_{A}(d,n,n-r)}.$

######  Definition 6.7.

We define type $A$ Lascoux coefficients $d_{I,J}$ as follows. For
$X=(x_{1},\ldots,x_{k})$ and $Y=(y_{1},\dots,y_{l})$ two sets of
indeterminates, we denote by $X+Y$ the set of indeterminates $x_{i}+y_{j}$,
$1\leq i\leq k$, $1\leq j\leq l$. Then the $d_{I,J}$’s are defined by the
formal identity

$s_{(d)}(X+Y)=\sum_{\begin{subarray}{c}\\#I=k,\\#J=l\\\
|\lambda(I)|+|\lambda(J)|=d\end{subarray}}d_{I,J}s_{\lambda(I)}(X)s_{\lambda(J)}(Y).$

Equivalently, for the product of the universal bundles
$\mathcal{U}_{1}\otimes\mathcal{U}_{2}$ over a product of Grassmannians
$G(k,m)\times G(l,n)$:

$Seg_{d}(\mathcal{U}_{1}\otimes\mathcal{U}_{2})=\sum_{\begin{subarray}{c}\\#I=k,\\#J=l\\\
|\lambda(I)|+|\lambda(J)|=d\end{subarray}}{d_{I,J}\sigma^{1}_{\lambda(I)}\sigma^{2}_{\lambda(J)}}$

Analogously to 3.7, we have the following formula for $\delta_{A}$:

######  Theorem 6.8.

$\delta_{A}(m,n,r)=\sum_{\begin{subarray}{c}I,J\subset[n]\\\ \\#I=\\#J=n-r\\\
\sum{I}+\sum{J}=m-n+r\end{subarray}}{d_{I,J}d_{[n]\setminus I,[n]\setminus
J}}$

###  6.C. Induction relations and polynomiality

We denote by $D(t)$ the infinite matrix with entries
$D(t)_{ij}={{t+i+j}\choose i}$. This matrix gives us a formula for $d_{I,J}$
[16, Proposition 2.8].

######  Proposition 6.9.

Let $I=\\{i_{1},\dots,i_{r}\\},\ J=\\{j_{1},\dots,j_{s}\\}$ be two sets of
nonnegative integers with $r\leq s$. Then

$d_{I,J}=\begin{cases}\mathop{\rm det}\nolimits
D(s-r)_{I,\\{j_{s-r+1}-(s-r),\dots,j_{s}-(s-r)\\}}&\text{ if }j_{i}=i-1\text{
for all }1\leq i\leq s-r\\\ 0&\text{ otherwise .}\end{cases}$

In particular, if $\\#I=\\#J$ then $d_{I,J}=\mathop{\rm det}\nolimits
D(0)_{I,J}$.

######  Lemma 6.10.

  1. (1)

Let $I=\\{i_{1},\dots,i_{s}\\},\ J=\\{j_{1},\dots,j_{s}\\}$ with
$i_{1},j_{1}>1$. Write $I_{0}=\\{0\\}\cup I$ and $J_{0}=\\{0\\}\cup J$. Then

$d_{I,J}=(s+1)d_{I_{0},J_{0}}-\sum_{p=1}^{s}{d_{I_{0}\setminus\\{i_{p}\\}\cup\\{i_{p}-1\\},J_{0}}}-\sum_{q=1}^{s}{d_{I_{0},J_{0}\setminus\\{j_{q}\\}\cup\\{j_{q}-1\\}}}.$

(Here, if $I_{0}\setminus\\{i_{p}\\}\cup\\{i_{p}-1\\}$ is a multiset, then
$d_{I_{0}\setminus\\{i_{p}\\}\cup\\{i_{p}-1\\},J_{0}}=0$.)

  2. (2)

For $i_{1}=0$ or $j_{1}=0$ we have:

$d_{\\{i_{1},\ldots,i_{s}\\},\\{j_{1},j_{2},\ldots,j_{s}\\}}=\sum_{\begin{subarray}{c}i_{\ell}\leq
i^{\prime}_{\ell}<i_{\ell+1}\\\ j_{\ell}\leq
j^{\prime}_{\ell}<j_{\ell+1}\end{subarray}}{d_{\\{i^{\prime}_{1},\ldots,i^{\prime}_{s-1}\\},\\{j^{\prime}_{1},\ldots,j^{\prime}_{s-1}\\}}}.$

###### Proof.

  1. (1)

We expand the determinant $\mathop{\rm det}\nolimits D(0)_{I_{0},J_{0}}$ in
each row, and sum up:

$\displaystyle(s+1)d_{I_{0},J_{0}}=$
$\displaystyle\sum_{p,q=0}^{s}{(-1)^{p+q}\binom{i_{p}+j_{q}}{i_{p}}d_{I_{0}\setminus\\{i_{p}\\},J_{0}\setminus\\{j_{q}\\}}}$
$\displaystyle=$ $\displaystyle
d_{I,J}+\sum_{p=1}^{s}{(-1)^{p}d_{I_{0}\setminus\\{i_{p}\\},J}}+\sum_{q=1}^{s}{(-1)^{q}d_{I,J_{0}\setminus\\{j_{q}\\}}}$
$\displaystyle+\sum_{p,q=1}^{s}{(-1)^{p+q}\left(\binom{i_{p}+j_{q}-1}{i_{p}}+\binom{i_{p}+j_{q}-1}{i_{p}-1}\right)d_{I_{0}\setminus\\{i_{p}\\},J_{0}\setminus\\{j_{q}\\}}}$
$\displaystyle=$ $\displaystyle
d_{I,J}+\sum_{p=1}^{s}\sum_{q=0}^{s}{(-1)^{p+q}\binom{i_{p}+j_{q}-1}{i_{p}-1}d_{I_{0}\setminus\\{i_{p}\\},J_{0}\setminus\\{j_{q}\\}}}$
$\displaystyle+\sum_{q=1}^{s}\sum_{p=0}^{s}{(-1)^{p+q}\binom{i_{p}+j_{q}-1}{i_{p}}d_{I_{0}\setminus\\{i_{p}\\},J_{0}\setminus\\{j_{q}\\}}}$
$\displaystyle=$ $\displaystyle
d_{I,J}+\sum_{p=1}^{s}{d_{I_{0}\setminus\\{i_{p}\\}\cup\\{i_{p}-1\\},J_{0}}}+\sum_{q=1}^{s}{d_{I_{0},J_{0}\setminus\\{j_{q}\\}\cup\\{j_{q}-1\\}}}.$

  2. (2)

The proof of the second formula is similar to the proof of formula 4.3 in
Lemma 4.5. We only consider the case $i_{1}=0$ and in
$s_{(d)}(\\{x_{i}+y_{j}\mid 1\leq i,j\leq s\\})$ we subsitute $x_{s}=0$. This
yields

$d_{\\{i_{1},\ldots,i_{s}\\},\\{j_{1},j_{2},\ldots,j_{s}\\}}=\sum_{j_{\ell-1}<j^{\prime}_{\ell}\leq
j_{\ell}}{d_{\\{i_{2}-1,\ldots,i_{s}-1\\},\\{j^{\prime}_{1},\ldots,j^{\prime}_{s}\\}}}.$

Then by Proposition 6.9 all summands with $j^{\prime}_{1}>0$ are zero. This
allows to substitute $y_{s}=0$ in $s_{(d)}(\\{x_{i}+y_{j}\mid 1\leq i\leq
s-1,1\leq j\leq s\\})$ and conclude the lemma analogously to formula 4.3 in
Lemma 4.5.

######  Theorem 6.11.

Let $I=\\{i_{1},\ldots,i_{r}\\},J=\\{j_{1},\dots,j_{r}\\}$ be two sets of
strictly increasing nonnegative integers. The function defined for $n\geq 0$
by

$LP^{A}_{I,J}(n):=\begin{cases}d_{[n]\setminus I,[n]\setminus J}&\text{ if
}I,J\subset[n],\\\ 0&\text{ otherwise},\end{cases}$

is polynomial in $n$.

###### Proof.

From Lemma 6.10 it follows that

$LP^{A}_{I,J}(n)=(n-r+1)LP^{A}_{I\setminus\\{0\\},J\setminus\\{0\\}}(n)\\\
-\sum_{\ell:i_{\ell+1}>i_{\ell}+1}{LP^{A}_{I\setminus\\{0,i_{\ell}\\}\sqcup\\{i_{\ell}+1\\},J\setminus\\{0\\}}(n)}-\sum_{\ell:j_{\ell+1}>j_{\ell}+1}{LP^{A}_{I\setminus\\{0\\},J\setminus\\{0,j_{\ell}\\}\sqcup\\{j_{\ell}+1\\}}(n)}$

if $i_{0}=j_{0}=0$, and otherwise

$LP^{A}_{I,J}(n)=\sum_{I^{\prime},J^{\prime}}{LP^{A}_{I^{\prime},J^{\prime}}(n-1)},$

where the sum is over all pairs $(I^{\prime},J^{\prime})$ of the form
$(\\{i_{1}-\epsilon_{1},\ldots,i_{r}-\epsilon_{r}\\},\\{j_{1}-\mu_{1},\ldots,j_{r}-\mu_{r}\\})$
with $\epsilon_{\ell},\mu_{\ell}\in\\{0,1\\}$. As in the first proof of 4.3,
it follows by induction that $LP^{A}_{I,J}$ is polynomial.

######  Theorem 6.12.

For every fixed $m,s$, the function $\delta_{A}(m,n,n-s)$ is a polynomial in
$n$.

###### Proof.

Follows from Theorems 6.8 and 6.11.

######  Theorem 6.13.

For any fixed $d$, the function $\phi_{A}(n,d)$ is a polynomial for $n>0$.

###### Proof.

Follows from eq. 6.1 and 6.12.

###  6.D. Proof of the NRS conjecture for type A

We start with the following analogue of Lemma 5.12.

######  Lemma 6.14.

Let $K,L$ be sets of $r$ nonnegative integers, with $\sum K+\sum L\leq m-r$.
Then

$\sum_{I\geq K}d_{I,L}(-1)^{\sum I-\sum K}s_{I,K}\binom{m-1}{m-r-\sum I-\sum
L}=d_{K,L}$

if $\sum K+\sum L=m-r$, while this sum vanishes if $\sum K+\sum L<m-r.$

###### Proof.

Using Lemma 2.4 we compute $s_{(m-r^{2})}(X+Y)$ as

$\displaystyle\sum_{k=r^{2}}^{m}\binom{m-1}{m-k}s_{(k-r^{2})}(X+Y-1)=\hskip
170.71652pt$ $\displaystyle=\sum_{k=r^{2}}^{m}\binom{m-1}{m-k}\sum_{\sum
I+\sum L=k-r}d_{I,L}s_{\lambda(I)}(X-1)s_{\lambda(L)}(Y)=\hskip 28.45274pt$
$\displaystyle\hskip 28.45274pt=\sum_{k=r^{2}}^{m}\binom{m-1}{m-k}\sum_{\sum
I+\sum L=k-r}d_{I,L}\sum_{K\leq I}(-1)^{\sum I-\sum
K}s_{I,K}s_{\lambda(K)}(X)s_{\lambda(L)}(Y).$

Comparing this expansion with that of Definition 6.7 yields the claim.

######  Definition 6.15.

For two partitions $\lambda=\lambda(I)$, $\mu=\lambda(J)$, of length $r$, we
define the polynomial

$a_{I,J}(n):=s_{(r)^{r}+\lambda,\tilde{\mu}}(\underbrace{1,\ldots,1}_{n\;\rm
times}).$

######  Lemma 6.16.

Let $I,J$ be two sets of $r$ nonnegative integers. Then

$a_{I,J}(n)=\sum_{L\leq I}s_{I,L}d_{[n]\setminus L,[n]\setminus J},$
$d_{[n]\setminus I,[n]\setminus J}=\sum_{L\leq I}(-1)^{\sum I-\sum
L}s_{I,L}a_{L,J}(n).$

###### Proof.

These two formulas being equivalent, we will prove the first one. We use the
projection formula given in [26, Lemma 3.1], [13, Proposition 1, page 51],
where one considers two vector bundles $E,F$ of respective ranks $n,m$ over
some variety $X$. The Grassmann bundles $G_{s}E$ and $G^{s}F$ (parametrizing
rank $s$ subspaces and rank $s$ quotients respectively) over $X$ admit
tautological and quotient bundles $S_{E},Q_{E}$ and $S_{F},Q_{F}$, where
$S_{E}$ and $Q_{F}$ have rank $s$. Let
$\tau:G:=G_{s}E\times_{X}G^{s}F\rightarrow X$ denote the total projection. The
variety $G$ is endowed with the vector bundle
$H:=\tau^{*}Hom(F,E)/Hom(Q_{F},S_{E})$. In our situation we will suppose that
$m=n$, and let $r=n-s$, which is the rank of both $S_{F}$ and $Q_{E}$. Then
the projection formula asserts that for any two partitions $\lambda,\mu$ of
length at most $r$,

$s_{(r)^{r}+\lambda,\tilde{\mu}}(E-F)=\tau_{*}\Big{(}s_{\lambda}(Q_{E})s_{\mu}(S_{F})c_{top}(H)\Big{)}.$

(In the original formula $s_{\mu}(S_{F})$ is replaced by
$s_{\tilde{\mu}}(-S_{F})$, but they are equal.) We can replace $c_{top}(H)$ by
$c(H)$ and keep only the term of the correct degree, which yields

$c(Hom(F,E))\tau_{*}\Big{(}s_{\lambda}(Q_{E})s_{\mu}(S_{F})s(Hom(Q_{F},S_{E})\Big{)}_{|\mathrm{deg}=r^{2}+|\lambda|+|\mu|}.$

Now we specialize to the case where $E=E_{0}\otimes L$ and $F=F_{0}$, where
$E_{0},F_{0}$ are trivial vector bundles of rank $n$, and $L$ is a line bundle
on $X$ with $c_{1}(L)=\delta$. In this case

$c(Hom(F,E))=(1+\delta)^{s^{2}}.$

The Grassmann bundles $G_{s}E$ and $G^{s}F$ are then the trivial bundles
$G_{s}E_{0}\times X$ and $G^{s}F=G^{s}F_{0}\times X$ respectively, while the
tautological and quotient bundles are $S_{E}=S_{E_{0}}\otimes L$,
$Q_{E}=Q_{E_{0}}\otimes L$, $S_{F}=S_{F_{0}}$ and $Q_{F}=Q_{F_{0}}$, where we
omit the obvious pullbacks. In this situation,

$s(Hom(Q_{F},S_{E}))=\sum_{\ell\geq
0}(1+\delta)^{-s^{2}-\ell}s_{\ell}(Hom(Q_{F_{0}},S_{E_{0}})).$

Let $\lambda=\lambda(I)$ and $\mu=\lambda(J)$. Using Definition 6.7,
Definition 2.3 and the duality properties of Schubert classes, we deduce that

$s_{(r)^{r}+\lambda,\tilde{\mu}}(E-F)=a_{I,J}(n)\delta^{r^{2}+|\lambda|+|\mu|}$

can be computed by picking the term of the correct degree in

$\sum_{\ell\geq 0}(1+\delta)^{n^{2}-s^{2}-\ell}\sum_{L}\delta^{\sum I-\sum
L}s_{I,L}d_{[n]/J,[n]/L},$

where the size of $L$ is constrained by the relation $\ell+\sum
L=n(n-1)-s(s-1)-\sum J$. This implies that $n^{2}-s^{2}-\ell+\sum I-\sum
L=r+\sum I+\sum J=r^{2}+|\lambda|+|\mu|$. So the term of the correct degree is
actually the term of maximal degree in $\delta$, and the claim follows.

We are now ready to prove the NRS Conjecture in type A.

######  Theorem 6.17.

$\delta_{A}(m,n,n-r)=\sum_{\begin{subarray}{c}\\#I=\\#L=r,\\\ \sum I+\sum
L\leq m-r\end{subarray}}d_{I,L}(-1)^{m-r-\sum I-\sum L}\binom{m-1}{m-r-\sum
I-\sum L}a_{I,L}(n).$

###### Proof.

Using Theorem 6.8, Lemma 6.14 and Lemma 6.16, we get

$\displaystyle\delta_{A}(m,n,n-r)=\sum_{\begin{subarray}{c}K,L\\\ \sum K+\sum
L=m-r\end{subarray}}d_{K,L}d_{[n]/K,[n]/L}=$ $\displaystyle\hskip
56.9055pt=\sum_{\begin{subarray}{c}K,L\\\ \sum K+\sum L\leq
m-r\end{subarray}}(-1)^{m-r-\sum K-\sum L}\sum_{I\geq K}\sum_{L}d_{I,L}\times$
$\displaystyle\hskip 113.81102pt\times(-1)^{\sum I-\sum
K}s_{I,K}\binom{m-1}{m-r-\sum I-\sum L}d_{[n]/K,[n]/L}$ $\displaystyle\hskip
56.9055pt=\sum_{\begin{subarray}{c}I,L\\\ \sum I+\sum L\leq
m-r\end{subarray}}(-1)^{m-r-\sum I-\sum L}d_{I,L}\times$ $\displaystyle\hskip
113.81102pt\times\binom{m-1}{m-r-\sum I-\sum L}\sum_{K\leq I}(-1)^{\sum I-\sum
K}s_{I,K}d_{[n]/K,[n]/L}$ $\displaystyle\hskip
56.9055pt=\sum_{\begin{subarray}{c}I,L\\\ \sum I+\sum L\leq
m-r\end{subarray}}(-1)^{m-r-\sum I-\sum L}d_{I,L}\binom{m-1}{m-r-\sum I-\sum
L}a_{I,L}(n).$

##  7\. Skew-symmetric matrices

###  7.A. Codegrees of smooth skew-symmetric determinantal loci

Let $A_{n}$ denote the space of skew-symmetric complex matrices of size $n$,
and $AD^{n-r,n}\subset\mathbb{P}(A_{n})$ the locus of matrices of rank at most
$n-r$, where $n-r$ is always supposed to be even. Denote by $AD^{n-r,n}_{m}$
its intersection with a general $m$-dimensional projective space. Its
dimension is $d=m-\binom{r}{2}$ when this is non negative, otherwise it is
empty. The analogs of the Pataki’s inequalities are given by:

######  Proposition 7.1.

The dual variety of $AD^{n-r,n}_{m}$ is a hypersurface if and only if

$\binom{r}{2}\leq m\leq\binom{n}{2}-\binom{n-r}{2}.$

As in the previous case, the degree of this dual variety can be computed by
classical means when $AD^{n-r,n}_{m}$ is smooth, which is equivalent to
$\binom{r}{2}\leq m\leq\binom{r}{2}+2r$. The class formula gives, in terms of
topological Euler characteristics,

$\mathop{\rm
deg}\nolimits(AD^{n-r,n}_{m})^{*}=(-1)^{d}\Big{(}\chi(AD^{n-r,n}_{m})-2\chi(AD^{n-r,n}_{m-1})+\chi(AD^{n-r,n}_{m-2}\Big{)}.$

Euler characteristics of smooth skew-symmetric degeneracy loci have also been
computed by Pragacz [26]. For $E$ a vector bundle of rank $e$ over a variety
$X$, and $\varphi:E^{*}\rightarrow E$ a skew-symmetric morphism, the formula
given in [13, page 64] is

$\chi(D_{s}(\varphi))=\int_{X}P_{s}(E)c(X),$

where $c(X)$ denotes the total Chern class of $X$, while $P_{s}(E)$ is a
universal polynomial in the Chern classes of $E$. Explicitely,

$P_{s}(E)=\sum_{\ell(\lambda)\leq
n-s}(-1)^{|\lambda|}[\lambda+\rho(n-s-1)]P_{\lambda+\rho(n-s-1)}(E),$

where the coefficients $[\lambda+\rho(n-s-1)]$ are those appearing in the
Segre class of the skew-symmetric square of a vector bundle of rank $n-s$.
These coefficients were denoted $\alpha_{I}$ in [16], that we will rather
follow, where $I$ is a set of $r=n-s$ nonnegative integers.

Let us apply this formula to $AD^{n-r,n}_{m}$, which we consider formally as
the degeneracy locus $D_{s}(\varphi)$ of the tautological skew-symmetric
morphism $\phi:F=\mathcal{O}(-\frac{1}{2})^{\oplus
n}\longrightarrow\mathcal{O}(\frac{1}{2})^{\oplus n}$ over $X=\mathbb{P}^{m}$.
Since
$c(\mathbb{P}^{m})-2hc(\mathbb{P}^{m-1})+h^{2}c(\mathbb{P}^{m-2})=(1+h)^{m-1}$,
where $h$ denotes the hyperplane class, we get the formula

$\mathop{\rm deg}\nolimits(AD^{n-r,n}_{m})^{*}=\sum_{I}\binom{m-1}{m-\sum
I}\alpha_{I}P_{I}(\underbrace{1,\ldots,1}_{n\;\mathrm{times}}),$

where the sum goes over the sets $I$ of $r$ nonnegative integers. Once again
the dependence on $n$ for $r$ fixed is only in the last term, more precisely
in the number of one’s on which the P-Schur functions are evaluated. We have
already seen that this dependence is well known to be polynomial in $n$.

A priori this formula is only valid in the range $\binom{r}{2}\leq
m\leq\binom{r}{2}+2r$, when $AD^{n-r,n}_{m}$ is smooth. That it should be true
in general would be an analogue of the NRS conjecture in type D. We will prove
below that this statement is correct. Our notations for the dual degrees will
be as follows

######  Definition 7.2.

Define $\delta_{D}(m,n,r)$ to be the degree of the variety
$(AD_{m}^{2r,2n})^{*}$ if it is a hypersurface, and zero otherwise. Here
$AD_{m}^{2r,2n}$ is the variety of rank at most $2r$ skew-symmetric $2n\times
2n$ matrices, intersected with a general (projective) $m$ dimensional
subspace. Equivalently, if we let
$Z_{r}\subset\mathbb{P}(\wedge^{2}V^{*})\times\mathbb{P}(\wedge^{2}V)$ be the
variety of pairs of matrices $(X,Y)$, up to scalars, with $X\cdot Y=0$,
$\mathop{\rm rk}\nolimits X\leq 2r$, $\mathop{\rm rk}\nolimits Y\leq n-2r$.
Then the multidegree of $Z_{r}$ is equal to

$[Z_{r}]=\sum_{m}{\delta_{D}(m,n,r)H_{1}^{\binom{n}{2}-m}H_{n-1}^{m}},$

where $H_{1}$ and $H_{n-1}$ are the pullbacks of the hyperplane classes from
$\mathbb{P}(\wedge^{2}V^{*})$ and $\mathbb{P}(\wedge^{2}V)$.

###  7.B. Complete skew-symmetric forms

A well-known particularity of skew-symmetric forms is that the cases of odd
and even sizes are quite different. In particular the following definition
only makes sense in the even case.

######  Definition 7.3.

The number $\phi_{D}(n,d)$ is the degree of the variety $\mathcal{L}^{-1}$,
where $\mathcal{L}\subseteq\mathbb{P}(A_{2n})$ is a general linear subspace of
dimension $d-1$.

In this section, we will be only working with skew-symmetric matrices of even
size $2n\times 2n$. The relevant space to deal with is then the _space of
complete skew-forms_. Just as with complete quadrics, there are many ways of
constructing this space. Here we give just two, referring the reader to the
literature [2, 41, 21] for other equivalent definitions.

######  Definition 7.4.

Let $V$ be a $2n$-dimensional vector space. The space of complete skew-forms
$CS(V)$ is defined as the closure of
$\phi(\mathbb{P}(\bigwedge^{2}(V))^{\circ})$, where

$\phi:\mathbb{P}\left(\bigwedge^{2}V\right)^{\circ}\to\mathbb{P}\left(\bigwedge^{2}V\right)\times\mathbb{P}\left(\bigwedge^{4}V\right)\times\ldots\times\mathbb{P}\left(\bigwedge^{2n-2}V\right),$

given by

$[A]\mapsto([A],[\wedge^{2}A],\ldots,[\wedge^{n-1}A]).$

We note that here $\wedge^{i}A$ is viewed as an element of $\bigwedge^{2i}V$,
see also [2, Section 3]. In coordinates, the map
$\bigwedge^{2}V\to\bigwedge^{2i}V$ sends the entries of a skew-symmetric
matrix to the Pfaffians of its principal $2i\times 2i$ submatrices.

For simplicity we will also use the notation $CS_{2n}=CS(\mathbb{C}^{2n})$.

As in the smmetric case, the space of complete skew-forms can be constructed
by blowing up ${\mathbb{P}}(A_{2n})$ along the subvariety of rank two
matrices, then the strict transform of the subvariety of matrices of rank at
most four, and so on. As such, it admits a first series $S_{1},\ldots,S_{n-1}$
of special classes of divisors: the classes of (the strict transforms of) the
exceptional divisors $E_{1},\ldots,E_{n-1}$ of these successive blow-ups. A
second natural series $L_{1},\ldots,L_{n-1}$ of classes of divisors can be
obtained by pulling back the hyperplane classes under the projections
$\pi_{i}:CS(V)\to\mathbb{P}\left(\bigwedge^{2i}V\right)$.

The analogue of 3.4 holds:

######  Proposition 7.5.

The classes $L_{1},\ldots,L_{n-1}$ for a basis $\mathop{\rm
Pic}\nolimits(CS(V))$, in which the $S_{i}$’s are given by

$S_{i}=-L_{i-1}+2L_{i}-L_{i+1},$

with $L_{0}=L_{n}:=0$.

###### Proof.

Follows from [21, Proposition 3.6, Theorem 3.9].

As with symmetric matrices, the numbers $\phi_{D}$ and $\delta_{D}$ can be
expressed as intersection products in the Chow ring of $CS_{2n}$:

######  Proposition 7.6.

$\phi_{D}(n,d)=\int_{CS_{2n}}L_{1}^{\binom{2n}{2}-d}L_{n-1}^{d-1}$
$\delta_{D}(m,n,r)=\int_{CS_{2n}}S_{r}L_{1}^{\binom{2n}{2}-m-1}L_{n-1}^{m-1}=\int_{E_{r}}L_{1}^{\binom{2n}{2}-m-1}L_{n-1}^{m-1}.$

###### Proof.

Analogous to the proof of 3.5.

From the two propositions above and the Pataki inequalities, we deduce that

(7.1) $\phi_{D}(n,d)=\frac{1}{n}\sum_{\binom{r}{2}\leq
d}{r\delta_{D}(d,n,n-r)},$

the analogue of section 3.B.

######  Definition 7.7.

We define type $D$ Lascoux coefficients $\alpha_{I}$ as follows. For
$X=(x_{1},\ldots,x_{k})$ a set of indeterminates, we denote by $\lambda(X)$
the set of indeterminates $x_{i}+x_{j}$, $1\leq i<j\leq k$. Then the
$\alpha_{I}$’s are defined by the formal identity

$s_{(d)}(\lambda(X))=\sum_{\begin{subarray}{c}\\#I=k,\\\
|\lambda(I)|=d\end{subarray}}\alpha_{I}s_{\lambda(I)}(X).$

Equivalently, for the universal bundle $\mathcal{U}$ over a Grassmannian
$G(k,m)$,

$Seg_{d}\left(\bigwedge^{2}\mathcal{U}\right)=\sum_{\begin{subarray}{c}\\#I=k\\\
|\lambda(I)|=d\end{subarray}}\alpha_{I}\sigma_{\lambda(I)}.$

For more about these coefficients, see [16, Proposition A.16].

######  Theorem 7.8.

$\delta_{D}(m,n,r)=\sum_{\begin{subarray}{c}I\subset[2n]\\\ \\#I=2n-2r\\\
\sum{I}=m\end{subarray}}{\alpha_{I}\alpha_{[2n]\setminus I}}$

###### Proof.

Analogous to the proof of 3.7.

###  7.C. Induction relations and polynomiality

We will now prove the polynomiality (or more precisely, quasipolynomiality) of
$\alpha_{[k]\setminus I}$. The following recursive relations will be central
to our proof:

######  Lemma 7.9.

  1. (1)

For $j_{1}>0$ we have:

(7.2)
$\alpha_{\\{j_{1},\dots,j_{s}\\}}=\begin{cases}\alpha_{\\{0,j_{1},\dots,j_{s}\\}}&\text{
if }s\text{ is even }\\\ 0&\text{ if }s\text{ is odd }\end{cases}$

  2. (2)

For $j_{1}=0$ we have:

(7.3) $\alpha_{\\{j_{1},j_{2},\ldots,j_{s}\\}}=\sum_{j_{\ell}\leq
j^{\prime}_{\ell}<j_{\ell+1}}{\alpha_{\\{j^{\prime}_{1},\ldots,j^{\prime}_{s-1}\\}}}.$

###### Proof.

First formula is [26, p. 446], [16, (A.16.3)] and [28, p. 163-166]. The proof
of the second formula is analogous to the proof of eq. 4.3 in Lemma 4.5.

######  Theorem 7.10.

Let $I=\\{i_{1},\ldots,i_{s}\\}$ be a set of strictly increasing nonnegative
integers. For $k\geq 0$ the function:

$LP^{D}_{I}(k):=\begin{cases}\alpha_{[k]\setminus I}&\text{ if
}I\subset[k],\\\ 0&\text{ otherwise}.\end{cases}$

is a quasi-polynomial in $k$ with period 2, i.e. for both even $k$ and odd $k$
it is a polynomial.

###### Proof.

We proceed as in the first proof of Theorem 4.3 by induction on $\\#I$ and
then on $\sum I$ using relations from Lemma 7.9. The difference is that in the
case $i_{0}=0$ we have

$LP^{D}_{I}(n)=\begin{cases}LP^{D}_{I\setminus{0}}(n)&\text{ if }n-\\#I\text{
is even }\\\ 0&\text{ if }n-\\#I\text{ is odd }\end{cases}$

which is clearly by induction hypothesis a quasipolynomial in $n$ with period
2. The rest is analogous as in the proof of Theorem 4.3.

From 7.8 and 7.10 we deduce the polynomiality of $\delta_{D}$:

######  Theorem 7.11.

For every fixed $m,s$, the function $\delta_{D}(m,n,n-s)$ is a polynomial in
$n$.

Using eq. 7.1,we also get the polynomiality of $\phi_{D}$:

######  Theorem 7.12.

For any fixed $d$, the function $\phi_{D}(n,d)$ is a polynomial for $n>0$.

###  7.D. Proof of the NRS conjecture in type $D$

The proof of Theorem 7.16 will be extremely similar to that of the original
NRS Conjecture.

######  Lemma 7.13.

Let $J$ be a set of $r$ nonnegative integers, with $\sum J\leq m$. Then

$\sum_{\begin{subarray}{c}I\geq J\\\ \sum I\leq
m\end{subarray}}\alpha_{I}\left(-\frac{1}{2}\right)^{\sum I-\sum
J}s_{I,J}\binom{m-1}{m-\sum I}=\begin{cases}0&\text{ if }\sum J<m\\\
\alpha_{J}&\text{ if }\sum J=m\par\end{cases}$

###### Proof.

Given a set of $r$ variables $X=(x_{1},\dots,x_{r})$, we denote by
$\lambda(X)$ the set of variables $(x_{i}+x_{j},i<j)$. Using Lemma 2.4 we
compute $s_{(m-\binom{r}{2})}(\lambda(X))$ as

$\displaystyle\sum_{k=\binom{r}{2}}^{m}\binom{m-1}{m-k}s_{(k-\binom{r}{2})}(\lambda(X)-1)=\sum_{k=\binom{r}{2}}^{m}\binom{m-1}{m-k}s_{(k-\binom{r}{2})}(\lambda(X-\frac{1}{2}))=$
$\displaystyle\hskip
28.45274pt=\sum_{k=\binom{r}{2}}^{m}\binom{m-1}{m-k}\sum_{\sum
I=k}\alpha_{I}s_{\lambda(I)}(X-\frac{1}{2})$ $\displaystyle\hskip
56.9055pt=\sum_{k=\binom{r}{2}}^{m}\binom{m-1}{m-k}\sum_{\sum
I=k}\alpha_{I}\sum_{J\leq I}s_{I,J}(-\frac{1}{2})^{\sum I-\sum
J}s_{\lambda(J)}(X).$

Comparing this expression the expansion in Definition 6.7 yields the claim.

######  Definition 7.14.

For $I$ a set of nonnegative integers, we define $d_{I}(n)$ by the formula:

$d_{I}(n):=P_{I}(\underbrace{1/2,\dots,1/2}_{n\;\rm times}).$

Like the Schur $P$-polynomial themselves [19, Section III.8], these
polynomials may be computed recursively. The following lemma describes the
relation between $d_{I}(n)$ and the Lascoux coefficients $\alpha_{J}$.

######  Lemma 7.15.

Let $I$ be a set of $r$ nonnegative integers. Then

$d_{I}(n)=\sum_{J\leq I}\left(\frac{1}{2}\right)^{\sum I-\sum
J}s_{I,J}\alpha_{[n]\setminus J},$ $\alpha_{[n]\setminus I}=\sum_{J\leq
I}\left(-\frac{1}{2}\right)^{\sum I-\sum J}s_{I,J}d_{J}(n).$

###### Proof.

The two formulas are equivalent; we shall prove the first one. For a vector
bundle $\mathcal{E}$ of rank $n$ over some base $X$, we consider the relative
Grassmannian $G^{r}(\mathcal{E})$ of rank $r$ quotients of $\mathcal{E}$, with
its projection $\pi$ to $X$. We denote by $\mathcal{S}$ and $\mathcal{Q}$ the
relative tautological subbundle and quotient bundle of $\pi^{*}\mathcal{E}$,
of respective ranks $s=n-r$ and $r$. Then for $I=(i_{1}<\cdots<i_{r})$,

$P_{I}(\mathcal{E})=\pi_{*}(c_{top}(\mathcal{S}\otimes\mathcal{Q})P_{I}(\mathcal{Q})),$

(cf. [13, Example 2, p. 50]). Moreover, [13, (4.5)], [27, Proposition 2.2] can
be written as

$P_{I}(\mathcal{Q})=c_{top}(\wedge^{2}\mathcal{Q})s_{\lambda(I)}(\mathcal{Q}).$

Since $\pi^{*}\mathcal{E}$ is an extension of $\mathcal{Q}$ by $\mathcal{S}$,
the bundle $\pi^{*}(\wedge^{2}\mathcal{E})$ admits a filtration whose
successive quotients are $\wedge^{2}\mathcal{Q}$,
$\mathcal{S}\otimes\mathcal{Q}$ and $\wedge^{2}\mathcal{S}$. Hence the
identity

$c(\mathcal{S}\otimes\mathcal{Q})c(\wedge^{2}\mathcal{Q})=s(\wedge^{2}\mathcal{S}^{*})\pi^{*}c(\wedge^{2}\mathcal{E}).$

Equation (5.1) can thus be rewritten as

$P_{I}(\mathcal{E})=c(\wedge^{2}\mathcal{E})\pi_{*}(s(\wedge^{2}\mathcal{S}^{*})s_{\lambda(I)}(\mathcal{Q}))_{|deg=\Sigma
I},$

where the last symbols mean we only keep the component of degree $\sum I$.

Now suppose that $\mathcal{E}=\mathcal{E}_{0}\otimes L$ for a line bundle $L$
and a trivial vector bundle $\mathcal{E}_{0}$. Then

$c(\wedge^{2}\mathcal{E})=(1+2\delta)^{\binom{s}{2}}.$

Moreover $G^{r}(\mathcal{E})$ is a trivial bundle over $X$, while
$\mathcal{S}=\mathcal{S}_{0}\otimes L$ and $\mathcal{Q}=\mathcal{Q}_{0}\otimes
L$ are obtained by pulling-back the tautological and quotient bundles
$\mathcal{S}_{0}$, $\mathcal{Q}_{0}$ over a fixed Grassmannian
$G^{r}(\mathbf{C}^{n})$ (we omit the pull-backs for simplicity). By Definition
2.3 we have:

$s_{\lambda(I)}(\mathcal{Q})=\sum_{J\leq
I}s_{I,J}s_{\lambda(J)}(\mathcal{Q}_{0})\delta^{\Sigma I-\Sigma J},$

where $\delta=c_{1}(L)$. Moreover, the Segre classes of
$\wedge^{2}\mathcal{S}_{0}^{*}$ and $\wedge^{2}\mathcal{S}^{*}$are related by
the formula

$s(\wedge^{2}\mathcal{S}^{*})=\sum_{\ell\geq
0}(1+2\delta)^{-\binom{r}{2}-\ell}s_{\ell}(\wedge^{2}\mathcal{S}_{0}^{*}).$

Plugging these two formulas into the previous one, we get $P_{I}(\mathcal{E})$
as

$\sum_{J\leq
I}\sum_{L}(1+2\delta)^{\binom{n}{2}-\binom{r}{2}-|\lambda(L)|}\delta^{\Sigma
I-\Sigma
J}s_{I,J}\alpha_{L}\pi_{*}(s_{\lambda(L)}(\mathcal{S}_{0}^{*})s_{\lambda(J)}(\mathcal{Q}_{0}))_{|deg=\Sigma
I}.$

Now recall that the Schur classes $s_{\alpha}(\mathcal{S}_{0}^{*})$ and
$s_{\beta}(\mathcal{Q}_{0})$, for partitions $\alpha\subset(r^{s})$ and
$\beta\subset(s^{r})$, that are non zero, give dual bases of Schubert cycles
on the Grassmannian $G^{r}(\mathbf{C}^{n})$. This can be expressed as

$\pi_{*}(s_{\lambda(L)}(\mathcal{S}_{0}^{*})s_{\lambda(J)}(\mathcal{Q}_{0}))=\delta_{L,[n]/J},$

where $\delta_{L,[n]/J}$ is the Kronecker delta. Note that $L=[n]/J$ implies
that $|\lambda(L)|+|\lambda(J)|=rs$. We thus get the formula

$P_{I}(\mathcal{E})=\Big{(}\sum_{J\leq I}(1+2\delta)^{\Sigma J}\delta^{\Sigma
I-\Sigma J}s_{I,J}\alpha_{[n]/J}\Big{)}_{|deg=\Sigma I}.$

But since the degree of the polynomial into brackets is exactly $\Sigma I$, we
just need to keep its top degree component, that is

$P_{I}(\mathcal{E})=\sum_{J\leq I}2^{\Sigma
J}s_{I,J}\alpha_{[n]/J}\delta^{\Sigma I}.$

We conclude by applying formally this formula to the bundle
$E=\mathcal{O}(1/2)^{\oplus n}$ over the projective space.

######  Theorem 7.16.

Let $m,n,r$ be positive integers. Then

$\delta_{D}(m,n,n-r)=\sum_{\sum I\leq m}(-1)^{m-\sum
I}\alpha_{I}d_{I}(n)\binom{m-1}{m-\sum I}$

where the sum goes through all sets of nonnegative integers of cardinality
$r$.

###### Proof.

We replace $d_{I}(n)$ by the expression from Lemma 5.4, change the order of
summation and use Lemma 7.13 in the last step:

$\displaystyle\sum_{\sum I\leq m}(-1)^{m-\sum
I}\alpha_{I}d_{I}(n)\binom{m-1}{m-\sum I}=$ $\displaystyle\hskip
56.9055pt=\sum_{\sum I\leq m}\sum_{J\leq I}s_{I,J}\alpha_{[2n]\setminus
J}\left(\frac{1}{2}\right)^{\sum I-\sum J}(-1)^{m-\sum
I}\alpha_{I}\binom{m-1}{m-\sum I}$ $\displaystyle\hskip 56.9055pt=\sum_{\sum
J\leq m}(-1)^{m-\sum J}\alpha_{[2n]\setminus J}\sum_{\begin{subarray}{c}I\geq
J\\\ \sum I\leq m\end{subarray}}s_{I,J}\left(-\frac{1}{2}\right)^{\sum I-\sum
J}\alpha_{I}\binom{m-1}{m-\sum I}$ $\displaystyle\hskip 142.26378pt=\sum_{\sum
J=m}\alpha_{[2n]\setminus J}\alpha_{J}=\delta_{D}(m,n,n-s).$

##  8\. Future directions and Conjectures

###  8.A. Dual degrees of defective determinantal loci

For symmetric matrices, 4.1 asserts that when $SD^{n-s,n}_{m}$ is not dual
defective, that is, when Pataki’s inequalities (3.2) are satisfied, its
codegree depends polynomially on $n$ when $s$ and $m$ are fixed. By the
fundamental duality relation (2.10), this also means that the codegree of
$SD^{s,n}_{\binom{n+1}{2}-m}$ depends polynomially on $n$ for $s$ and $m$
fixed.

What does happen in the defective cases? For example, by [15], one has the
formula

$\mathrm{codegree}(SD^{s,n}_{\binom{n+1}{2}-1})=\mathrm{degree}(SD^{n-s,n}_{\binom{n+1}{2}-1})=\prod_{j=1}^{s}\frac{\binom{n+j-1}{s-j+1}}{\binom{2j-1}{j-1}},$

which is clearly polynomial in $n$. Note that in this case the dual defect is
$\binom{s+1}{2}-1$, independently of $n$. Could it happen that the following
holds?

######  Conjecture 8.1.

For any fixed $m>0$,

  1. (1)

the dual defect of $SD^{s,n}_{\binom{n+1}{2}-m}$ is equal to
$\max(0,\binom{s+1}{2}-m)$, independently of $n$,

  2. (2)

the codegree of $SD^{s,n}_{\binom{n+1}{2}-m}$ depends polynomially on $n$.

Similar statements should hold for general and skew-symmetric matrices. For
general matrices, the degrees of the determinantal loci have been computed in
[15]. In particular, this implies that

$\mathrm{codegree}(D^{s,n}_{n^{2}-1})=\mathrm{degree}(D^{n-s,n}_{n^{2}-1})=\prod_{j=1}^{s}\frac{\binom{n+j-1}{2j-1}}{\binom{s+j-1}{2j-1}}$

is polynomial in $n$. In this case the dual defect is $s^{2}-1$, independently
of $n$. Could it happen that the following holds?

######  Conjecture 8.2.

For any fixed $m>0$,

  1. (1)

the dual defect of $D^{s,n}_{n^{2}-m}$ is equal to $\max(0,s^{2}-m)$,
independently of $n$,

  2. (2)

the codegree of $D^{s,n}_{n^{2}-m}$ depends polynomially on $n$.

Finally, for skew-symmetric matrices, the degrees of the determinantal loci
have also been computed in [15]. In particular, this implies that

$\mathrm{codegree}(AD^{2s,2n}_{\binom{2n}{2}-1})=\mathrm{degree}(AD^{2n-2s,2n}_{\binom{2n}{2}-1})=\frac{1}{2^{2s-1}}\prod_{j=1}^{2s-1}\frac{\binom{2n+j-1}{2s-j}}{\binom{2j-1}{j-1}}$

is polynomial in $n$. In this case the dual defect is $\binom{2s}{2}-1$,
independently of $n$. Could it happen that the following holds, and a similar
statement in odd dimensions?

######  Conjecture 8.3.

For any fixed $m>0$,

  1. (1)

the dual defect of $AD^{2s,2n}_{\binom{2n}{2}-m}$ is equal to
$\max(0,\binom{2n}{2}-m)$, independently of $n$,

  2. (2)

the codegree of $AD^{2s,2n}_{\binom{2n}{2}-m}$ depends polynomially on $n$.

###  8.B. Dual degrees of singular varieties

There is a general Plücker formula for the degree of the dual variety $X^{*}$
of a possibly singular projective variety $X\subset{\mathbb{P}}^{n}$ [11,
Theorem 1.1]:

$\mathop{\rm deg}\nolimits
X^{*}=(-1)^{n}\left(\raisebox{2.0pt}{$\chi$}(\mathop{\rm
Eu}\nolimits_{X})-2\raisebox{2.0pt}{$\chi$}(\mathop{\rm
Eu}\nolimits_{X_{1}})+\raisebox{2.0pt}{$\chi$}(\mathop{\rm
Eu}\nolimits_{X_{2}})\right),$

where $X_{1}$ (resp. $X_{2}$) is a general hyperplane section (resp.
codimension two linear section) of $X$. Here $\mathop{\rm Eu}\nolimits(X)$
denotes the _Euler obstruction_ of $X$ [12, Example 4.2.9 and Example 19.1.7].
Our Theorem 5.2 confirming the NRS conjecture, and their analogues for the
other types of determinantal loci, would immediately follow if one could prove
that Pragacz’s formulas [26] for the topological Euler characteristics of
determinantal loci, which a priori are only valid in the smooth case, can also
be used in the singular case, and compute the Euler characteristics of the
Euler obstruction. This is an approach we plan to investigate in the near
future, and could potentially apply also to the defective cases.

###  8.C. Polynomiality of intersection products

4.2 shows that for fixed $d$, the product

$\phi(n,d)=\int_{CQ_{n}}L_{1}^{d-1}L_{n-1}^{\binom{n+1}{2}-d}$

is a polynomial in $n$. Can this result be generalized to other intersection
products on $CQ$? More precisely, we have the following question:

######  Question 8.4.

Let $d_{1},\ldots,d_{k}\in{\mathbb{N}}$ with $\sum{d_{i}}=d-1$. Is

$\int_{CQ_{n}}L_{1}^{d_{1}}L_{2}^{d_{2}}\cdots
L_{k}^{d_{k}}L_{n-1}^{\binom{n+1}{2}-d}$

a polynomial in $n>k+1$?

More generally: by which cohomology classes can one replace
$L_{1}^{d_{1}}L_{2}^{d_{2}}\cdots L_{k}^{d_{k}}$ for the polynomiality
property to hold?

###  8.D. Representation theory

The version of the NRS conjecture that we proved in type A (6.17) expresses
the codegree $\delta_{A}(m,n,n-s)$ as a linear combination of dimensions of
Schur modules of $GL(n,\mathbb{C})$, with highest weights and multiplicities
depending only on $m$ and $r$. In other words, it is obtained as the
dimensional evaluation of a fixed character, depending only on $m$ and $s$. A
natural question is: what is really this character? Is there a natural
(combinatorial, or geometric) interpretation of the corresponding
representations?

The same question can be raised both in types B and D, where the dependence in
$n$ of the codegrees $\delta(m,n,n-s)$ and $\delta_{D}(m,n,n-s)$ only appears
through the number of one’s one which a certain combination of Q-Schur or
P-Schur functions are evaluated. We mentionned in the introduction that these
evaluations of Q-Schur or P-Schur functions count certain types of shifted
tableaux. An alternative interpretation is that they give the dimensions of
certain representations of the queer Lie super-algebra $\mathfrak{q}(n)$ (see
for example [5, Theorem 4.11]). So $\delta(m,n,n-s)$ and $\delta_{D}(m,n,n-s)$
can also be interpreted as dimensions of certain representations of
$\mathfrak{q}(n)$, whose characters only depend on $m$ and $s$. What are these
representations? Do they admit natural constructions or interpretations?

###  8.E. Noncommutative matroids

We believe that our results may also be viewed in the context of
noncommutative matroids. Indeed, suppose we restrict to diagonal matrices,
instead of symmetric. One may still consider the rational map given by
inverting matrices, which gives the classical Cremona transformation. The
famous resolution of that graph given by the permutohedral variety is the
analog of the variety of complete quadrics. A representable matroid may be
viewed as a subspace of the space of diagonal matrices. Many interesting
invariants of this matroid may be read from the cohomology class of its strict
transform in the permutohedral variety. In this analogy, the representable
symmetric noncommutative matroid, whould be a subspace of symmetric matrices.
Its crucial invariants should come from the cohomology class of the strict
transform of that subspace to the variety of complete quadrics. For future
work, it would be interesting to dare to define noncommutative matroids as
special cohomology classes of the variety of complete quadrics.

###  8.F. Algebraic statistics

In terms of algebraic statistics, the number $\phi(n,d)$ is equal to the
maximum likelihood degree of a general _linear concentration model_. A related
quantity is the maximum likelihood degree of a general _linear covariance
model_. For $\mathcal{L}\subseteq S^{2}({\mathbb{C}}^{n})$, the ML-degree of
the associated linear concentration model is the number of pairs
$(\Sigma,K)\in S^{2}({\mathbb{C}}^{n})^{2}$ satisfying

$\Sigma\cdot K=\mathop{\rm Id}\nolimits_{n}\text{, }K\in\mathcal{L}\text{,
}\Sigma-S\in\mathcal{L}^{\perp},$

where $S$ is generic. In contrast, the ML-degree of the associated linear
covariance model is the number of pairs $(\Sigma,K)\in
S^{2}({\mathbb{C}}^{n})^{2}$ for which

$\Sigma\cdot K=\mathop{\rm Id}\nolimits_{n}\text{,
}\Sigma\in\mathcal{L}\text{, }KSK-K\in\mathcal{L}^{\perp}.$

In [37], it was conjectured that for generic $\mathcal{L}$, these ML-degrees
also are a polynomial in $n$. In future work, we plan to apply our geometric
methods in order to prove this conjecture.

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