1.2pt Reciprocal ML degrees of diagonal linear concentration models 14C17,
05B35, 62R01 C. Eur - T. Fife - J. SAMPER - T. SEYNNAEVE

# Reciprocal maximum likelihood degrees of diagonal linear concentration
models

Christopher Eur<EMAIL_ADDRESS>Department of Mathematics
Stanford University
Tara Fife<EMAIL_ADDRESS>Max Planck Institute
for Mathematics in the Sciences
José Alejandro Samper<EMAIL_ADDRESS>Departamento de Matemáticas
Pontificia Universidad Católica de Chile
Tim Seynnaeve<EMAIL_ADDRESS>Mathematical Institute
University of Bern

###### Abstract

We show that the reciprocal maximal likelihood degree (rmld) of a diagonal
linear concentration model $\mathcal{L}\subseteq\mathbb{C}^{n}$ of dimension
$r$ is equal to $(-2)^{r}\chi_{M}(\textstyle\frac{1}{2}),$ where $\chi_{M}$ is
the characteristic polynomial of the matroid $M$ associated to $\mathcal{L}$.
In particular, this establishes the polynomiality of the rmld for general
diagonal linear concentration models, positively answering a question of
Sturmfels, Timme, and Zwiernik.

###### keywords:

maximum likelihood degrees, reciprocal spaces, matroids, characteristic
polynomials

##  1 Introduction

Let $\mathbb{S}^{n}$ be the space of (real or complex) $n\times n$ symmetric
matrices, and $\mathbb{S}^{n}_{>0}$ the subset consisting of real positive
definite symmetric matrices. For a fixed $S\in\mathbb{S}^{n}_{>0}$, the log-
likelihood function $\ell_{S}:\mathbb{S}^{n}_{>0}\to\mathbb{R}$ is defined by

$\ell_{S}(K):=\log\det K-\operatorname{trace}(S\cdot K).$

For a subvariety $\mathcal{L}\subseteq\mathbb{S}^{n}$, the maximum likelihood
(ML) degree $\operatorname{mld}(\mathcal{L})$ is the number of invertible
complex critical points of $\ell_{S}$ on the smooth locus of $\mathcal{L}$,
counted with multiplicity, for a general choice of $S$. Writing
$\mathcal{L}^{-1}\subseteq\mathbb{S}^{n}$ for the subvariety obtained as the
closure of $\\{K^{-1}\in\mathbb{S}^{n}\mid K\in\mathcal{L}\text{
invertible}\\}$, one defines the reciprocal maximum likelihood degree
$\operatorname{rmld}(\mathcal{L})$ as the number of invertible complex
critical points of $\ell_{S}$ on the smooth loci of $\mathcal{L}^{-1}$,
counted with multiplicity.

Computing (reciprocal) ML degrees arises in statistical applications, where
$\mathbb{S}^{n}_{>0}$ is often considered as the set of concentration matrices
of multivariate normal distributions [17]. We caution that the terminology
here regarding reciprocal vs. non-reciprocal ML degree is the opposite of that
in [16, 1], where $\mathbb{S}^{n}_{>0}$ is considered as the set of
_covariance_ matrices (inverses of concentration matrices). In particular, our
$\operatorname{rmld}$ is the ML degree of a linear covariance model. Our
convention here agrees with [17, 10, 5, 8].

Let $[n]=\\{1,\ldots,n\\}$. A diagonal linear concentration model is a linear
subspace $\mathcal{L}\subseteq\mathbb{C}^{[n]}$, where $\mathbb{C}^{[n]}$ is
identified with the space of diagonal matrices in $\mathbb{S}^{n}$. Let $M$ be
the matroid on $[n]$ whose independent subsets are $I\subseteq[n]$ such that
the composition
$\mathbb{C}^{I}\hookrightarrow\mathbb{C}^{[n]}\twoheadrightarrow\mathcal{L}^{\vee}$
is injective. Without loss of generality, we always assume that $\mathcal{L}$
is not contained in a coordinate hyperplane, or equivalently, that $M$ is
loopless, since otherwise $\operatorname{rmld}(\mathcal{L})=0$ from the
definition. Our main result is the formula for the reciprocal ML degree of
$\mathcal{L}$ in terms of $M$.

######  Theorem 1.

Let $\mathcal{L}\subseteq\mathbb{C}^{[n]}$ be a diagonal linear concentration
model of dimension $r$, and $M$ the associated matroid of rank $r$ on $[n]$.
Then we have

$\operatorname{rmld}(\mathcal{L})=(-2)^{r}\chi_{M}(\textstyle\frac{1}{2}),$

where $\chi_{M}$ is the characteristic polynomial of $M$.

In [17, 16], the (non-reciprocal) ML degree of $\mathcal{L}$ was shown to be
$|\chi_{M}(0)|$. Computing the reciprocal ML degree presents fundamentally new
challenges; see Remark 11 for a comparison.

From computational experiments, the authors of [16] asked whether the
reciprocal ML degree of a _general_ diagonal linear concentration model of
dimension $r$ in $\mathbb{C}^{[n]}$ is a polynomial in $n$ of degree $r-1$.
Evaluating our Theorem 1 at uniform matroids answers their question
positively.

######  Corollary 2.

Let $\mathcal{L}\subseteq\mathbb{C}^{[n]}$ be a general linear concentration
model of dimension $r$. Then we have

$\operatorname{rmld}(\mathcal{L})=\sum_{i=1}^{r}\textstyle\binom{n-i-1}{r-i}2^{r-i}.$

For instance, when $r=3$ we have $2n^{2}-8n+7$, and when $r=4$ we have
$4/3n^{3}-10n^{2}+68/3n-15$, as predicted in [16] from numerical computations.

To prove Theorem 1, we use the following alternate description of the
reciprocal ML degree, obtained by a standard computation in multivariable
calculus. Let $\mathcal{L}^{\perp}$ denote the orthogonal complement of a
subspace $\mathcal{L}\subseteq\mathbb{C}^{[n]}$ under the standard pairing

$\langle(x_{1},\ldots,x_{n}),(y_{1},\ldots,y_{n})\rangle:=\sum_{i=1}^{n}x_{i}y_{i}.$

######  Proposition 3.

[16, Proposition 4.3] The reciprocal ML degree of a linear subspace
$\mathcal{L}\subseteq\mathbb{C}^{[n]}$ is equal to the number of solutions
$(x_{1},\ldots,x_{n})\in(\mathbb{C}^{*})^{[n]}$, counted with multiplicity, to
the following system of equations, where $s_{1},\ldots,s_{n}\in\mathbb{C}$ are
generic parameters:

$(x_{1}^{-1},\ldots,x_{n}^{-1})\in\mathcal{L}\quad\text{and}\quad(s_{1}x_{1}^{2}-x_{1},\ldots,s_{n}x_{n}^{2}-x_{n})\in\mathcal{L}^{\perp}.$

Thus, we prove Theorem 1 by establishing the following generalization.

######  Theorem 4.

For an $r$-dimensional linear subspace $\mathcal{L}\subseteq\mathbb{C}^{[n]}$,
a generic choice of parameters $s_{1},\ldots,s_{n}\in\mathbb{C}$, and any
integer $d\geq 1$, the number of solutions
$(x_{1},\ldots,x_{n})\in(\mathbb{C}^{*})^{[n]}$, counted with multiplicity, to
the system of equations

$(x_{1}^{-1},\ldots,x_{n}^{-1})\in\mathcal{L}\quad\text{and}\quad(s_{1}x_{1}^{d}-x_{1},\ldots,s_{n}x_{n}^{d}-x_{n})\in\mathcal{L}^{\perp}$
($\dagger$)

is equal to

$(-d)^{r}\chi_{M}(\textstyle\frac{1}{d}),\quad\text{or equivalently,}\quad
d^{r}T_{M}(1-\frac{1}{d},0),$

where $\chi_{M}$ is the characteristic polynomial and $T_{M}$ is the Tutte
polynomial of the matroid $M$ associated to $\mathcal{L}$.

######  Remark 5.

For an $r$-dimensional subspace $\mathcal{L}\subseteq\mathbb{C}^{[n]}$, let
$U(\mathcal{L}):=\mathcal{L}\cap(\mathbb{C}^{*})^{[n]}$ be the hyperplane
arrangement complement, and $M$ the associated matroid. Then the Poincaré
polynomial of $U(\mathcal{L})$

$P_{U(\mathcal{L})}(q):=\sum_{i\geq
0}\big{(}\operatorname{rank}H_{i}(U(\mathcal{L});\mathbb{Z})\big{)}q^{i}$

coincides with polynomial $(-q)^{r}\chi_{M}(-\frac{1}{q})$ [11]. In
particular, Theorem 1 states that
$\operatorname{rmld}(\mathcal{L})=(-1)^{r}P_{U(\mathcal{L})}(-2)$. This echoes
the result of [7], which showed that, for a different log-likelihood function
(from discrete statistical models), the ML degree of a smooth very affine
variety $U$ is its signed topological Euler characteristic $(-1)^{\dim
U}P_{U}(-1)$. However, ML degrees in our case are not topological invariants
of very affine varieties: Observe that
$\mathcal{L}^{-1}\cap(\mathbb{C}^{*})^{[n]}\simeq\mathcal{L}\cap(\mathbb{C}^{*})^{[n]}$
but in general
$\operatorname{rmld}(\mathcal{L}^{-1})=\operatorname{mld}(\mathcal{L})=\chi_{M}(0)\neq(-2)^{r}\chi_{M}(\frac{1}{2})=\operatorname{rmld}(\mathcal{L})$.
It may still be interesting to find other families of subvarieties
$\mathcal{L}\subseteq\mathbb{S}^{n}$ such that
$\operatorname{rmld}(\mathcal{L})=(-1)^{\dim\mathcal{L}}P_{U(\mathcal{L})}(-2)$,
where $U(\mathcal{L}):=\\{K\in\mathcal{L}\mid K\text{ invertible}\\}$. For
example, general pencils of conics form one such family [3, 5].

Outline. In Section 2 we review properties of reciprocal linear spaces
$\mathcal{L}^{-1}$, and introduce score varieties, which together with
$\mathcal{L}^{-1}$ encode the system of equations ($\dagger$ ‣ 4). After
establishing two key technical lemmas in Section 3, in Section 4 we compute
the number of solutions to the system of equations ($\dagger$ ‣ 4) in
$\mathbb{C}^{[n]}$, instead of in $(\mathbb{C}^{*})^{[n]}$, in two different
ways: One is a Bézout-like computation, and the other is a summation, with
each summand corresponding to a set of solutions with specified support (non-
zero coordinates). An inclusion-exclusion argument in Section 5 then yields
the proof of Theorem 4.

### Notation

For an affine subvariety $X\subseteq\mathbb{C}^{n}$, we write
$\overline{X}\subseteq\mathbb{P}^{n}$ for its projective closure. If
$X\subseteq\mathbb{C}^{n}$ is defined by a homogeneous ideal, then we write
$\mathbb{P}X\subseteq\mathbb{P}^{n-1}$ for its projectivization. For a point
$p\in X\subseteq\mathbb{C}^{[n]}$, we write $TC_{p}X\subseteq\mathbb{C}^{[n]}$
for the tangent cone of $X$ at $p$. For
$p=(p_{1},\ldots,p_{n})\in\mathbb{C}^{[n]}$, we write
$\operatorname{supp}(p):=\\{i\in[n]\mid p_{i}\neq 0\\}$ for its support, and
for $I\subseteq[n]$, write $p_{|I}$ for the projection of $p$ onto
$\mathbb{C}^{I}\subseteq\mathbb{C}^{[n]}$.

##  2 Reciprocal linear spaces and score varieties

We set notations concerning matroids associated to linear subspaces, and
review necessary facts about reciprocal linear spaces. We assume familiarity
with matroid theory, and refer to [19, 12] as standard references.

Let us fix a linear subspace $\mathcal{L}\subseteq\mathbb{C}^{[n]}$ of
dimension $r$. Let $A$ be an $r\times n$ matrix whose row-span equals
$\mathcal{L}$. We will often use the fact that the minimal sets among supports
of elements in the row-span of $A$ form the cocircuits of $M$.

For a subset $I\subseteq[n]$, let $\mathcal{L}_{|I}\subseteq\mathbb{C}^{I}$ be
the image of $\mathcal{L}$ under the coordinate projection
$\mathbb{C}^{[n]}\twoheadrightarrow\mathbb{C}^{I}$, and let $\mathcal{L}_{/I}$
be the intersection of $\mathcal{L}$ with the coordinate subspace
$\\{0\\}^{I}\times\mathbb{C}^{[n]\setminus I}$, considered as a subspace of
$\mathbb{C}^{[n]\setminus I}$. The matroid of $\mathcal{L}_{|I}$ is the
restriction $M|I$, whereas the matroid of $\mathcal{L}_{/I}$ is the
contraction $M/I$.

The reciprocal linear space $\mathcal{L}^{-1}$ of $\mathcal{L}$ is the Zariski
closure in $\mathbb{C}^{[n]}$ of
$\\{(x_{1},\ldots,x_{n})\in(\mathbb{C}^{*})^{[n]}\mid(x_{1}^{-1},\ldots,x_{n}^{-1})\in\mathcal{L}\\}$.
Note that $\mathcal{L}^{-1}\cap(\mathbb{C}^{*})^{[n]}$ is smooth, being
isomorphic to $\mathcal{L}\cap(\mathbb{C}^{*})^{[n]}$. For $I\subseteq[n]$, we
write $\mathcal{L}^{-1}_{|I}$ for $(\mathcal{L}_{|I})^{-1}$, and likewise
write $\mathcal{L}^{-1}_{/I}=(\mathcal{L}_{/I})^{-1}$. We collect together in
the following theorem the known properties of $\mathcal{L}^{-1}$ that we will
need.

######  Theorem 6.

Let $\mathcal{L}^{-1}\subseteq\mathbb{C}^{[n]}$ be the reciprocal linear space
of $\mathcal{L}\subseteq\mathbb{C}^{[n]}$.

  1. (a)

[13, Lemma 2] The ideal of $\mathcal{L}^{-1}$ is homogeneous, and
$\mathbb{P}\mathcal{L}^{-1}$ has degree $|\mu(M)|$, where
$\mu(M):=\chi_{M}(0)$ is the Möbius invariant of the matroid $M$.

  2. (b)

[13, Proposition 7] The ideal of $\mathcal{L}^{-1}$ is Cohen-Macaulay, with
any basis of $\mathcal{L}^{\perp}$ forming a system of parameters, i.e.
$\mathcal{L}^{-1}\cap\mathcal{L}^{\perp}=\\{\mathbf{0}\\}$.

  3. (c)

[13, Proposition 5] The intersection
$\mathcal{L}^{-1}\cap((\mathbb{C}^{*})^{F}\times\\{0\\}^{[n]\setminus F})$ is
nonempty if and only if $F\subseteq[n]$ is a flat of $M$, and in that case,
one has

$\mathcal{L}^{-1}\cap((\mathbb{C}^{*})^{F}\times\\{0\\}^{[n]\setminus
F})=(\mathcal{L}_{|F}^{-1}\cap(\mathbb{C}^{*})^{F})\times\\{0\\}^{[n]\setminus
F}.$

  4. (d)

[14, Theorem 24] For a flat $F\subseteq[n]$ and a point $p\in\mathcal{L}^{-1}$
with $\operatorname{supp}(p)=F$, the tangent cone of $\mathcal{L}^{-1}$ at $p$
is the product

$TC_{p}\mathcal{L}^{-1}=TC_{p_{|F}}\mathcal{L}_{|F}^{-1}\times\mathcal{L}_{/F}^{-1}\simeq\mathcal{L}_{|F}\times\mathcal{L}_{/F}^{-1}.$

All four statements in Theorem 6 can be derived easily from the Gröbner basis
for the defining ideal of $\mathcal{L}^{-1}$ computed in [13, Theorem 4]. In
[14], the statement of Theorem 6.(d) originally reads
$TC_{p}\mathcal{L}^{-1}=p_{|F}^{2}\mathcal{L}_{|F}\times\mathcal{L}_{/F}^{-1}$,
where $p^{2}\mathcal{L}$ denotes the linear subspace
$\\{(p_{1}^{2}x_{1},\ldots,p_{n}^{2}x_{n})\mid(x_{1},\ldots,x_{n})\in\mathcal{L}\\}$.
It is straightforward to verify that
$TC_{p_{|F}}\mathcal{L}_{|F}^{-1}=p_{|F}^{2}\mathcal{L}_{|F}$. Theorem 6.(a)
also follows from [18, Theorem 1.2], which expressed the Hilbert series of the
ideal of $\mathcal{L}^{-1}$ in terms of the characteristic polynomial
$\chi_{M}$.

The reciprocal linear space $\mathcal{L}^{-1}$ encodes the left half of the
system of equations in Equation ($\dagger$ ‣ 4). Let us now consider the
variety encoding the condition
$(s_{1}x_{1}^{d}-x_{1},\ldots,s_{n}x_{n}^{d}-x_{n})\in\mathcal{L}^{\perp}$.
For an integer $d\geq 1$ and a parameter
$s=(s_{1},\ldots,s_{n})\in\mathbb{C}^{n}$, we define the score variety as

$Y(\mathcal{L},s,d):=\\{(x_{1},\ldots,x_{n})\in\mathbb{C}^{[n]}\mid(s_{1}x_{1}^{d}-x_{1},\ldots,s_{n}x_{n}^{d}-x_{n})\in\mathcal{L}^{\perp}\\}.$

We will simply write $Y$ when we trust that no confusion will arise. We note
here that score varieties are smooth for a generic choice of
$s\in\mathbb{C}^{n}$.

######  Lemma 2.1.

For $d\geq 1$ and a generic choice of $(s_{1},\ldots,s_{n})\in\mathbb{C}^{n}$,
the score variety $Y$ is smooth.

######  Proof 2.2.

If $d=1$, then $Y$ is linear, so suppose $d\geq 2$. Let $A$ be the $r\times n$
matrix whose row span equals $\mathcal{L}$, and let $g_{1},\ldots,g_{r}$ be
the polynomials obtained by multiplying the rows of $A$ with
$(s_{1}x_{1}^{d}-x_{1},\ldots,s_{n}x_{n}^{d}-x_{n})^{T}$. These minimally
generate the defining ideal $I_{Y}\subseteq\mathbb{C}[x_{1},\ldots,x_{n}]$ of
$Y$. The Jacobian matrix with respect to these minimal generators is

$\operatorname{Jac}(x)=A\cdot\operatorname{diag}(ds_{1}x_{1}^{d-1}-1,\ldots,ds_{n}x_{n}^{d-1}-1),$
(1)

i.e. matrix $A$ whose $i$-th column is scaled by $ds_{i}x_{i}^{d-1}-1$ for
each $1\leq i\leq n$. Suppose now that $\operatorname{Jac}(x)$ has rank $<r$
for some $x\in\mathbb{C}^{n}$, that is, the restriction $M|I$ of the matroid
$M$ to the set $I=\\{i\in[n]\mid ds_{i}x_{i}^{d-1}-1\neq 0\\}$ has rank $<r$.
This happens if and only if $I$ is contained in a hyperplane flat of $M$, or
equivalently, the subset $J:=[n]\setminus I$ contains a cocircuit of $M$. As
the minimal supports of the row-space of $A$ constitute the cocircuits of $M$,
let $v=(v_{1},\ldots,v_{n})\in\mathbb{C}^{n}$ be the element in the row-space
of $A$ whose support $C^{*}=\operatorname{supp}(v)\subseteq[n]$ is a cocircuit
of $M$ contained in $J$. Then we have

$v\cdot(s_{1}x_{1}^{d}-x_{1},\ldots,s_{n}x_{n}^{d}-x_{n})^{T}=\sum_{i\in
C^{*}}v_{i}(s_{i}x_{i}^{d}-x_{i})=(\textstyle{\frac{1}{d}}-1)\displaystyle\sum_{i\in
C^{*}}v_{i}x_{i}$

where last equality follows from $ds_{i}x_{i}^{d-1}-1=0$ for $i\in J$. This
quantity needs to be zero if $x\in Y$. We claim that for a general choice of
$(s_{1},\ldots,s_{n})$ this quantity can never be zero: Consider the set

$Z:=\\{(\zeta_{1},\ldots,\zeta_{n})\in\mathbb{C}^{n}\mid\zeta_{i}\text{ is a
$(d-1)$-th root of }\textstyle{\frac{1}{ds_{i}}}\text{ if $i\in C^{*}$}\\}.$

For a general choice of $(s_{1},\ldots,s_{n})$, no element of $Z$ satisfies
$\sum_{i\in C^{*}}v_{i}x_{i}=0$.

##  3 Two genericity lemmas

We now present the two key technical lemmas for our future intersection
multiplicity computations. Both make essential use of the fact that the
parameter $s\in\mathbb{C}^{n}$ can be chosen generically, and the second lemma
uses that $\mathbb{C}$ has characteristic zero. To state the first lemma, let
us define a subscheme of $\mathbb{C}^{[n]}$

$Y_{\infty}(\mathcal{L},s,d):=\\{(x_{1},\ldots,x_{n})\in\mathbb{C}^{[n]}\mid(s_{1}x_{1}^{d},\ldots,s_{n}x_{n}^{d})\in\mathcal{L}^{\perp}\\}.$

######  Lemma 3.1.

For a generic choice of $s\in\mathbb{C}^{n}$ and any integer $d\geq 1$, one
has

$\mathcal{L}^{-1}\cap Y_{\infty}(\mathcal{L},s,d)=\\{\mathbf{0}\\}.$

######  Proof 3.2.

Let us define a subscheme $V\subseteq\mathbb{C}^{[n]}\times\mathbb{C}^{[n]}$
by

$V=\\{(x,s)\in\mathbb{C}^{[n]}\times\mathbb{C}^{[n]}\mid
x\in\mathcal{L}^{-1}\text{ and
}(s_{1}x_{1}^{d},\ldots,s_{n}x_{n}^{d})\in\mathcal{L}^{\perp}\\}.$

For any point $x$ in the dense open loci
$\mathcal{L}^{-1}\cap(\mathbb{C}^{*})^{[n]}$ of $\mathcal{L}^{-1}$, the set
$\\{s\in\mathbb{C}^{[n]}\mid(s_{1}x_{1}^{d},\ldots,s_{n}x_{n}^{d})\in\mathcal{L}^{\perp}\\}$
is a linear subspace of dimension $n-r$, so the dimension of $V$ is
$r+(n-r)=n$. Moreover, the subscheme $V$ is bi-homogeneous, and thus the bi-
projectivization $\overline{V}\subseteq\mathbb{P}^{n-1}\times\mathbb{P}^{n-1}$
has dimension $n-2$. Writing $\pi_{2}$ for the projection of $\overline{V}$ to
the second $\mathbb{P}^{n-1}$, we hence find that the loci
$\mathbb{P}^{n-1}\setminus\pi_{2}(\overline{V})$ is nonempty and open in
$\mathbb{P}^{n-1}$. That is, the affine cone over
$\mathbb{P}^{n-1}\setminus\pi_{2}(\overline{V})$ is dense open in
$\mathbb{C}^{[n]}$, and $\mathcal{L}^{-1}\cap
Y_{\infty}(\mathcal{L},s,d)=\\{\mathbf{0}\\}$ for any $s\in\mathbb{C}^{[n]}$
in the affine cone.

######  Remark 7.

When $s=(1,\ldots,1)$ and $d=1$, Lemma 3.1 is the second half of Theorem
6.(b), which was established by an explicit Gröbner basis computation. For
$d\geq 2$ however, the lemma fails in general with $s=(1,\ldots,1)$.

######  Lemma 3.3.

For a generic choice of $s\in\mathbb{C}^{n}$ and any integer $d\geq 1$, the
intersection $\mathcal{L}^{-1}\cap
Y(\mathcal{L},s,d)\cap(\mathbb{C}^{*})^{[n]}$ is either empty or smooth of
dimension 0.

######  Proof 3.4.

Without loss of generality, we assume that the $r\times n$ matrix $A$ whose
row-span equals $\mathcal{L}$ is of the form $[I_{r}\ |\ A^{\prime}]$, where
$I_{r}$ is the $r\times r$ identity matrix. For $1\leq i\leq r$, let
$a^{\prime}_{i}$ be the $i$-th row of $A^{\prime}$. The ideal of $Y$ is
minimally generated by

$\mathfrak{d}=\\{(s_{i}x_{i}^{d}-x_{i})-a^{\prime}_{i}\cdot(s_{r+1}x_{r+1}^{d}-x_{r+1},\ldots,s_{n}x_{n}^{d}-x_{n})^{T}\mid
1\leq i\leq r\\}.$

Fixing a generic choice of $s_{r+1},\ldots,s_{n}$, and letting
$s_{1},\ldots,s_{r}$ vary freely, for each $i=1,\ldots,r$ we may consider
$(s_{i}x_{i}^{d}-x_{i})-a^{\prime}_{i}\cdot(s_{r+1}x_{r+1}^{d}-x_{r+1},\ldots,s_{n}x_{n}^{d}-x_{n})^{T}$
as a pencil $\mathfrak{d}_{i}$ of hypersurfaces in $\mathbb{C}^{[n]}$. Note
that the union of the base loci of $\mathfrak{d}_{1},\ldots,\mathfrak{d}_{r}$
is contained in the union of the coordinate hyperplanes. Thus, we obtain a map
$\mathcal{L}^{-1}\cap(\mathbb{C}^{*})^{[n]}\to(\mathbb{P}^{1})^{r}$ of smooth
varieties. By generic smoothness [6, III.10.7], the general fiber, which is
the intersection $\mathcal{L}^{-1}\cap Y\cap(\mathbb{C}^{*})^{[n]}$ for a
general choice of $s\in\mathbb{C}^{n}$, is either empty or smooth of dimension
0.

Let us now denote

$\begin{split}\mathcal{D}(\mathcal{L},d):=&\text{ the degree of the (empty or
0-dimensional) subscheme}\\\ &\ \mathcal{L}^{-1}\cap
Y(\mathcal{L},s,d)\cap(\mathbb{C}^{*})^{[n]}\subset\mathbb{C}^{[n]}\end{split}$

for a generic choice of $s\in\mathbb{C}^{n}$, which is equal to the number of
points in the intersection since it is smooth by Lemma 3.3. Theorem 4 is now
equivalently stated as
$\mathcal{D}(\mathcal{L},d)=(-d)^{r}\chi_{M}(\frac{1}{d})$, and Proposition 3
states that $\mathcal{D}(\mathcal{L},2)=\operatorname{rmld}(\mathcal{L})$.

##  4 Total intersection multiplicity

We now compute the degree of the intersection $\mathcal{L}^{-1}\cap
Y(\mathcal{L},s,d)$ as a subscheme of $\mathbb{C}^{[n]}$ in two different
ways. First, we have a Bézout-like computation.

######  Proposition 8.

For a generic choice of $s\in\mathbb{C}^{n}$, the intersection
$\mathcal{L}^{-1}\cap Y(\mathcal{L},s,d)$ is a 0-dimensional scheme of degree
$d^{r}|\mu(M)|$.

######  Proof 4.1.

For $i=1,\ldots,r$, let
$\overline{f}_{i}\in\mathbb{C}[x_{0},x_{1},\ldots,x_{n}]$ be the homogeneous
polynomial obtained as $i$-th row of $A$ times
$(s_{1}x_{1}^{d}-x_{1}x_{0}^{d-1},\ldots,s_{n}x_{n}^{d}-x_{n}x_{0}^{d-1})^{T}$.
We first claim that $(\overline{f}_{1},\ldots,\overline{f}_{r})$ forms a
regular sequence on the projective closure
$\overline{\mathcal{L}^{-1}}\subset\mathbb{P}^{n}$. Since the projective
variety $\overline{\mathcal{L}^{-1}}$ is (arithmetically) Cohen-Macaulay by
Theorem 6.(b), it suffices to show that the intersection
$\overline{\mathcal{L}^{-1}}\cap V(\overline{f}_{1},\ldots,\overline{f}_{r})$
is 0-dimensional, as every system of parameters in a standard graded Cohen-
Macaulay ring is a regular sequence [2, Theorem 2.1.2].

At the hyperplane at infinity, the intersection
$V(x_{0})\cap\overline{\mathcal{L}^{-1}}\cap
V(\overline{f}_{1},\ldots,\overline{f}_{r})$ is isomorphic to
$\mathbb{P}\mathcal{L}^{-1}\cap\mathbb{P}Y_{\infty}$, which is empty for a
generic $s\in\mathbb{C}^{n}$ by Lemma 3.1. On the complement of the hyperplane
at infinity, the intersection is equal to $\mathcal{L}^{-1}\cap Y$, since the
dehomogenizations of the polynomials
$\overline{f}_{1},\ldots,\overline{f}_{r}$ give the defining equations of $Y$.
From Theorem 6.(c) and the definition of $Y(\mathcal{L},s,d)$, it follows that

$\mathcal{L}^{-1}\cap Y\cap((\mathbb{C}^{*})^{F}\times\\{0\\}^{[n]\setminus
F})=(\mathcal{L}_{|F}^{-1}\cap
Y(\mathcal{L}_{|F},s_{|F},d)\cap(\mathbb{C}^{*})^{F})\times\\{0\\}^{[n]\setminus
F}$

if $F\subseteq[n]$ is a flat or empty otherwise, and thus Lemma 3.3 applied to
each flat $F$ implies that $\mathcal{L}^{-1}\cap Y$ is 0-dimensional. Thus,
the degree $d$ polynomials $(\overline{f}_{1},\ldots,\overline{f}_{r})$ form a
regular sequence on $\overline{\mathcal{L}^{-1}}$, and hence the degree of
$\overline{\mathcal{L}^{-1}}\cap V(\overline{f}_{1},\ldots,\overline{f}_{r})$
is $d^{r}\deg(\overline{\mathcal{L}^{-1}})$. As the ideal of
$\mathcal{L}^{-1}$ homogeneous, the degrees of $\mathbb{P}\mathcal{L}^{-1}$
and $\overline{\mathcal{L}^{-1}}$ are equal, with the value being $|\mu(M)|$
by Theorem 6.(a). Lastly, since $\overline{\mathcal{L}^{-1}}\cap
V(\overline{f}_{1},\ldots,\overline{f}_{r})$ is empty at the hyperplane at
infinity, the degree of the intersection is equal to the degree of
$\mathcal{L}^{-1}\cap Y(\mathcal{L},s,d)$.

We now compute the degree of $\mathcal{L}^{-1}\cap Y(\mathcal{L},s,d)$ as the
sum of contributions from the various strata of $\mathcal{L}^{-1}$. First, we
need the following notation. Note that a 0-dimensional subscheme
$X\subset\mathbb{C}^{[n]}$ is a union $\bigcup_{\alpha}X_{\alpha}$ of
irreducible (possibly non-reduced) components $X_{\alpha}$, each of which is
topologically a point $(X_{\alpha})_{red}$ in $\mathbb{C}^{[n]}$. For a subset
$I\subseteq[n]$, we write $X^{F}$ to be the subscheme of $X$ defined as the
union of components of $X$ whose support is $F$, i.e.

$X^{F}:=\bigcup\\{X_{\alpha}\mid(X_{\alpha})_{red}\in(\mathbb{C}^{*})^{F}\times\\{0\\}^{[n]\setminus
F}\\}.$

Moreover, recall the notation that $\mathcal{D}(\mathcal{L},d)$ denotes the
degree of the subscheme $\mathcal{L}^{-1}\cap
Y(\mathcal{L},s,d)\cap(\mathbb{C}^{*})^{[n]}\subset\mathbb{C}^{[n]}$ for a
generic choice of $s\in\mathbb{C}^{n}$.

######  Proposition 9.

For a generic choice of $s\in\mathbb{C}^{n}$, and for a flat $F\subseteq[n]$
of $M$, the degree of $(\mathcal{L}^{-1}\cap
Y(\mathcal{L},s,d))^{F}\subset\mathbb{C}^{[n]}$ is equal to
$\mathcal{D}(\mathcal{L}_{|F},d)\cdot|\mu(M/F)|$.

######  Proof 4.2.

As topological spaces, the subscheme $(\mathcal{L}^{-1}\cap Y)^{F}$ is equal
to the intersection $\mathcal{L}^{-1}\cap
Y\cap((\mathbb{C}^{*})^{F}\times\\{0\\}^{[n]\setminus F})$, which is by
Theorem 6.(c) isomorphic to $\mathcal{L}_{|F}^{-1}\cap
Y(\mathcal{L}_{|F},s_{|F},d)\cap(\mathbb{C}^{*})^{F}$, which as a scheme is a
disjoint union of $\mathcal{D}(\mathcal{L}_{|F},d)$ many smooth points by
Lemma 3.3. It remains only to show that if $\widetilde{p}$ is an irreducible
component in $(\mathcal{L}^{-1}\cap Y)^{F}$, then the degree of
$\widetilde{p}$ is $|\mu(M/F)|$.

For this end, we recall [4, Theorem 1.26 & Proposition 1.29]: Suppose two
Cohen-Macaulay subvarieties $X$ and $X^{\prime}$ of complementary dimensions
in a smooth variety $Z$ intersect dimensionally properly. Then, the degree of
the intersection $X\cap X^{\prime}$ at a point $q$ in the intersection is
equal to the product of the degrees of projectivized tangent cones
$\mathbb{P}TC_{q}X$ and $\mathbb{P}TC_{q}X^{\prime}$, provided that
$\mathbb{P}TC_{q}X$ and $\mathbb{P}TC_{q}X^{\prime}$ are disjoint in
$\mathbb{P}T_{q}Z$.

We apply this to $\mathcal{L}^{-1}$ and $Y$, which are Cohen-Macaulay
respectively by Theorem 6.(b) and Lemma 2.1, and they intersect dimensionally
properly by Proposition 8. Topologically $\widetilde{p}$ is a point
$p\in(\mathbb{C}^{*})^{F}\times\\{0\\}^{[n]\setminus F}$. Combining Theorem
6.(d) with Theorem 6.(a), one has that the degree of
$\mathbb{P}TC_{p}\mathcal{L}^{-1}$ is equal to $|\mu(M/F)|$. The degree of
$\mathbb{P}TC_{p}Y$ is 1 since $Y$ is smooth (Lemma 2.1). Thus, we are done
once we show that $\mathbb{P}TC_{p}\mathcal{L}^{-1}$ and $\mathbb{P}TC_{p}Y$
are disjoint. This is done in the following lemma.

######  Lemma 4.3.

Let $p$ be a point in $\mathcal{L}^{-1}\cap
Y\cap((\mathbb{C}^{*})^{F}\times\\{0\\}^{[n]\setminus F})$ for a generic
choice of $s\in\mathbb{C}^{n}$, and for a flat $F\subseteq[n]$. Then we have
$\mathbb{P}TC_{p}\mathcal{L}^{-1}\cap\mathbb{P}TC_{p}Y=\emptyset$.

######  Proof 4.4.

Let $A$ be an $r\times n$ matrix whose row-span is $\mathcal{L}$. As computed
in the proof of Lemma 2.1 in Equation (1), the tangent cone $TC_{p}Y$ is equal
to

$\ker\big{(}A\cdot\operatorname{diag}(ds_{1}p_{1}^{d-1}-1,\ldots,ds_{n}p_{n}^{d-1}-1)\big{)}\subset\mathbb{C}^{[n]}.$

Since $p_{i}=0$ for $i\in[n]\setminus F$, and since the cocircuits of the
matroid $M/F$ are cocircuits of $M$ contained in $[n]\setminus F$, if $x\in
TC_{p}Y$ then $x_{|[n]\setminus
F}\in\mathcal{L}_{/F}^{\perp}\subset\mathbb{C}^{[n]\setminus F}$. On the other
hand, by Theorem 6.(d) we have
$TC_{p}\mathcal{L}^{-1}=TC_{p_{|F}}\mathcal{L}_{|F}^{-1}\times\mathcal{L}_{/F}^{-1}.$
Let us now consider $x\in TC_{p}\mathcal{L}^{-1}\cap TC_{p}Y$. We have
$x_{|[n]\setminus
F}\in\mathcal{L}_{/F}^{\perp}\cap\mathcal{L}_{/F}^{-1}=\\{\mathbf{0}\\}$,
where the equality follows from Theorem 6.(b), and thus
$x=x_{|F}\times\mathbf{0}$. That $x\in TC_{p}Y$ now implies that $x_{|F}\in
TC_{p_{|F}}Y(\mathcal{L}_{|F},s_{|F},d)$. Thus, we conclude $x_{|F}=0$, since
by Lemma 3.3 the intersection $\mathcal{L}_{|F}^{-1}\cap
Y(\mathcal{L}_{|F},s_{|F},d)$ is smooth, and in particular transversal at
$p_{|F}$, i.e. $TC_{p_{|F}}\mathcal{L}_{|F}^{-1}\cap
TC_{p_{|F}}Y(\mathcal{L}_{|F},s_{|F},d)=\\{\mathbf{0}\\}$.

Combining Propositions 8 and 9 yields the following.

######  Corollary 10.

For a generic choice of $s\in\mathbb{C}^{n}$, we have

$d^{r}|\mu(M)|=\deg\big{(}\mathcal{L}^{-1}\cap
Y(\mathcal{L},s,d)\big{)}=\sum_{\begin{subarray}{c}F\subseteq[n]\\\ \text{a
flat of $M$}\end{subarray}}\mathcal{D}(\mathcal{L}_{|F},d)|\mu(M/F)|.$

##  5 Inclusion-exclusion

We now finish the proof of the main theorem by combining Corollary 10 with an
inclusion-exclusion argument. For the facts regarding the lattice of flats of
a matroid and the Möbius invariant used here, see [21].

######  Proof 5.1 (Proof of Theorem 4).

Write $\operatorname{rk}:2^{[n]}\to\mathbb{Z}$ for the rank function of the
matroid $M$ associated to $\mathcal{L}$. Let us recall that for a matroid
$M^{\prime}$ of rank $r^{\prime}$, one has
$(-1)^{r^{\prime}}\mu(M^{\prime})=|\mu(M^{\prime})|$. Then, Corollary 10
states that

$(-d)^{\operatorname{rk}[n]}\mu(M)=\sum_{\begin{subarray}{c}F\subseteq[n]\\\
\text{a flat of
$M$}\end{subarray}}(-1)^{\operatorname{rk}[n]-\operatorname{rk}F}\mathcal{D}(\mathcal{L}_{|F},d)\mu(M/F),$

or more generally, one has, for any flat $F\subseteq[n]$ of $M$,

$(-d)^{\operatorname{rk}F}\mu(M|F)=\sum_{\begin{subarray}{c}F^{\prime}\subseteq
F\\\ \text{a flat of
$M$}\end{subarray}}(-1)^{\operatorname{rk}F-\operatorname{rk}F^{\prime}}\mathcal{D}(\mathcal{L}_{|F^{\prime}},d)\mu(M|F/F^{\prime}).$

As $\mu(M|F/F^{\prime})$ is the value of the Möbius function
$\mu(F^{\prime},F)$ on the lattice of flats of $M$, applying the Möbius
inversion formula [15, Proposition 3.7.1] (with
$f(F)=d^{\operatorname{rk}F}\mu({M}|{F})$ and
$g(F)=(-1)^{\operatorname{rk}F}\mathcal{D}(\mathcal{L}_{|F},d)$) yields

$(-1)^{\operatorname{rk}F}\mathcal{D}(\mathcal{L}_{|F},d)=\sum_{\begin{subarray}{c}F^{\prime}\subseteq
F\\\ \text{a flat of
$M$}\end{subarray}}d^{\operatorname{rk}F^{\prime}}\mu({M}|{F^{\prime}}).$

Now, letting $F=[n]$ and noting $\operatorname{rk}[n]=r$, we have

$(-1)^{r}\mathcal{D}(\mathcal{L},d)=d^{r}\sum_{\begin{subarray}{c}F^{\prime}\subseteq[n]\\\
\text{a flat of
$M$}\end{subarray}}(\textstyle{\frac{1}{d}})^{r-\operatorname{rk}F^{\prime}}\mu(M|F^{\prime})=d^{r}\chi_{M}(\frac{1}{d}),$

so that
$\mathcal{D}(\mathcal{L},d)=(-d)^{r}\chi_{M}(\frac{1}{d})=d^{r}T_{M}(1-\frac{1}{d},0)$
as desired.

######  Remark 11.

For the non-reciprocal ML degree $\operatorname{mld}(\mathcal{L})$, a standard
computation similar to the one that gives Proposition 3 (see [17, 16]) yields

$\operatorname{mld}(\mathcal{L})=\deg\big{(}\mathcal{L}^{-1}\cap
Y(\mathcal{L},s,d=0)\cap(\mathbb{C}^{*})^{[n]}\big{)}=\mathcal{D}(\mathcal{L},d=0).$

One can hence recover [17, Corollary 3], which states
$\operatorname{mld}(\mathcal{L})=|\mu(M)|$, by minor modifications of our
arguments here. This case is in fact simpler, with no need for the
consideration of tangent cones as was done in Proposition 9, because the
intersection $\mathcal{L}^{-1}\cap Y(\mathcal{L},s,d)$ lies entirely in
$(\mathbb{C}^{*})^{[n]}$ when $d=0$. We emphasize that $\mathcal{L}^{-1}\cap
Y(\mathcal{L},s,d)$ _never_ lies entirely in $(\mathbb{C}^{*})^{[n]}$ when
$d\geq 1$.

######  Remark 12.

Combining Theorem 4 with the “recipe formula” for Tutte-Grothendieck
invariants of matroids (see for instance [20, Theorem 2.16]), one has that
$\mathcal{D}(\mathcal{L},d)$ for $d\geq 1$ satisfies the following deletion-
contraction relation given an element $e\in[n]$:

$\mathcal{D}(\mathcal{L},d)=\begin{cases}0&\text{if $e$ is a loop,}\\\
(d-1)\cdot\mathcal{D}(\mathcal{L}_{/e},d)&\text{if $e$ is a coloop,}\\\
\mathcal{D}(\mathcal{L}_{\setminus
e},d)+d\cdot\mathcal{D}(\mathcal{L}_{/e},d)&\text{if $e$ neither loop nor
coloop,}\end{cases}$

where the base cases are $\mathcal{D}(\mathcal{L}=\mathbb{C}^{[1]},d)=1$ and
$\mathcal{D}(\mathcal{L}=\\{0\\}\subset\mathbb{C}^{[1]},d)=0$. From this
deletion-contraction relation, one can verify that the following statements
are equivalent:

  1. (i)

$\mathcal{D}(\mathcal{L},2)=\operatorname{rmld}(\mathcal{L})=1$,

  2. (ii)

$M$ is a partition matroid (i.e. every component of $M$ has rank 1),

  3. (iii)

$\operatorname{mld}(\mathcal{L})=|\mu(M)|=1$, and

  4. (iv)

$\mathcal{L}^{-1}$ is linear.

It may be interesting to find a proof of Theorem 4 that directly reflects the
deletion-contraction relation above geometrically.

######  Proof 5.2 (Proof of Corollary 2).

For an $r$-dimensional general subspace
$\mathcal{L}\subseteq\mathbb{C}^{[n]}$, the associated matroid is the uniform
matroid $U_{r,n}$, for which the Tutte polynomial has the following formula
(see for instance [9])

$T_{U_{r,n}}(x,y)=\sum_{i=1}^{r}\textstyle\binom{n-i-1}{r-i}x^{i}+\displaystyle\sum_{j=1}^{n-r}\textstyle\binom{n-j-1}{r-1}y^{j}.$

Thus, for a general $\mathcal{L}$, Theorem 4 implies that

$D(\mathcal{L},d)=d^{r}T_{U_{r,n}}\left(\textstyle{1-\frac{1}{d}},0\right)=\displaystyle
d^{r}\sum_{1=1}^{r}\textstyle\binom{n-i-1}{r-i}\left(1-\frac{1}{d}\right)^{i}.$

Evaluating at $d=2$ gives us

$\operatorname{rmld}(\mathcal{L})=2^{r}\sum_{i=1}^{r}\textstyle\binom{n-i-1}{r-i}\left(\frac{1}{2}\right)^{i}\displaystyle=\sum_{i=1}^{r}\textstyle\binom{n-i-1}{r-i}2^{r-i},$

as desired.

######  Remark 13.

It is interesting to note that, for diagonal linear concentration models, our
formula Theorem 1 implies that $\operatorname{rmld}(\mathcal{L})$ is always
odd unless it is zero. The same pattern seems to persist for reciprocal ML
degrees of general linear concentration models [16, Table 1].

### Acknowledgements

We thank Yairon Cid Ruiz and Bernd Sturmfels for helpful conversations, and we
thank the referee for simplifying the proof of Lemma 3.1. We also thank the
organizers of the Linear Spaces of Symmetric Matrices working group at MPI MiS
Leipzig. C.E. is partially supported by the US National Science Foundation
(DMS-2001854).

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