Maximal rank in matrix spaces via graph matchings

TL;DR

This paper investigates the maximum possible rank in affine subspaces of symmetric or alternating matrices by relating it to graph matching numbers, providing new bounds and proofs.

Contribution

It introduces a novel approach connecting matrix rank problems with graph matchings, offering simplified proofs and bounds for subspace dimensions.

Findings

Established bounds on the dimension of matrix subspaces based on maximal rank.

Linked matrix rank properties to graph matching numbers.

Provided simplified proofs for existing bounds.

Abstract

We study the maximal rank in affine subspaces of symmetric or alternating matrices, in terms of the matching numbers of certain associated graphs. Applications include simple proofs of upper bounds on the dimension of such subspaces in terms of their maximal rank.

Equations

$$G_{\mathcal{S}}=\{\tilde{q}(B):0\ne B\in \mathcal{B}\}=\{\tilde{q}(S_1-S_2):S_1\ne S_2\in \mathcal{S}\}\subset {\tilde{K}}_n.$$

$$\mathcal{S}=\Big{\{}\left[\begin{array}[]{ll}x&x\\ x&1\end{array}\right]:x\in{\twelvebb F}_{2}\Big{\}}\subset H_{2}({\twelvebb F}_{2})$$

$$\mathcal{S}=\Big{\{}\left[\begin{array}[]{lll}x&x&0\\ x&0&y\\ 0&y&y\end{array}\right]:x,y\in{\twelvebb F}_{2}\Big{\}}\subset H_{3}({\twelvebb F}_{2})$$

$$U_{1}^{a}(n,k)=\big{\{}\big{(}A(i,j)\big{)}_{i,j=1}^{n}\in A_{n}({\twelvebb F}):A(i,j)=0\text{~{}if~{}}\max\{i,j\}>k+1\big{\}}$$

$$U_{2}^{a}(n,k)=\big{\{}\big{(}A(i,j)\big{)}_{i,j=1}^{n}\in A_{n}({\twelvebb F}):A(i,j)=0\text{~{}if~{}}\min\{i,j\}>\frac{k}{2}\big{\}}.$$

$$\begin{split}u_{a}(n,k)&=\max\big{\{}\dim U_{1}^{a}(n,k),\dim U_{2}^{a}(n,k)\big{\}}\\ &=\max\Big{\{}\binom{2t+1}{2},tn-\binom{t+1}{2}\Big{\}}.\end{split}$$

$$U_{1}^{s}(n,k)=\big{\{}\big{(}A(i,j)\big{)}_{i,j=1}^{n}\in H_{n}({\twelvebb F}):A(i,j)=0\text{~{}if~{}}\max\{i,j\}>k\big{\}}.$$

$$U_{2}^{s}(n,k)=\big{\{}\big{(}A(i,j)\big{)}_{i,j=1}^{n}\in H_{n}({\twelvebb F}):A(i,j)=0\text{~{}if~{}}\min\{i,j\}>t\big{\}},$$

$$U_{2}^{s}(n,k)=\big{\{}\big{(}A(i,j)\big{)}_{i,j=1}^{n}\in H_{n}({\twelvebb F}):A(i,j)=0\text{~{}if~{}}\min\{i,j\}>t~{}\&~{}(i,j)\neq(t+1,t+1)\big{\}}.$$

$$\begin{split}u_{s}(n,k)&=\max\big{\{}\dim U_{1}^{s}(n,k),\dim U_{2}^{s}(n,k)\big{\}}\\ &=\left\{\begin{array}[]{ll}\max\big{\{}\binom{2t+1}{2},tn-\binom{t}{2}\big{\}}&k=2t,\\ \max\big{\{}\binom{2t+2}{2},tn-\binom{t}{2}+1\big{\}}&k=2t+1.\end{array}\right.\end{split}$$

$$q(B_{i})=\left\{\begin{array}[]{cc}(\alpha_{i},\alpha_{i})&i\in I_{1},\\ (\beta_{i},\gamma_{i})&i\in I_{2}.\end{array}\right.$$

$$f(x)=\det\Big{(}A+\sum_{i=1}^{t}x_{i}B_{i}\Big{)}\in{\twelvebb F}[x_{1},\ldots,x_{t}].$$

$$\begin{split}f(x)&=\det(A+\sum_{i=1}^{t}x_{i}B_{i})\\ &=\sum_{\sigma\in{\mathbb{S}}_{n}}\text{sgn}(\sigma)\prod_{j=1}^{n}\big{(}A(j,\sigma(j))+\sum_{i=1}^{t}x_{i}B_{i}(j,\sigma(j))\big{)}\\ &=\sum_{\sigma\in{\mathbb{S}}_{n}}\text{sgn}(\sigma)\sum_{P=(C_{0},\ldots,C_{t})\in\mathcal{P}}\big{(}\prod_{j\in C_{0}}A(j,\sigma(j))\prod_{i=1}^{t}\prod_{j\in C_{i}}B_{i}(j,\sigma(j))\big{)}\prod_{i=1}^{t}x_{i}^{|C_{i}|}.\end{split}$$

$$g(Q,\sigma )=\prod _{i=1}^t\prod _{j\in C_i}{B}_i(j,\sigma (j)).$$

$$\sum _{\sigma \in {\mathbb{S}}_n}\text{sgn}(\sigma )\sum _{Q=(C_1,\dots ,C_t)\in \mathcal{Q}}g(Q,\sigma ).$$

$$\sigma_{0}(j)=\left\{\begin{array}[]{cc}\alpha_{i}&j=\alpha_{i},\\ \gamma_{i}&j=\beta_{i},\\ \beta_{i}&j=\gamma_{i}.\end{array}\right.$$

$$C_{i}=\left\{\begin{array}[]{cc}\{k_{i}\}&i\in I_{1},\\ \{\ell_{i},m_{i}\}&i\in I_{2}.\end{array}\right.$$

$$\begin{split}g(Q,\sigma)&=\prod_{i=1}^{t}\prod_{j\in C_{i}}B_{i}(j,\sigma(j))\\ &=\prod_{i\in I_{1}}B_{i}(k_{i},\sigma(k_{i}))\prod_{i\in I_{2}}\Big{(}B_{i}(\ell_{i},\sigma(\ell_{i}))\cdot B_{i}(m_{i},\sigma(m_{i}))\Big{)}.\end{split}$$

$$\big{(}\min(k_{i},\sigma(k_{i})),\max(k_{i},\sigma(k_{i})\big{)}\preceq q(B_{i})=(\alpha_{i},\alpha_{i}),$$

$$2^{k_i}+2^{\sigma (k_i)}\le 2^{{\alpha }_i}+2^{{\alpha }_i}.$$

$$\begin{split}&\Big{(}\min\big{(}\ell_{i},\sigma(\ell_{i})\big{)},\max\big{(}\ell_{i},\sigma(\ell_{i})\big{)}\Big{)}\preceq q(B_{i})=(\beta_{i},\gamma_{i}),\\ &\Big{(}\min\big{(}m_{i},\sigma(m_{i})\big{)},\max\big{(}m_{i},\sigma(m_{i})\big{)}\Big{)}\preceq q(B_{i})=(\beta_{i},\gamma_{i}),\end{split}$$

$$\begin{array}{rl}& 2^{{\ell }_i}+2^{\sigma ({\ell }_i)}\le 2^{{\beta }_i}+2^{{\gamma }_i},\\ & 2^{m_i}+2^{\sigma (m_i)}\le 2^{{\beta }_i}+2^{{\gamma }_i}.\end{array}$$

$$\begin{split}&\sum_{i\in I_{1}}2^{\sigma(k_{i})}+\sum_{i\in I_{2}}\big{(}2^{\sigma(\ell_{i})}+2^{\sigma(m_{i})}\big{)}\\ &=\sum_{i\in I_{1}}2^{k_{i}}+\sum_{i\in I_{2}}\big{(}2^{\ell_{i}}+2^{m_{i}}\big{)}\\ &=\sum_{i\in I_{1}}2^{\alpha_{i}}+\sum_{i\in I_{2}}\big{(}2^{\beta_{i}}+2^{\gamma_{i}}\big{)}\\ &=\sum_{j=1}^{n}2^{j}=2^{n+1}-2\end{split}$$

$$\begin{split}2^{n+2}-4&=\sum_{i\in I_{1}}\big{(}2^{k_{i}}+2^{\sigma(k_{i})}\big{)}+\sum_{i\in I_{2}}\Big{(}\big{(}2^{\ell_{i}}+2^{\sigma(\ell_{i})}\big{)}+\big{(}2^{m_{i}}+2^{\sigma(m_{i})}\big{)}\Big{)}\\ &\leq\sum_{i\in I_{1}}\big{(}2^{\alpha_{i}}+2^{\alpha_{i}}\big{)}+\sum_{i\in I_{2}}\Big{(}\big{(}2^{\beta_{i}}+2^{\gamma_{i}}\big{)}+\big{(}2^{\beta_{i}}+2^{\gamma_{i}}\big{)}\Big{)}\\ &=2^{n+2}-4.\end{split}$$

$$2^{k_i}+2^{\sigma (k_i)}=2^{{\alpha }_i}+2^{{\alpha }_i},$$

$$\begin{array}{rl}{2}^{{\ell }_i}+2^{\sigma ({\ell }_i)}& =2^{{\beta }_i}+2^{{\gamma }_i},\\ 2^{m_i}+2^{\sigma (m_i)}& =2^{{\beta }_i}+2^{{\gamma }_i}.\end{array}$$

$$\{{\ell }_i,\sigma ({\ell }_i)\}=\{{\beta }_i,{\gamma }_i\}=\{m_i,\sigma (m_i)\}.$$

$$C_i=\{{\ell }_i,m_i\}=\{{\beta }_i,{\gamma }_i\}=\tilde{q}(B_i)$$

$$\det (A+\sum _{i=1}^t{\lambda }_i{B}_i)=f(\lambda )\ne 0.$$

$$\theta(M)=\text{sgn}\left(\begin{array}[]{ccccc}1&2&\cdots&n-1&n\\ k_{1}&\ell_{1}&\cdots&k_{t}&\ell_{t}\end{array}\right)$$

Full text

It is shown that if $\mathcal{S}$ is a translate of a linear space contained in $W_{n}({\twelvebb F})$ and $|{\twelvebb F}|\geq 3$ then $\rho(\mathcal{S})\geq\mu(G_{\mathcal{S}})$. The restriction on ${\twelvebb F}$ can be removed if $\mathcal{S}$ is an affine subspace of $A_{n}({\twelvebb F})$. Applications include simple proofs of upper bounds on the dimension of affine subspaces of symmetric and alternating matrices of bounded rank.

2000 MSC: 05C50; 47L05

Keywords: Spaces of matrices of bounded rank, Graph matching.

1 Introduction

Let $M_{n}({\twelvebb F})$ be the space of $n\times n$ matrices over a field ${\twelvebb F}$. For a subset $\mathcal{S}\subset M_{n}({\twelvebb F})$ let $\rho(\mathcal{S})=\max\{{\rm rank}(A):A\in\mathcal{S}\}$ denote the maximum rank of a matrix in $\mathcal{S}$. Let $H_{n}({\twelvebb F})$ denote the space of symmetric matrices in $M_{n}({\twelvebb F})$. A matrix A=\big{(}A(i,j)\big{)}_{i,j=1}^{n}\in M_{n}({\twelvebb F}) is alternating if $A=-A^{T}$ and $A(i,i)=0$ for $1\leq i\leq n$. Let $A_{n}({\twelvebb F})$ denote the space of alternating matrices in $M_{n}({\twelvebb F})$. A matrix $A\in M_{n}({\twelvebb F})$ is weakly symmetric if $A(i,j)\neq 0$ iff $A(j,i)\neq 0$ holds for all $i,j$. Let $W_{n}({\twelvebb F})$ denote the set of all weakly symmetric matrices in $M_{n}({\twelvebb F})$. Note that $A_{n}({\twelvebb F}),H_{n}({\twelvebb F})\subset W_{n}({\twelvebb F})$ and $W_{n}({\twelvebb F}_{2})=H_{n}({\twelvebb F}_{2})$. In this note we study lower bounds on $\rho(\mathcal{S})$ for affine translates $\mathcal{S}$ of linear spaces of weakly symmetric matrices, in terms of matching numbers of a certain graph associated with $\mathcal{S}$.

Let $[n]=\{1,\ldots,n\}$ and let $[n]_{\leq}^{2}=\{(i,j)\in[n]^{2}:i\leq j\}$. The colexicographic order on $[n]_{\leq}^{2}$ is given by $(i,j)\prec(i^{\prime},j^{\prime})$ iff $j<j^{\prime}$ or $j=j^{\prime}$ and $i<i^{\prime}$. Equivalently, $(i,j)\prec(i^{\prime},j^{\prime})$ iff $2^{i}+2^{j}<2^{i^{\prime}}+2^{j^{\prime}}$. Let $K_{n}=\{e\subset[n]:|e|=2\}$ denote the edge set of the complete graph on $[n]$ and let $\tilde{K}_{n}=\{e\subset[n]:0<|e|\leq 2\}$ denote the edge set of the complete graph with loops on $[n]$. A subset $M\subset\tilde{K}_{n}$ is a matching if $e\cap e^{\prime}=\emptyset$ for all $e\neq e^{\prime}\in M$. For a graph with loops $G\subset\tilde{K}_{n}$ let ${\cal M}(G)$ denote the set of all matchings $M\subset G$. Let $\nu(G)=\max\{|M|:M\in{\cal M}(G)\}$ and $\mu(G)=\max\{\sum_{e\in M}|e|:M\in{\cal M}(G)\}$. A matching $M\subset\tilde{K}_{n}$ is perfect if $\mu(M)=n$. Note that if $G$ is loopless, i.e. $G\subset K_{n}$, then $\nu(G)$ is the usual matching number of $G$ and $\mu(G)=2\nu(G)$.

For 0\neq A=\big{(}A(i,j)\big{)}_{i,j=1}^{n}\in W_{n}({\twelvebb F}) let $q(A)=\max\{(i,j)\in[n]_{\leq}^{2}:A(i,j)\neq 0\}$ where the maximum is taken with respect to the colexicographic order. For $A$ such that $q(A)=(i,j)$ let $\tilde{q}(A)=\{i,j\}\in\tilde{K}_{n}$. Let $\mathcal{S}$ be a translate of a linear space $\mathcal{B}\subset W_{n}({\twelvebb F})$, i.e. $\mathcal{S}=A+\mathcal{B}$ for some $A\in M_{n}({\twelvebb F})$. Associate with $\mathcal{S}$ a graph with loops

[TABLE]

Our main results provide a link between the maximum rank in $\mathcal{S}$ and matchings in $G_{\mathcal{S}}$.

Theorem 1.1**.**

Suppose $|{\twelvebb F}|\geq 3$ and let $\mathcal{S}=A+\mathcal{B}$ where $A\in M_{n}({\twelvebb F})$ and $\mathcal{B}$ is a linear space contained in $W_{n}({\twelvebb F})$. Then $\rho(\mathcal{S})\geq\mu(G_{\mathcal{S}})$.

The restriction on ${\twelvebb F}$ is superfluous if $\mathcal{S}$ is an affine space of alternating matrices:

Theorem 1.2**.**

Let ${\twelvebb F}$ be an arbitrary field and let $\mathcal{S}$ be an affine subspace of $A_{n}({\twelvebb F})$. Then $\rho(\mathcal{S})\geq\mu(G_{\mathcal{S}})$.

Remarks:

1. The case $A=0$ and $|{\twelvebb F}|\geq\mu(G_{\mathcal{B}})+1$ of Theorem 1.1 had (essentially) been proved in [5]. The approach to the present improved result is somewhat different and uses additional ideas. The proof of Theorem 1.2 utilizes Pfaffians of alternating matrices.

2. Theorem 1.1 does not hold for ${\twelvebb F}={\twelvebb F}_{2}$ as the following examples show.

(i) The affine subspace

[TABLE]

satisfies $\mu(G_{\mathcal{S}})=2>1=\rho(\mathcal{S})$.

(ii) The linear subspace

[TABLE]

satisfies $\mu(G_{\mathcal{S}})=3>2=\rho(\mathcal{S})$.

Note however that $\rho(\mathcal{S})=\nu(G_{\mathcal{S}})$ holds in both examples. In fact, the following consequence of Theorem 1.2 holds.

Corollary 1.3**.**

Let $\mathcal{S}$ be an affine subspace of $H_{n}({\twelvebb F}_{2})$. Then $\rho(\mathcal{S})\geq\nu(G_{\mathcal{S}})$.

Theorems 1.1 and 1.2 can be used to provide short combinatorial proofs of upper bounds on the maximal dimensions of affine subspaces of $A_{n}({\twelvebb F})$ and of translates of linear spaces contained in $W_{n}({\twelvebb F})$, that consist of matrices of bounded rank. We first consider the alternating case. For $k=2t$ even and $n\geq k$ let

[TABLE]

and

[TABLE]

Then $\rho(U_{1}^{a}(n,k))=\rho(U_{2}^{a}(n,k))=k$. Define

[TABLE]

The following result was first obtained in [5] under the assumption that $\mathcal{S}$ is a linear subspace of $A_{n}({\twelvebb F})$ and $|{\twelvebb F}|\geq n+1$. The general case has recently been proved by de Seguins Pazzis [6]. Here it is obtained as a direct consequence of Theorem 1.2.

Theorem 1.4** ([5, 6]).**

Let ${\twelvebb F}$ be any field and let $\mathcal{S}$ be an affine subspace of $A_{n}({\twelvebb F})$ such that $\rho(\mathcal{S})=k<n$. Then $\dim\mathcal{S}\leq u_{a}(n,k)$.

We next consider the weakly symmetric case. Let $k\leq n$. Define

[TABLE]

For $k=2t$ even let

[TABLE]

and for $k=2t+1$ odd let

[TABLE]

Then $\rho(U_{1}^{s}(n,k))=\rho(U_{2}^{s}(n,k))=k$. Define

[TABLE]

The following result was first obtained in [5] under the assumption that $\mathcal{S}$ is a linear subspace of $H_{n}({\twelvebb F})$ and $|{\twelvebb F}|\geq n+1$. It has recently been proved by de Seguins Pazzis [6] for affine subspaces $\mathcal{S}$ of $H_{n}({\twelvebb F})$ with no restrictions on ${\twelvebb F}$. Here it is established for slightly more general affine spaces provided that ${\twelvebb F}\neq{\twelvebb F}_{2}$.

Theorem 1.5**.**

Suppose $|{\twelvebb F}|\geq 3$ and let $\mathcal{S}=A+{\cal B}$ where $A\in M_{n}({\twelvebb F})$ and ${\cal B}$ is a linear space contained in $W_{n}({\twelvebb F})$. If $\rho(\mathcal{S})=k$ then $\dim\mathcal{S}\leq u_{s}(n,k)$.

The paper is organized as follows. Theorem 1.1 is proved in Section 2. In Section 3 we use Pfaffians to establish Theorem 1.2 and then deduce Corollary 1.3. In Section 4 we recall the Erdős-Gallai theorem [2] on the maximal number of edges in simple graphs with bounded matching number, as well as the analogous result for graphs with loops [5]. Combining these with Theorems 1.2 and 1.1, directly implies Theorems 1.4 and 1.5. We conclude in Section 5 with some remarks and open problems.

2 Translates of Weakly Symmetric Subspaces

In this section we prove Theorem 1.1. Let $\mathcal{S}=A+\mathcal{B}$ be an affine subspace of $M_{n}({\twelvebb F})$ where $A\in M_{n}({\twelvebb F})$ and $\mathcal{B}\subset W_{n}({\twelvebb F})$ is a linear space. We first note that by restricting to the principal submatrix determined by a matching $M\subset G_{\mathcal{S}}$ such that $\sum_{e\in M}|e|=\mu(G_{\mathcal{S}})$, it suffices to show that if $\mu(G_{\mathcal{S}})=n$, then $\mathcal{S}$ contains a nonsingular matrix. Choose $B_{1},\ldots,B_{t}\in\mathcal{B}$ such that $M=\{\tilde{q}(B_{1}),\ldots,\tilde{q}(B_{t})\}$ is a prefect matching of $\tilde{K}_{n}$, and let $|\tilde{q}(B_{i})|=\delta_{i}$. Then $\sum_{i=1}^{t}\delta_{i}=n$. For $j=1,2$ let $I_{j}=\{1\leq i\leq t:\delta_{i}=j\}$. For $1\leq i\leq t$ let

[TABLE]

Let $x=(x_{1},\ldots,x_{t})$ be a vector of variables and let

[TABLE]

We have to show that there exists an $\lambda=(\lambda_{1},\ldots,\lambda_{t})\in{\twelvebb F}^{t}$ such that $f(\lambda)\neq 0$. The main ingredient in the proof is the following

Proposition 2.1**.**

The monomial $\prod_{i=1}^{t}x_{i}^{\delta_{i}}$ appears in $f(x)$ with a nonzero coefficient.

Proof: Let ${\mathbb{S}}_{n}$ be the symmetric group on $[n]$ and let $\mathcal{P}$ denote the set of all ordered partitions $P=(C_{0},C_{1},\ldots,C_{t})$ of $[n]$ into $t+1$ disjoint sets. Then:

[TABLE]

Let $\mathcal{Q}$ denote the set of all ordered partitions $Q=(C_{1},\ldots,C_{t})$ of $[n]$ into $t$ disjoint sets such that $|C_{i}|=\delta_{i}$. For $Q=(C_{1},\ldots,C_{t})\in\mathcal{Q}$ and $\sigma\in{\mathbb{S}}_{n}$ let

[TABLE]

As $\sum_{i=1}^{t}\delta_{i}=n$, it follows from (1) that the coefficient of $\prod_{i=1}^{t}x_{i}^{\delta_{i}}$ in $f(x)$ is

[TABLE]

Consider the partition $Q_{0}=(\tilde{q}(B_{1}),\ldots,\tilde{q}(B_{t}))\in\mathcal{Q}$ and the permutation $\sigma_{0}\in{\mathbb{S}}_{n}$ given by

[TABLE]

Note that $g(Q_{0},\sigma_{0})\neq 0$. Hence, Proposition 2.1 will follow from (2) and the next observation.

Claim 2.2**.**

Let $(Q,\sigma)\in\mathcal{Q}\times{\mathbb{S}}_{n}$ such that $g(Q,\sigma)\neq 0$. Then $(Q,\sigma)=(Q_{0},\sigma_{0})$.

Proof: Write $Q=(C_{1},\ldots,C_{t})$ where

[TABLE]

Then

[TABLE]

As $g(Q,\sigma)\neq 0$ it follows that for $i\in I_{1}$:

[TABLE]

hence

[TABLE]

Similarly, for $i\in I_{2}$:

[TABLE]

hence

[TABLE]

Next note that $Q_{0},Q$ and $\sigma(Q)=(\sigma(C_{1}),\ldots,\sigma(C_{t}))$ are all partitions of $[n]$, therefore

[TABLE]

Summing (3) over $i\in I_{1}$ and (4) over $i\in I_{2}$ it follows by (5) that

[TABLE]

By (6), all inequalities in (3) and (4) are in fact equalities. Thus, for all $i\in I_{1}$

[TABLE]

and for all $i\in I_{2}$

[TABLE]

It follows by (7) that $k_{i}=\sigma(k_{i})=\alpha_{i}$ for all $i\in I_{1}$. Similarly, (8) implies that for all $i\in I_{2}$

[TABLE]

Therefore

[TABLE]

and $\sigma(\ell_{i})=m_{i},\sigma(m_{i})=\ell_{i}$. Thus $(Q,\sigma)=(Q_{0},\sigma_{0})$.

$\Box$

Recall Alon’s Combinatorial Nullstellensatz (Theorem 1.2 in [1]).

Theorem 2.3** (Alon [1]).**

Let ${\twelvebb F}$ be an be an arbitrary field and let $g=g(x_{1},\ldots,x_{t})\in{\twelvebb F}[x_{1}\ldots,x_{t}]$. Suppose the total degree $\deg(g)$ of $g$ is $\sum_{i=1}^{t}d_{i}$ where each $d_{i}$ is a nonnegative integer, and suppose the coefficient of $\prod_{i=1}^{t}x_{i}^{d_{i}}$ in $g$ is nonzero. Then, if $\Lambda_{1},\ldots,\Lambda_{t}$ are subsets of ${\twelvebb F}$ with $|\Lambda_{i}|>d_{i}$, there exist $\lambda_{1}\in\Lambda_{1},\ldots,\lambda_{t}\in\Lambda_{t}$ such that $g(\lambda_{1},\ldots,\lambda_{t})\neq 0$.

Proof of Theorem 1.1: By Proposition 2.1, the coefficient of $\prod_{i=1}^{t}x_{i}^{\delta_{i}}$ in $f(x)$ is nonzero. Applying Theorem 2.3 with $g=f$, $d_{i}=\delta_{i}$ and $\Lambda_{i}={\twelvebb F}$, and noting that $\sum_{i=1}^{t}\delta_{i}=n=\deg(f)$ and $|\Lambda_{i}|=|{\twelvebb F}|\geq 3>2\geq\delta_{i}$, it follows that there exists a $\lambda=(\lambda_{1},\ldots,\lambda_{t})\in{\twelvebb F}^{t}$ such that

[TABLE]

$\Box$

3 Affine Subspaces of Alternating Matrices

We first recall the definition of the Pfaffian of an alternating matrix C=\big{(}C(i,j)\big{)}_{i,j=1}^{n}\in A_{n}({\twelvebb F}) of even order $n=2t$ over an arbitrary field ${\twelvebb F}$. Let $\mathcal{M}_{n}$ denote the set of all perfect matchings in $K_{n}$. For $M=\{e_{1},\ldots,e_{t}\}\in\mathcal{M}_{n}$, where $e_{1}\prec\cdots\prec e_{t}$ and $e_{i}=\{k_{i}<\ell_{i}\}$ for $1\leq i\leq t$, let

[TABLE]

and let

[TABLE]

The Pfaffian of $C$ is defined by

[TABLE]

It is well known that $\det(C)=\text{Pf}(C)^{2}$ (see e.g. Exercise 4.24 in [4]).  

Proof of Theorem 1.2: As in the proof of theorem 1.1, it suffices to show that if $\mathcal{S}$ is an affine subspace of $A_{n}({\twelvebb F})$ and $\nu(G_{\mathcal{S}})=\frac{n}{2}=t$, then $\mathcal{S}$ contains a nonsingular matrix.

Choose $B_{1},\ldots,B_{t}\in\mathcal{B}$ such that $M_{0}=\{\tilde{q}(B_{1}),\ldots,\tilde{q}(B_{t})\}$ is a perfect matching of $K_{n}$. For $1\leq i\leq t$ write $q(B_{i})=(\alpha_{i},\beta_{i})$. By reordering we may assume that

[TABLE]

Let

[TABLE]

Claim 3.1**.**

The coefficient of $x_{1}\cdots x_{t}$ in $f(x_{1},\ldots,x_{t})$ is nonzero.

Proof: Let $M=\{e_{1},\ldots,e_{t}\}$ be a matching in $\mathcal{M}_{n}$ where $e_{i}=\{k_{i}<\ell_{i}\}$ for $1\leq i\leq t$ and

[TABLE]

Then

[TABLE]

Let $\lambda(M)$ denote the coefficient of $x_{1}\cdots x_{t}$ in $\mu(A+\sum_{i=1}^{t}x_{i}B_{i},M)$. Then

[TABLE]

Next note that if a permutation $\pi\in{\mathbb{S}}_{t}$ satisfies

[TABLE]

then $(k_{j},\ell_{j})\preceq q(B_{\pi(j)})=(\alpha_{\pi(j)},\beta_{\pi(j)})$ and hence

[TABLE]

for all $1\leq j\leq t$. Since

[TABLE]

it follows that $2^{k_{j}}+2^{\ell_{j}}=2^{\alpha_{\pi(j)}}+2^{\beta_{\pi(j)}}$ and hence $\{k_{j},\ell_{j}\}=\{\alpha_{\pi(j)},\beta_{\pi(j)}\}$ for all $1\leq j\leq t$, i.e. $M=M_{0}$. The monotonicity assumptions (9) and (10) further imply that $\pi$ is the identity permutation. It follows that the coefficient of $x_{1}\cdots x_{t}$ in $f(x)$ is

[TABLE]

$\Box$

We now complete the proof of Theorem 1.2. By Claim 3.1 the monomial $x_{1}\cdots x_{t}$ appears in $f$ with a nonzero coefficient. Applying Theorem 2.3 with $g=f$, $d_{i}=1$ and $\Lambda_{i}=\{0,1\}$ for all $i$, and noting that $\deg(x_{1}\cdots x_{t})=t=\deg(f)$ and $|\Lambda_{i}|=2>1=d_{i}$, it follows that there exists an $\epsilon=(\epsilon_{1},\ldots,\epsilon_{t})\in\{0,1\}^{t}$ such that $f(\epsilon)\neq 0$. Hence $C=A+\sum_{i=1}^{t}\epsilon_{i}B_{i}\in\mathcal{S}$ satisfies

[TABLE]

$\Box$

Proof of Corollary 1.3: Let $\mathcal{S}=A+{\cal B}$ where $A\in H_{n}({\twelvebb F}_{2})$ and ${\cal B}$ is a linear subspace of $H_{n}({\twelvebb F}_{2})$. Consider the affine subspace

[TABLE]

If $0\neq B\in{\cal B}$ satisfies $q(B)=(i,j)$ then

[TABLE]

It follows that $\nu(G_{\mathcal{S}^{\prime}})=\nu(G_{\mathcal{S}})$. Hence by Theorem 1.2

[TABLE]

$\Box$

4 Maximal Rank and Dimension

In this section we prove Theorems 1.4 and 1.5. For the alternating case we recall the following extremal graph theoretic result.

Theorem 4.1** (Erdős-Gallai [2]).**

Let $G\subset K_{n}$ be a simple graph that satisfies $\mu(G)=k<n$. Then

[TABLE]

Proof of Theorem 1.4: Let $\mathcal{S}\subset A_{n}({\twelvebb F})$ be an affine subspace of $A_{n}({\twelvebb F})$ such that $\rho(\mathcal{S})=k<n$. Note that $k$ is even. Write $\mathcal{S}=A+\mathcal{B}$ where $A\in A_{n}({\twelvebb F})$ and $\mathcal{B}$ is a linear subspace of $A_{n}({\twelvebb F})$. By performing Gaussian elimination on a basis of $\mathcal{B}$ it is clear that $|G_{\mathcal{S}}|=\dim\mathcal{S}$. Theorem 1.2 implies that

[TABLE]

It thus follows from Theorem 4.1 that $\dim\mathcal{S}=|G_{\mathcal{S}}|\leq u_{a}(n,k)$.

$\Box$

For the weakly symmetric case we need the following version of Theorem 4.1 for graphs with loops.

Theorem 4.2** ([5]).**

Let $G\subset\tilde{K}_{n}$ satisfies $\mu(G)=k$, then $|G|\leq u_{s}(n,k)$.

Proof of Theorem 1.5: Suppose $|{\twelvebb F}|\geq 3$ and let $\mathcal{S}=A+{\cal B}$ where $A\in M_{n}({\twelvebb F})$ and ${\cal B}$ is a linear space contained in $W_{n}({\twelvebb F})$, and let $\rho(\mathcal{S})=k$. As before $|G_{\mathcal{S}}|=\dim\mathcal{S}$. Hence, by Theorem 1.1

[TABLE]

It thus follows from Theorem 4.2 that $\dim\mathcal{S}=|G_{\mathcal{S}}|\leq u_{s}(n,k)$.

$\Box$

5 Concluding Remarks

In this note we showed that if $\mathcal{S}$ is a translate of a linear space contained in $W_{n}({\twelvebb F})$ and $|{\twelvebb F}|\geq 3$ then $\rho(\mathcal{S})\geq\mu(G_{\mathcal{S}})$, and that the same holds with no restriction on ${\twelvebb F}$ if $\mathcal{S}$ is an affine subspace of $A_{n}({\twelvebb F})$. These results improve on some earlier work in [5] and provide a simple approach to upper bounds on the dimension of affine spaces of symmetric and alternating matrices of bounded rank. We conclude with the following two remarks.

1. As noted above, de Seguins Pazzis [6] proved that Theorem 1.5 holds also over ${\twelvebb F}_{2}$ if $\mathcal{S}$ is an affine subspace of $H_{n}({\twelvebb F}_{2})$. It would be interesting to decide whether this case can also be handled using the combinatorial approach of the present paper.

2. When $\mathcal{S}$ is a linear subspace of $A_{n}({\twelvebb F})$ and the field ${\twelvebb F}$ satisfies $|{\twelvebb F}|\geq n+1$, Theorem 1.2 is the case $p=2$ of Theorem 2.1 in [3] that gives a lower bound on the maximal rank of a $p$-vector in a linear subspace $\mathcal{S}$ of the $p$-th exterior power $\bigwedge^{p}{\twelvebb F}^{n}$ in terms of the weak matching number of a certain $p$-uniform hypergraph associated to $\mathcal{S}$. It would be interesting to determine whether this result and its consequences remain true over arbitrary fields.