On the Smallest Support Size of Integer Solutions to Linear Equations

TL;DR

This work bounds the smallest support size of integer solutions to $A x = b$ with $A\in\mathbb{Z}^{m\times n}$ and $b\in\mathcal{L}(A)$ by relating $f(A)$ to the ratio $\Gamma(A)/\gcd(A)$ of largest $m\times m$ subdeterminants to the gcd of all such subdeterminants. Using a linear-algebra toolkit (Hermite normal form and Jacobi-type relations), the authors prove $\frac{\Gamma(A)}{\gcd(A)} \ge p_2^m p_3^m \cdots p_{\left\lfloor f(A)/m\right\rfloor}^m p_{\left\lceil f(A)/m\right\rceil}^{f(A)-m\left\lfloor f(A)/m\right\rfloor}$, and derive an upper bound on $f(A)$ via $f(A) \le m + \min_{\tau}\Omega_m(|\det(A_\tau)|/\gcd(A))$, aligning with the state-of-the-art bound up to a factor and establishing asymptotic tightness. The Hadamard inequality then yields a sharp bound on $h(m,t)=\max f(A)$ as $O\big( m\log(\sqrt{m}t)/\log\log(\sqrt{m}t) \big)$. A concrete, tight example demonstrates equality in the main bound, confirming asymptotic optimality. Overall, the paper provides a simpler, self-contained linear-algebraic route to near-optimal bounds on the sparse integer solutions of linear systems, with clear implications for the computational complexity of these problems.

Abstract

In this note, we study the size of the support of integer solutions to linear equations $Ax=b, ~x\in\Z^n$ where $A\in\Z^{m\times n}, b\in\Z^n$. We give an upper bound on the smallest support size as a function of $A$, taken as a worst case over all $b$ such that the above system has a solution. This bound is asymptotically tight, and in fact matches the bound given in Aliev, Averkov, De Loera and Oertel Mathematical Programming 2022, while the proof presented here is simpler, relying only on linear algebra.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Lemma 4
• proof
• Theorem 5
• Lemma 6
• Corollary 7
• proof : Proof of Theorem \ref{['thm:integer:equation:support']}