Testing systems of real quadratic equations for approximate solutions

TL;DR

This paper presents an efficient method to differentiate between systems of quadratic equations that have many solutions or near-solutions and those that are far from solvable, using a novel expectation-based test with specific penalty functions.

Contribution

It introduces a quasi-polynomial time algorithm to approximate the expectation of a specially chosen penalty function for systems of quadratic forms, enabling classification of their solvability.

Findings

Efficient quasi-polynomial time approximation of expectation for quadratic systems.

Ability to distinguish solvable from unsolvable systems under certain conditions.

Applicable to systems with bounded variable dependence and size constraints.

Abstract

Consider systems of equations $q_i(x)=0$, where $q_i: {\Bbb R}^n \longrightarrow {\Bbb R}$, $i=1, \ldots, m$, are quadratic forms. Our goal is to tell efficiently systems with many non-trivial solutions or near-solutions $x \ne 0$ from systems that are far from having a solution. For that, we pick a delta-shaped penalty function $F: {\Bbb R} \longrightarrow [0, 1]$ with $F(0)=1$ and $F(y) < 1$ for $y \ne 0$ and compute the expectation of $F(q_1(x)) \cdots F(q_m(x))$ for a random $x$ sampled from the standard Gaussian measure in ${\Bbb R}^n$. We choose $F(y)=y^{-2}\sin^2 y$ and show that the expectation can be approximated within relative error $0< \epsilon < 1$ in quasi-polynomial time $(m+n)^{O(\ln (m+n)-\ln \epsilon)}$, provided each form $q_i$ depends on not more than $r$ real variables, has common variables with at most $r-1$ other forms and satisfies $|q_i(x)| \leq \gamma \|x\|^2/r$, where $\gamma >0$ is an absolute constant. This allows us to distinguish between "easily solvable" and "badly unsolvable" systems in some non-trivial situations.