When a system of real quadratic equations has a solution

TL;DR

This paper establishes a polynomial-time checkable sufficient condition for the solvability of systems of real quadratic equations, using spectral properties of associated matrices and tools from analysis.

Contribution

It introduces a new algebraic condition based on operator norms that guarantees solutions for quadratic systems, extending previous results and applicable to random instances.

Findings

The condition is efficiently checkable via semidefinite programming.

It guarantees solutions when the number of equations is up to a constant times the square root of the dimension.

The approach combines algebraic and analytic techniques, including Fourier analysis.

Abstract

We provide a sufficient condition for solvability of a system of real quadratic equations $p_i(x)=y_i$, $i=1, \ldots, m$, where $p_i: {\mathbb R}^n \longrightarrow {\mathbb R}$ are quadratic forms. By solving a positive semidefinite program, one can reduce it to another system of the type $q_i(x)=\alpha_i$, $i=1, \ldots, m$, where $q_i: {\mathbb R}^n \longrightarrow {\mathbb R}$ are quadratic forms and $\alpha_i=\mathrm{tr\ } q_i$. We prove that the latter system has solution $x \in {\mathbb R}^n$ if for some (equivalently, for any) orthonormal basis $A_1,\ldots, A_m$ in the space spanned by the matrices of the forms $q_i$, the operator norm of $A_1^2 + \ldots + A_m^2$ does not exceed $\eta/m$ for some absolute constant $\eta > 0$. The condition can be checked in polynomial time and is satisfied, for example, for random $q_i$ provided $m \leq \gamma \sqrt{n}$ for an absolute constant $\gamma >0$. We prove a similar sufficient condition for a system of homogeneous quadratic equations to have a non-trivial solution. While the condition we obtain is of an algebraic nature, the proof relies on analytic tools including Fourier analysis and measure concentration.