When a system of real quadratic equations has a solution

Abstract

We provide a sufficient condition for solvability of a system of real quadratic equations pi(x)=yi, i=1,…,m, where pi:Rn⟶R are quadratic forms. By solving a positive semidefinite program, one can reduce it to another system of the type qi(x)=αi, i=1,…,m, where qi:Rn⟶R are quadratic forms and αi=traceqi. We prove that the latter system has solution x∈Rn if for some (equivalently, for any) orthonormal basis A1,…,Am in the space spanned by the matrices of the forms qi, the operator norm of A12+…+Am2 does not exceed η/m for some absolute constant η>0. The condition can be checked in polynomial time and is satisfied, for example, for random qi provided m≤γn for an absolute constant γ>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.