A Note on Property Testing Sum of Squares and Multivariate Polynomial Interpolation

TL;DR

This paper explores the complexity of property testing for whether multivariate polynomials are sums of squares, establishing near-determination sample bounds and analyzing specific non-negative polynomials using pseudo-expectations.

Contribution

It provides a lower bound on sample complexity for property testing sum of squares in multivariate polynomials and introduces techniques involving polynomial interpolation and pseudo-expectations.

Findings

Testing requires nearly as many samples as full polynomial determination.

Polynomial interpolation on random points produces small-norm polynomials.

Certain non-negative polynomials are proven far from sums of squares using pseudo-expectation methods.

Abstract

In this paper, we investigate property testing whether or not a degree d multivariate poly- nomial is a sum of squares or is far from a sum of squares. We show that if we require that the property tester always accepts YES instances and uses random samples, $n^{\Omega(d)}$ samples are required, which is not much fewer than it would take to completely determine the polynomial. To prove this lower bound, we show that with high probability, multivariate polynomial in- terpolation matches arbitrary values on random points and the resulting polynomial has small norm. We then consider a particular polynomial which is non-negative yet not a sum of squares and use pseudo-expectation values to prove it is far from being a sum of squares.