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

In this paper, we investigate property testing whether or not a degree d multivariate polynomial 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, nΩ(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 interpolation 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. . Institute for Advanced Study. Supported by the Simons Collaboration for Algorithms and Geometry and by the NSF under agreement No. CCF-1412958. Part of this work was done while at Cornell University. Yale University