OPTIMAL TESTING OF MULTIVARIATE POLYNOMIALS OVER SMALL PRIME FIELDS
We consider the problem of testing whether a given function f : F-q(n) -> F-q is close to an n-variate degree d polynomial over the finite field F-q of q elements. The natural, low-query test for this property would be to first pick the smallest dimension t = t(q,d) approximate to d/q such that every function of degree greater than d reveals this aspect on some t-dimensional affine subspace of F-q(n). Then, one would test that f when restricted to a random t-dimensional affine subspace is a polynomial of degree at most d on this subspace. Such a test makes only q(t) queries, independent of n. Previous works, by Alon et al. [IEEE Trans. Inform. Theory, 51 (2005), pp. 4032-4039], Kaufman and Ron [SIAM J. Comput., 36 (2006), pp. 779-802], and Jutla et al. [Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004, pp. 423-432], showed that this natural test rejected functions that were Omega(1)-far from degree d-polynomials with probability at least Omega(q(-t)). (The initial work [IEEE Trans. Inform. Theory, 51 (2005), pp. 4032-4039] considered only the case of q = 2, while the work [Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004, pp. 423-432] considered only the case of prime q. The results in [SIAM J. Comput., 36 (2006), pp. 779-802] hold for all fields.) Thus to get a constant probability of detecting functions that are at a constant distance from the space of degree d polynomials, the tests made q(2t) queries. Kaufman and Ron also noted that when q is prime, then q(t) queries are necessary. Thus these tests were off by at least a quadratic factor from known lower bounds. Bhattacharyya et al. [Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, 2010, pp. 488-497] gave an optimal analysis of this test for the case of the binary field and showed that the natural test actually rejects functions that were Omega(1)-far from degree d-polynomials with probability Omega(1). In this work we extend this result for all fields showing that the natural test does indeed reject functions that are Omega(1)-far from degree d polynomials with Omega(1)-probability, where the constants depend only on q the field size. Thus our analysis shows that this test is optimal (matches known lower bounds) when q is prime. The main technical ingredient in our work is a tight analysis of the number of "hyperplanes" (affine subspaces of co-dimension 1) on which the restriction of a degree d polynomial has degree less than d. We show that the number of such hyperplanes is at most O(q(tq,d))-which is tight to within constant factors.
property testing low-degree polynomials reed-muller sublinear time algorithms