Optimal Testing of Multivariate Polynomials over Small Prime Fields

We consider the problem of testing if a given function f : F_qⁿ→F_q is close to a 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 pick the smallest dimension t = t_q,d≈ d/q such that every function of degree greater than d reveals this aspect on some i-dimensional affine subspace of F_qⁿ and to test that f when restricted to a random i-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. [1], and Kaufman and Ron [7] and Jutla et al. [6], showed that this natural test rejected functions that were Ω(1)-far from degree d-polynomials with probability at least Ω,(q^-t). (The initial work [1] considered only the case of q = 2, while the work [6] only considered the case of prime q. The results in [7] hold for all fields.) Thus to get a constant probability of detecting functions that are at 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. [2] 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 Ω(1)-far from degree d- polynomials with probability Ω(1). In this work we extend this result for all fields showing that the natural test does indeed reject functions that are Ω(1)-far from degree d polynomials with Ω(1)-probability, where the constants depend only on q the field size. Thus our analysis thus shows that this test is optimal (matches known lower bounds) when q is prime. The main technical ingredi- nt 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.