Optimal testing of multivariate polynomials over small prime fields

We consider the problem of testing if a given function f:double-struck F _q ⁿ → double-struck F _q is close to a n-variate degree d polynomial over the finite field double-struck 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 dreveals this aspect on some t-dimensional affine subspace of double-struck F _q ⁿ and to 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. [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 ingredient in our work is a tight analysis of the number of "hyper planes" (affine subspaces of co-dimension 1) on which the restriction of a degree dpolynomial has degree less than d. We show that the number of such hyper planes is at most O(q ^tq,d) - which is tight to within constant factors.