Optimal Testing of Multivariate Polynomials over Small Prime Fields - Haramaty, Elad; Shpilka, Amir; Sudan, Madhu (SIAM journal on computing, 2013)
We consider the problem of testing whether a given function $f : {\mathbb F}_q^n → {\mathbb F}_q$ is close to an $n$-variate degree $d$ polynomial over the finite field ${\mathbb F}_q$ of $q$ elements. The natural, low-query test for this property would be to first pick the smallest dimension $t = t_{q,d}≈ d/q$ such that every function of degree greater than $d$ reveals this aspect on some $t$-dimensional affine subspace of ${\mathbb F}_q^n$. 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 $45$th Annual IEEE Symposium on Foundations of Computer Science, 2004, pp. 423--432], 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 [IEEE Trans. Inform. Theory, 51 (2005), pp. 4032--4039] considered only the case of $q=2$, while the work [Proceedings of the $45$th 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 $51$st 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 $Ω(1)$-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 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 $O(q^{t_{q,d}})$---which is tight to within constant factors. [PUBLICATION ABSTRACT]