Optimal Testing of Multivariate Polynomials over Small Prime Fields

We consider the problem of testing if a given function $f : \F_q^n \right arrow \F_q$ 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 $t = t_{q,d}\approx d/q$ such that every function of degree greater than $d$reveals this aspect on {\em some} $t$-dimensional affine subspace of $\F_q^n$ and to test that $f$ when restricted to a {\em 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.~\cite{AKKLR}, and Kaufman and Ron~\cite{KaufmanRon06} and Jutla et al.~\cite{JPRZ04}, 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~\cite{AKKLR} considered only the case of $q=2$, while the work~\cite{JPRZ04}only considered the case of prime $q$. The results in \cite{KaufmanRon06} 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.~\cite{BKSSZ10} 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 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 $d$polynomial has degree less than $d$. We show that the number of such hyper planes is at most $O(q^{t_{q,d}})$ -- which is tight to within constant factors.