Testing Polynomials over General Fields - Kaufman, Tali; Ron, Dana (SIAM journal on computing, 2006)
In this work we fill the knowledge gap concerning testing polynomials over finite fields. As previous works show, when the cardinality of the field, $q$, is sufficiently larger than the degree bound, $d$, then the number of queries sufficient for testing is polynomial or even linear in $d$. On the other hand, when $q=2$ then the number of queries, both sufficient and necessary, grows exponentially with $d$. Here we study the intermediate case where $2 < q = O(d)$ and show a smooth transition between the two extremes. Specifically, let $p$ be the characteristic of the field (so that $p$ is prime and $q = p^s$ for some integer $s ≥ 1$). Then the number of queries performed by the test grows like $\ell· q^{2\ell+1}$, where $\ell = \big\lceil \frac{d+1}{q-q/p}\big\rceil $. Furthermore, $q^{Ω(\ell)}$ queries are necessary when $q = O(d)$. The test itself provides a unifying view of the tests for these two extremes: it considers random affine subspaces of dimension $\ell$ and verifies that the function restricted to the selected subspaces is a polynomial of degree at most $d$. Viewed in the context of coding theory, our result shows that Reed-Muller codes over general fields (usually referred to as generalized Reed-Muller (GRM) codes) are locally testable. In the course of our analysis we provide a characterization of small-weight words that span the code. Such a characterization was previously known only when the field size is a prime or is sufficiently large, in which case the minimum-weight words span the code.