Robust Multiplication-Based Tests for Reed-Muller Codes

We consider the following multiplication-based tests to check if a given function f: F^n_q -> F_q is the evaluation of a degree-d polynomial over F_q for q prime. Test_{e,k}: Pick P_1,...,P_k independent random degree-e polynomials and accept iff the function f P_1 ... P_k is the evaluation of a degree-(d + ek) polynomial. We prove the robust soundness of the above tests for large values of e, answering a question of Dinur and Guruswami (FOCS 2013). Previous soundness analyses of these tests were known only for the case when either e = 1 or k = 1. Even for the case k = 1 and e > 1, earlier soundness analyses were not robust. We also analyze a derandomized version of this test, where (for example) the polynomials P_1 ,... , P_k can be the same random polynomial P. This generalizes a result of Guruswami et al. (STOC 2014). One of the key ingredients that go into the proof of this robust soundness is an extension of the standard Schwartz-Zippel lemma over general finite fields F_q, which may be of independent interest.