We consider the following multiplication-based tests to check if a given function f: \mathbb{F}_q^n\to \mathbb{F}_q is the evaluation of a degree-d polynomial over \mathbb{F}_q for q prime.

* \mathrm{Test}_{e,k}: Pick P_1,\ldots,P_k independent random degree-e polynomials and accept iff the function fP_1\cdots 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,\ldots,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 \mathbb{F}_q, which may be of independent interest.