  We consider the following multiplication-based tests to check if a given
function $f: \mathbb{F}_q^n\to \mathbb{F}_q$ is a codeword of the Reed-Muller
code of dimension $n$ and order $d$ over the finite field $\mathbb{F}_q$ for
prime $q$ (i.e., $f$ 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 (i.e., is a codeword of the Reed-Muller code of
dimension $n$ and order $(d+ek)$).
  We prove the robust soundness of the above tests for large values of $e$,
answering a question of Dinur and Guruswami [Israel Journal of Mathematics,
209:611-649, 2015]. 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 $P_1,\dots,P_k$ can be the same random polynomial $P$. This
generalizes a result of Guruswami et al. [SIAM J. Comput., 46(1):132-159,
2017].
  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.
