  In the $t$-online-erasure model in property testing, an adversary is allowed
to erase $t$ values of a queried function for each query the tester makes. This
model was recently formulated by Kalemaj, Raskhodnikova andVarma, who showed
that the properties of linearity of functions as well as quadraticity can be
tested in$O_t(1)$ many queries: $O(\log (t))$ for linearity and $2^{2^{O(t)}}$
for quadraticity. They asked whether the more general property of
low-degreeness can be tested in the online erasure model, whether better
testers exist for quadraticity, and if similar results hold when ``erasures''
are replaced with ``corruptions''.
  We show that, in the $t$-online-erasure model, for a prime power $q$, given
query access to a function $f: \mathbb{F}_q^n \xrightarrow[]{} \mathbb{F}_q$,
one can distinguish in $\mathrm{poly}(\log^{d+q}(t)/\delta)$ queries between
the case that $f$ is degree at most $d$, and the case that $f$ is $\delta$-far
from any degree $d$ function (with respect to the fractional hamming distance).
This answers the aforementioned questions and brings the query complexity to
nearly match the query complexity of low-degree testing in the classical
property testing model. Our results are based on the observation that the
property of low-degreeness admits a large and versatile family of query
efficient testers. Our testers operates by querying a uniformly random,
sufficiently large set of points in a large enough affine subspace, and finding
a tester for low-degreeness that only utilizes queries from that set of points.
  We believe that this tester may find other applications to algorithms in the
online-erasure model or other related models, and may be of independent
interest.
