  We show that lower bounds for explicit constant-variate polynomials over
fields of characteristic $p > 0$ are sufficient to derandomize polynomial
identity testing over fields of characteristic $p$. In this setting, existing
work on hardness-randomness tradeoffs for polynomial identity testing requires
either the characteristic to be sufficiently large or the notion of hardness to
be stronger than the standard syntactic notion of hardness used in algebraic
complexity. Our results make no restriction on the characteristic of the field
and use standard notions of hardness.
  We do this by combining the Kabanets-Impagliazzo generator with a white-box
procedure to take $p$-th roots of circuits computing a $p$-th power over fields
of characteristic $p$. When the number of variables appearing in the circuit is
bounded by some constant, this procedure turns out to be efficient, which
allows us to bypass difficulties related to factoring circuits in
characteristic $p$.
  We also combine the Kabanets-Impagliazzo generator with recent
"bootstrapping" results in polynomial identity testing to show that a
sufficiently-hard family of explicit constant-variate polynomials yields a
near-complete derandomization of polynomial identity testing. This result holds
over fields of both zero and positive characteristic and complements a recent
work of Guo, Kumar, Saptharishi, and Solomon, who obtained a slightly stronger
statement over fields of characteristic zero.
