Abstract
We show that lower bounds for explicit constantvariate 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 hardnessrandomness 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 KabanetsImpagliazzo generator with a whitebox procedure to take pth roots of circuits computing a pth 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 KabanetsImpagliazzo generator with recent "bootstrapping" results in polynomial identity testing to show that a sufficientlyhard family of explicit constantvariate polynomials yields a nearcomplete 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.
BibTeX  Entry
@InProceedings{andrews:LIPIcs:2020:12589,
author = {Robert Andrews},
title = {{Algebraic Hardness Versus Randomness in Low Characteristic}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {37:137:32},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771566},
ISSN = {18688969},
year = {2020},
volume = {169},
editor = {Shubhangi Saraf},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12589},
URN = {urn:nbn:de:0030drops125895},
doi = {10.4230/LIPIcs.CCC.2020.37},
annote = {Keywords: Polynomial identity testing, hardness versus randomness, low characteristic}
}
Keywords: 

Polynomial identity testing, hardness versus randomness, low characteristic 
Collection: 

35th Computational Complexity Conference (CCC 2020) 
Issue Date: 

2020 
Date of publication: 

17.07.2020 