Combinatorial Nullstellensatz modulo prime powers and the Parity Argument
Abstract
We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degreeconstrained subgraphs modulo $2^d$ such as $2^d$divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over $\mathbb{F}_2$ belong to the complexity class Polynomial Parity Argument (PPA).
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.4422
 Bibcode:
 2014arXiv1402.4422V
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Computational Complexity;
 Mathematics  Number Theory