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 degree-constrained subgraphs modulo 2 d such as 2 d -divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over F2 belong to the complexity class Polynomial Parity Argument (PPA).

On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz

“…Papadimitriou has proven that Explicit Chevalley is in PPA. Varga [27] has shown the same for the special case of CNSS where the input polynomial P is specified as the sum of a polynomial number of polynomials P i , where each P i is the product of explicitly given polynomials whose sum of degrees is at most n. In addition, the input also contains a polynomial time computable matching for all but one of the monomials x 1 • • • x n of P . However, the paper doesn't address the question why this doesn't make the problem a promise problem.…”

Functional degrees and arithmetic applications I: The set of functional degrees

“…The results of this section are not used elsewhere in this paper. However, integervalued polynomials and their reductions occur in Wilson's proof of Ax-Katz over F p [Wi06, Lemma 4], and the technique of representing functions between residue rings of Z via integer-valued polynomials also occurs in a work of Varga [Va14] generalizing Warning's Second Theorem. It seems potentially useful to know that these techniques can be viewed in terms of the Aichinger-Moosbauer calculus.…”

“…Remark 2.23. The fact that functions Z/p α Z → Z/p β Z can (after pullback via ε : Z → Z/p α Z) be represented by reductions of integer-valued polynomials is applied in work of Varga [Va14]. In [CW18] this work was generalized to maps of the form Z K /p α → Z K /p β where K is a number field, Z K is its ring of integers, and p is a nonzero prime ideal of Z K (so that Z K /p α and Z K /p β are finite rings of p-power order for some p ∈ P).…”