On the Complexity of Modulo-q Arguments and the Chevalley - Warning Theorem

We study the search problem class PPA_q defined as a modulo-q analog of the well-known polynomial parity argument class PPA introduced by Papadimitriou (JCSS 1994). Our first result shows that this class can be characterized in terms of PPA_p for prime p. Our main result is to establish that an explicit version of a search problem associated to the Chevalley - Warning theorem is complete for PPA_p for prime p. This problem is natural in that it does not explicitly involve circuits as part of the input. It is the first such complete problem for PPA_p when p ? 3. Finally we discuss connections between Chevalley-Warning theorem and the well-studied short integer solution problem and survey the structural properties of PPA_q

New bounds for Szemeredi's Theorem, I: Progressions of length 4 in finite field geometries

Let F be a fixed finite field of characteristic at least 5. Let G = F^n be the n-dimensional vector space over F, and write N := |G|. We show that if A is a subset of G with size at least c_F N(log N)^{-c}, for some absolute constant c > 0 and some c_F > 0, then A contains four distinct elements in arithmetic progression. This is equivalent, in the usual notation of additive combinatorics, to the assertion that r_4(G) <<_F N(log N)^{-c}.Comment: 30 pages, some irritating small errors correcte

Deterministic equation solving over finite fields

It is shown how to solve diagonal forms in many variables over finite fields by means of a deterministic efficient algorithm. Applications to norm equations, quadratic forms, and elliptic curves are given.Thomas Stieltjes Institute for MathematicsUBL - phd migration 201

A topological characterization of modulo-p arguments and implications for necklace splitting

The classes PPA-p have attracted attention lately, because they are the main candidates for capturing the complexity of Necklace Splitting with p thieves, for prime p. However, these classes were not known to have complete problems of a topological nature, which impedes any progress towards settling the complexity of the Necklace Splitting problem. On the contrary, topological problems have been pivotal in obtaining completeness results for PPAD and PPA, such as the PPAD-completeness of finding a Nash equilibrium [18, 15] and the PPA-completeness of Necklace Splitting with 2 thieves [24]. In this paper, we provide the first topological characterization of the classes PPA-p. First, we show that the computational problem associated with a simple generalization of Tucker's Lemma, termed p-polygon-Tucker, as well as the associated Borsuk-Ulam-type theorem, p-polygon-Borsuk-Ulam, are PPA-p-complete. Then, we show that the computational version of the well-known BSS Theorem [8], as well as the associated BSS-Tucker problem are PPA-p-complete. Finally, using a different generalization of Tucker's Lemma (termed Zp-star-Tucker), which we prove to be PPA-p-complete, we prove that p-thief Necklace Splitting is in PPA-p. This latter result gives a new combinatorial proof for the Necklace Splitting theorem, the only proof of this nature other than that of Meunier [42]. All of our containment results are obtained through a new combinatorial proof for Zp-versions of Tucker's lemma that is a natural generalization of the standard combinatorial proof of Tucker's lemma by Freund and Todd [27]. We believe that this new proof technique is of independent interest

Consensus Division in an Arbitrary Ratio

We consider the problem of partitioning a line segment into two subsets, so that n finite measures all have the same ratio of values for the subsets. Letting ? ? [0,1] denote the desired ratio, this generalises the PPA-complete consensus-halving problem, in which ? = 1/2. Stromquist and Woodall [Stromquist and Woodall, 1985] showed that for any ?, there exists a solution using 2n cuts of the segment. They also showed that if ? is irrational, that upper bound is almost optimal. In this work, we elaborate the bounds for rational values ?. For ? = ?/k, we show a lower bound of (k-1)/k ? 2n - O(1) cuts; we also obtain almost matching upper bounds for a large subset of rational ?. On the computational side, we explore its dependence on the number of cuts available. More specifically, 1) when using the minimal number of cuts for each instance is required, the problem is NP-hard for any ?; 2) for a large subset of rational ? = ?/k, when (k-1)/k ? 2n cuts are available, the problem is in PPA-k under Turing reduction; 3) when 2n cuts are allowed, the problem belongs to PPA for any ?; more generally, the problem belong to PPA-p for any prime p if 2(p-1)??p/2?/?p/2? ? n cuts are available