Publication

Conditional expanding bounds for two-variable functions over finite valuation rings

Van Thang Pham
2017
Journal Articles

Abstract

In this paper, we use methods from spectral graph theory to obtain some results on the sum-product problem over finite valuation rings R of order q(r) which generalize recent results given by Hegyvari and Hennecart (2013). More precisely, we prove that, for related pairs of two-variable functions f(x, y) and g(x, y), if A and B are two sets in R* with vertical bar A vertical bar = vertical bar B vertical bar = q(alpha), then max {vertical bar f(A, B)vertical bar, vertical bar g(A, B)vertical bar} >> vertical bar A vertical bar(1+Delta(alpha)), for some Delta(alpha) > 0. (C) 2016 Elsevier Ltd. All rights reserved.

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https://infoscience.epfl.ch/entities/publication/abf1fd75-92ce-4729-a4d2-d929d6e86263

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Van Thang Pham
2017
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/abf1fd75-92ce-4729-a4d2-d929d6e86263

About this result

Related concepts (26)

Valuation ring

In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x−1 belongs to D. Given a field F, if D is a subring of F such that either x or x−1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring.

Discrete valuation ring

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions: R is a local principal ideal domain, and not a field. R is a valuation ring with a value group isomorphic to the integers under addition. R is a local Dedekind domain and not a field. R is a Noetherian local domain whose maximal ideal is principal, and not a field.

Spectral graph theory

In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one.

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