Dissertation/Thesis Abstract

Refining Multivariate Value Set Bounds
by Smith, Luke Alexander, Ph.D., University of California, Irvine, 2015, 43; 3709756
Abstract (Summary)

Over finite fields, if the image of a polynomial map is not the entire field, then its cardinality can be bounded above by a significantly smaller value. Earlier results bound the cardinality of the value set using the degree of the polynomial, but more recent results make use of the powers of all monomials.

In this paper, we explore the geometric properties of the Newton polytope and show how they allow for tighter upper bounds on the cardinality of the multivariate value set. We then explore a method which allows for even stronger upper bounds, regardless of whether one uses the multivariate degree or the Newton polytope to bound the value set. Effectively, this provides an alternate proof of Kosters' degree bound, an improved Newton polytope-based bound, and an improvement of a degree matrix-based result given by Zan and Cao.

Indexing (document details)
Advisor: Wan, Daqing
Commitee: Rubin, Karl, Silverberg, Alice
School: University of California, Irvine
Department: Mathematics
School Location: United States -- California
Source: DAI-B 76/11(E), Dissertation Abstracts International
Subjects: Mathematics
Keywords: Finite fields, Multivariate, Newton polytope, Number theory, Polynomial maps, Value set
Publication Number: 3709756
ISBN: 9781321854787
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