Random Polynomials Over Finite Fields

Sharkey, Andrew (1999) Random Polynomials Over Finite Fields. PhD thesis, University of Glasgow.

Abstract

The idea of this thesis is to take some questions about polynomials over finite fields and 'answer' them using probability theory; that is, we give the average behaviour of certain properties of polynomials. We tend to deal with multivariate polynomials, so questions about factorisation are not considered. Questions which are considered are ones concerning images and pre-images under a random polynomial mapping, and more generic questions which lead to results on the distributions of certain character sums over finite fields. The methods used are based on those used by Odoni (details in Chapter 2). The probability space from which our random polynomial is chosen is essentially the set of all polynomials up to a given degree d, and we define a random variable associated with this space (for example, the number of zeros of a random polynomial). Once we have enough information about the random variable in question, we obtain asymptotic results about the distribution of this variable by letting both d and the size of the field, q, tend to infinty. The results in this work tend to rely on comparisons between random polynomials (of degree up to d) and random mappings. We therefore do a certain amount of work with random mappings, exploiting nice combinatorial properties which they exhibit, and also using some non-trivial results from the classical theory of random maps. The resulting theorems for random polynomials, when interpreted number-theoretically, are often what one would expect, but every once in a while they cough up a surprise.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Adviser: R W K Odoni
• Keywords: Mathematics
• Date of Award: 1999
• Depositing User: Enlighten Team
• Unique ID: glathesis:1999-76459
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 19 Nov 2019 14:19
• Last Modified: 19 Nov 2019 14:19
• URI: https://theses.gla.ac.uk/id/eprint/76459