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ELEMENTS OF LARGE ORDER IN PRIME FINITE FIELDS

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry Sequences and sets Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  16 October 2012

MEI-CHU CHANG
MEI-CHU CHANG*
Affiliation:
Department of Mathematics, University of California, Riverside, USA (email: [email protected])
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Abstract

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Given $f(x,y)\in \mathbb Z[x,y]$ with no common components with $x^a-y^b$ and $x^ay^b-1$, we prove that for $p$ sufficiently large, with $C(f)$ exceptions, the solutions $(x,y)\in \overline {\mathbb F}_p\times \overline {\mathbb F}_p$ of $f(x,y)=0$ satisfy $ {\rm ord}(x)+{\rm ord}(y)\gt c (\log p/\log \log p)^{1/2},$ where $c$ is a constant and ${\rm ord}(r)$ is the order of $r$ in the multiplicative group $\overline {\mathbb F}_p^*$. Moreover, for most $p\lt N$, $N$ being a large number, we prove that, with $C(f)$ exceptions, ${\rm ord}(x)+{\rm ord}(y)\gt p^{1/4+\epsilon (p)},$ where $\epsilon (p)$ is an arbitrary function tending to $0$ when $p$ goes to $\infty $.

Keywords

multiplicative ordermultiplicative groupfinite fieldsadditive combinatorics

MSC classification

Secondary: 11B75: Other combinatorial number theory 14G15: Finite ground fields 11G20: Curves over finite and local fields 11T06: Polynomials 11T30: Structure theory 11T55: Arithmetic theory of polynomial rings over finite fields 11G99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 1 , August 2013 , pp. 169 - 176
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

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