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Bounds for Multiplicative Cosets over Fields of Prime Order

Authors:	Corey Powell

Institution:	Department of Mathematics, University of California at Berkeley, Berkeley, California 94720

Abstract:	Let be a positive integer and suppose that is an odd prime with $p \equiv 1 \bmod m$ . Suppose that $a \in (\Bbb Z /p\Bbb Z )^{\ast }$ and consider the polynomial . If this polynomial has any roots in $(\Bbb Z /p\Bbb Z )^{\ast }$ , where the coset representatives for $\Bbb Z /p\Bbb Z$ are taken to be all integers with , then these roots will form a coset of the multiplicative subgroup $\mu _m$ of $(\Bbb Z /p\Bbb Z )^{\ast }$ consisting of the th roots of unity mod . Let be a coset of $\mu _m$ in $(\Bbb Z /p\Bbb Z )^{\ast }$ , and define $\|C\|=\max _{u \in C}{\|u\|}$ . In the paper ``Numbers Having Small th Roots mod ' (Mathematics of Computation, Vol. 61, No. 203 (1993),pp. 393-413), Robinson gives upper bounds for $M_1(m,p)=\min _{\tiny C \in (\Bbb Z /p\Bbb Z )^{\ast } /\mu _m }{\|C\|}$ of the form $M_1(m,p)<K_mp^{1-1/\phi (m)}$ , where $\phi$ is the Euler phi-function. This paper gives lower bounds that are of the same form, and seeks to sharpen the constants in the upper bounds of Robinson. The upper bounds of Robinson are proven to be optimal when is a power of or when

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