Bounds for multiplicative cosets over fields of prime order
HTML articles powered by AMS MathViewer
- by Corey Powell PDF
- Math. Comp. 66 (1997), 807-822 Request permission
Abstract:
Let $m$ be a positive integer and suppose that $p$ is an odd prime with $p \equiv 1 \bmod m$. Suppose that $a \in (\mathbb {Z}/p\mathbb {Z})^*$ and consider the polynomial $x^m-a$. If this polynomial has any roots in $(\mathbb {Z}/p\mathbb {Z})^*$, where the coset representatives for $\mathbb {Z}/p\mathbb {Z}$ are taken to be all integers $u$ with $|u|<p/2$, then these roots will form a coset of the multiplicative subgroup $\mu _m$ of $(\mathbb {Z}/p\mathbb {Z})^*$ consisting of the $m$th roots of unity mod $p$. Let $C$ be a coset of $\mu _m$ in $(\mathbb {Z}/p\mathbb {Z})^*$, and define $|C|=\max _{u \in C}{|u|}$. In the paper “Numbers Having $m$ Small $m$th Roots mod $p$” (Mathematics of Computation, Vol. 61, No. 203 (1993),pp. 393-413), Robinson gives upper bounds for $M_1(m,p)=\min _{\tiny C \in (\mathbb {Z}/p\mathbb {Z})^* /\mu _m }{|C|}$ 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 $m$ is a power of $2$ or when $m=6.$References
- J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
- John L. Kelley and T. P. Srinivasan, Measure and integral. Vol. 1, Graduate Texts in Mathematics, vol. 116, Springer-Verlag, New York, 1988. MR 918770, DOI 10.1007/978-1-4612-4570-4
- Sergey Konyagin and Igor Shparlinski, On the Distribution of Residues of Finitely Generated Multiplicative Groups and Some of Their Applications, to appear.
- Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723, DOI 10.1007/978-1-4612-0853-2
- C. G. Lekkerkerker, Geometry of numbers, Bibliotheca Mathematica, Vol. VIII, Wolters-Noordhoff Publishing, Groningen; North-Holland Publishing Co., Amsterdam-London, 1969. MR 0271032
- Raphael M. Robinson, Numbers having $m$ small $m$th roots mod $p$, Math. Comp. 61 (1993), no. 203, 393–413. MR 1189522, DOI 10.1090/S0025-5718-1993-1189522-0
- P. Stevenhagen and H.W. Lenstra, Jr., Chebotarev and his density theorem, Math. Intelligencer 18 (1996), 26–37.
- Jeffrey D. Vaaler, A geometric inequality with applications to linear forms, Pacific J. Math. 83 (1979), no. 2, 543–553. MR 557952, DOI 10.2140/pjm.1979.83.543
Additional Information
- Corey Powell
- Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
- Received by editor(s): May 30, 1995
- Received by editor(s) in revised form: January 26, 1996
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 807-822
- MSC (1991): Primary 11A07, 11A15; Secondary 11N05, 11R18, 11R44
- DOI: https://doi.org/10.1090/S0025-5718-97-00797-7
- MathSciNet review: 1372008