Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

Bounds for Multiplicative Cosets over Fields of Prime Order


Author: Corey Powell
Journal: Math. Comp. 66 (1997), 807-822
MSC (1991): Primary 11A07, 11A15; Secondary 11N05, 11R18, 11R44
MathSciNet review: 1372008
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 (\Bbb Z /p\Bbb Z )^{\ast } $ and consider the polynomial $x^m-a$. 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 $u$ with $|u|<p/2$, then these roots will form a coset of the multiplicative subgroup $\mu _m$ of $(\Bbb Z /p\Bbb Z )^{\ast } $ consisting of the $m$th roots of unity mod $p$. Let $C$ 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 $m$ Small $m$th Roots mod $p$'' (Mathematics of Computation, Vol. 61, No. 203 (1993),pp. 393-413), Robinson gives upper bounds for $M_1(m,p)=\min _{\tiny C \in (\Bbb Z /p\Bbb Z )^{\ast } /\mu _m }{|C|}$ 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 $m$ is a power of $2$ or when $m=6.$


References [Enhancements On Off] (What's this?)

  • 1. Algebraic number theory, Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the Inter national Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
  • 2. 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
  • 3. Sergey Konyagin and Igor Shparlinski, On the Distribution of Residues of Finitely Generated Multiplicative Groups and Some of Their Applications, to appear.
  • 4. Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723
  • 5. C. G. Lekkerkerker, Geometry of numbers, Bibliotheca Mathematica, Vol. VIII, Wolters-Noordhoff Publishing, Groningen; North-Holland Publishing Co., Amsterdam-London, 1969. MR 0271032
  • 6. Raphael M. Robinson, Numbers having 𝑚 small 𝑚th roots mod 𝑝, Math. Comp. 61 (1993), no. 203, 393–413. MR 1189522, 10.1090/S0025-5718-1993-1189522-0
  • 7. P. Stevenhagen and H.W. Lenstra, Jr., Chebotarev and his density theorem, Math. Intelligencer 18 (1996), 26-37. CMP 96:14
  • 8. Jeffrey D. Vaaler, A geometric inequality with applications to linear forms, Pacific J. Math. 83 (1979), no. 2, 543–553. MR 557952

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 11A07, 11A15, 11N05, 11R18, 11R44

Retrieve articles in all journals with MSC (1991): 11A07, 11A15, 11N05, 11R18, 11R44


Additional Information

Corey Powell
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720

DOI: http://dx.doi.org/10.1090/S0025-5718-97-00797-7
Received by editor(s): May 30, 1995
Received by editor(s) in revised form: January 26, 1996
Article copyright: © Copyright 1997 American Mathematical Society