J P Buhler

Bio: J P Buhler is an academic researcher from Reed College. The author has contributed to research in topics: Phase problem & Structure (category theory). The author has an hindex of 8, co-authored 10 publications receiving 475 citations.

On the essential dimension of a finite group

Abstract: Let f(x) = Σaixi be a monic polynomial of degree n whosecoefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) isgenerating (for the symmetric group) if it can be obtained from f(x) by a nondegenerate Tschirnhaus transformation. We show that the minimal number dk(n) of algebraically independent coefficients of such a polynomial is at least [n/2]. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilbert‘s 13th problem.

On the evaluation of Euler sums

Abstract: Euler studied double sums of the form $$ \z(r,s)=\sum_{1\le m

Irregular primes and cyclotomic invariants to four million

Abstract: Recent computations of irregular primes, and associated cyclotomic invariants, were extended to all primes below four million using an enhanced multisectioning/convolution method. Fermat's \"Last Theorem\" and Vandiver's conjecture were found to be true for those primes, and the cyclotomic invariants behaved as expected. There is exactly one prime less than four million whose index of irregularity is equal to seven. An irregular pair (p, t) consists of an odd prime p and an even integer t such that 0 < t < p 1 and p divides (the numerator of) the Bernoulli number Bt. The index of irregularity rp for a prime p is the number of irregular pairs for p. Kummer computed the irregular pairs for odd primes p less than 165 by 1874. In the 1920s and 1930s, H. S. Vandiver used desk calculators and graduate students to find the irregular primes for p < 620, and used these computations to verify Fermat's \"Last Theorem\" (FLT) for those primes. Derrick and Emma Lehmer, together with H. S. Vandiver, used a computer in 1954 [7] to make the same computations for p < 2000 \"in a few hours.\" They indicated an awareness of the importance of these computations independent of FLT: \"Irrespective of whether Fermat's Last Theorem is proved or disproved, the contents of the table... constitute a permanent addition to our knowledge of cyclotomic fields.\" The wisdom of this remark is clear, for instance, from Iwasawa's subsequent theory of the structure of cyclotomic class groups. The enduring interest of these calculations is also evident from the fact that a sequence of papers in this journal over the last thirty years has progressively extended the upper limit; one paper [5] appeared in the volume celebrating Derrick Lehmer's seventieth birthday. Our goal here is to extend this sequence by announcing that the computations of irregular pairs, and verification of the usual conjectures, have been completed for all primes p between one and four million. The notation, and details of the underlying algorithms, can be found in [1] and [4] (or their predecessors), which describe the calculations for p less than one million. The most immediate applications are to FLT, Vandiver's conjecture, and cyclotomic invariants. Received by the editor October 29, 1992 and, in revised form, December 14, 1992. 1991 Mathematics Subject Classification. Primary 11B68, 11D41, 11R23, 11Y40; Secondary 65T20, 11R18, 11R29. The first author was partially supported by National Science Foundation Grant DMS-9012989. © 1993 American Mathematical Society 0025-5718/93 $1.00+ $.25 per page

Irregular primes to one million

Abstract: Using "fast" algorithms for power series inversion (based on the fast Fourier transform and multisectioning of power series), we have calculated all irregular primes up to one million, including their indices of irregularity and associated irregular pairs. Using this data, we verified that Fermat's "Last Theorem" and Vandiver's conjecture are true for these primes. Two primes with index of irregularity five were already known; we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six. A pair of integers (p, k) is said to be an irregular pair if p is a prime, k is an even integer satisfying 2 p are known (see [8]). Our calculations show that Vandiver's conjecture is true for all p < 106. The table of irregular pairs could also be used to calculate Iwasawa invariants for the corresponding primes. We did not do this as part of our calculations; see [9] or [4] for a discussion of this problem. Previous computations of irregular pairs have used algorithms that take 0(p2) arithmetic operations for each prime p. The Bernoulli numbers are defined, in Received by the editor April 30, 1991 and, in revised form, September 10, 1991. 1991 Mathematics Subject Classification. Primary 11B68; Secondary 1 1D41, 65T20, 1 1Y99. The first author was partially supported by National Science Foundation Grant DMS-9012989. (D 1992 American Mathematical Society 0025-5718/92 $1.00 + $.25 per page

Fermat’s Last Theorem

Abstract: The authors would like to give special thanks to N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this paper. They are also grateful to A. Agboola, M. Bertolini, B. Edixhoven, J. Fearnley, R. Gross, L. Guo, F. Jarvis, H. Kisilevsky, E. Liverance, J. Manoharmayum, K. Ribet, D. Rohrlich, M. Rosen, R. Schoof, J.-P. Serre, C. Skinner, D. Thakur, J. Tilouine, J. Tunnell, A. Van der Poorten, and L. Washington for their helpful comments. Darmon thanks the members of CICMA and of the Quebec-Vermont Number Theory Seminar for many stimulating conversations on the topics of this paper, particularly in the Spring of 1995. For the same reason Diamond is grateful to the participants in an informal seminar at Columbia University in 1993-94, and Taylor thanks those attending the Oxford Number Theory Seminar in the Fall of 1995.

Special values of multiple polylogarithms

Abstract: Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.

Modular forms, a computational approach

Abstract: Modular forms Modular forms of level $1$ Modular forms of weight $2$ Dirichlet characters Eisenstein series and Bernoulli numbers Dimension formulas Linear algebra General modular symbols Computing with newforms Computing periods Solutions to selected exercises Appendix A: Computing in higher rank Bibliography Index.

On the rapid computation of various polylogarithmic constants

Abstract: We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log(2) or π on a modest work station in a few hours run time.