Joe Buhler

On the essential dimension of a finite group

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.Our approach to this question (and generalizations) is basedon the idea of the ’essential dimension‘ of a finite group G:the smallest possible dimension of an algebraic G-variety over k to which one can ‘compress’ a faithful linear representation of G. We show that dk(n) is just the essential dimension of the symmetricgroup Sn. We give results on the essential dimension ofother groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.

Irregular primes and cyclotomic invariants to 12 million

Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to 12 million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes Shokrollahi (1996). The latter idea reduces the problem to that of finding zeros of a polynomial overFpof degree < (p? 1)/2 among the quadratic nonresidues mod p. Use of fast polynomial gcd-algorithms gives anO (p log2p loglog p)-algorithm for this task. A more efficient algorithm, with comparable asymptotic running time, can be obtained by using Schonhage?Strassen integer multiplication techniques and fast multiple polynomial evaluation algorithms; this approach is particularly efficient when run on primes p for whichp? 1 has small prime factors. We also give some improvements on previous implementations for verifying the Kummer?Vandiver conjecture and for computing the cyclotomic invariants of a prime.

On the evaluation of Euler sums

Euler studied double sums of the form for positive integers r and s, and inferred, for the special cases r = 1 or r + s odd, elegant identities involving values of the Riemann zeta function. Here we establish various series expansions of ζ(r, s) for real numbers r and s. These expansions generally involve infinitely many zeta values. The series of one type terminate for integers r and s with r + s odd, reducing in those cases to the Euler identities. Series of another type are rapidly convergent and therefore useful in numerical experiments.

Lattice basis reduction, Jacobi sums and hyperelliptic cryptosystems

Using the LLL-algorithm for finding short vectors in lattices, we show how to compute a Jacobi sum for the prime field F p in Q( e 2πi/n ) in time O (log 3 p ), where n is small and fixed, p is large, and p = 1 (mod n ). This result is useful in the construction of hyperelliptic cryptosystems.

A note on the binomial drop polynomial of a poset

Abstract Suppose (P, ≺) is a poset of size n and π: P → P is a permutation. We say that π has a drop at x if π(x)≺x. Let δP(k) denote the number of π having k drops 0 ⩽ k Δ p (λ)≔ ∑ k δ p (k) λ+k n Further, define the incomparability graph I(P) to have vertex set P and edges ij whenever i and j are incomparable in P, i.e., neither i ≺ j nor j ≺ i holds. In this note we give a short proof that ΔP(λ) is equal to the chromatic polynomial of I(P).

On the convergence problem for lattice sums

Three-dimensional lattice sums Sigma +or-r-s, where r is the distance to a +or- charged lattice point, arise in classical lattice point problems (s=0) and the Madelung problem of physics and chemistry (s=1). In the latter case the sum over an infinite lattice is purely formal and the value of the Madelung constant must be defined precisely, e.g., via related convergent sums or analytic continuation in s. Indeed, the partial sum over the sphere r(R does not converge as R becomes large. The authors verify a conjecture of J F Delord (1988) that convergence can be obtained by neutralising each sphere with an appropriate surface charge. Specifically, if LR(s) is the sum over the lattice points in the sphere r(R then, for neutrally charged lattices, they show that as R goes to infinity the difference LR(s)-R-sLR(0) approaches L(s), where L(s) is defined by analytic continuation. When s=1 the term L(1) is the Madelung constant and LR(0)/R is the Coulombic correction term.

On Tschirnhaus Transformations

We revisit the classical problem of simplifying polynomials by means of Tschirnhaus transformations. We consider Tschirnhaus transformations involving (i) no auxiliary radicals, (ii) arbitrary radicals, (iii) odd degree radicals, and (iv) odd degree radicals and the square root of the discriminant. In [BR] we showed that by using substitutions of type (i) one cannot reduce the general polynomial of degree n to a form with less than [n/2] independent coefficients. In this paper we give a new proof of this result and also extend it to transformations of types (iii) and (iv). In the last section we present alternative proofs, based on the cohomological approach shown to us by J.-P. Serre.

Isotropic subspaces for skewforms and maximal abelian subgroups of p-groups

will be called a skewform. A subspace W is isotropic for a if the restriction of a to W is null, i.e., a(x, y) =0 for all X, YE W. More generally, if a = b 1 ,..., ak} is a collection of skewforms on V then a subspace W is said to be isotropic for a if it is isotropic for each a,. Note that we could choose to think of the collection a as an alternating bilinear map a: Vx V+ Fk. Given a collection a of k skewforms as above let m(a) be the dimension of the largest isotropic subspace for a. Let d(F, n, k) denote the minimum possible value of m(a) as a ranges over all k-tuples of skewforms on an n-dimensional vector space:

Fast and precise Fourier transforms

Many applications of fast Fourier transforms (FFTs), such as computer tomography, geophysical signal processing, high-resolution imaging radars, and prediction filters, require high-precision output. An error analysis reveals that the usual method of fixed-point computation of FFTs of vectors of length 2/sup l/ leads to an average loss of l/2 bits of precision. This phenomenon, often referred to as computational noise, causes major problems for arithmetic units with limited precision which are often used for real-time applications. Several researchers have noted that calculation of FFTs with algebraic integers avoids computational noise entirely. We combine a new algorithm for approximating complex numbers by cyclotomic integers with Chinese remaindering strategies to give an efficient algorithm to compute b-bit precision FFTs of length L. More precisely, we approximate complex numbers by cyclotomic integers in Z[e(2/spl pi/i/2/sup n/)] whose coefficients, when expressed as polynomials in e(2/spl pi/i/2/sup n/), are bounded in absolute value by some integer M. For fixed n our algorithm runs in time O(log(M)), and produces an approximation with worst case error of O(1/M(2/sup n-2/-1)). We prove that this algorithm has optimal worst case error by proving a corresponding lower bound on the worst case error of any approximation algorithm for this task. The main tool for designing the algorithms is the use of the cyclotomic units, a subgroup of finite index in the unit group of the cyclotomic field. First implementations of our algorithms indicate that they are fast enough to be used for the design of low-cost high-speed/high-precision FFT chips.

Lines imply spaces in density Ramsey theory

Abstract Some results of geometric Ramsey theory assert that if F is a finite field (respectively, set) and n is sufficiently large, then in any coloring of the points of Fn there is a monochromatic k-dimensional affine (respectively, combinatorial) subspace (see [9]). We prove that the density version of this result for lines (i.e., k = 1) implies the density version for arbitrary k. By using results in [3, 6] we obtain various consequences: a “group-theoretic” version of Roths Theorem, a proof of the density assertion for arbitrary k in the finite field case when ∥F∥ = 3, and a proof of the density assertion for arbitrary k in the combinatorial case when ∥F∥ = 2.