Showing papers on "Generic polynomial published in 1997"

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.

Courbes elliptiques sur Q sans nombres premiers exceptionnels

Abstract: Let E be an elliptic curve over Q. A prime N is said to be exceptional for E if the mod N Galois representation of E is not surjective, i.e. if the Galois group of the N-th division field of E is not equal to GL(2, N). We show that, in terms of heights, almost all curves have no exceptional prime.

Accelerated polynomial approximation of finite order entire functions by growth reduction

Abstract: Let f be an entire function of positive order and finite type. The subject of this note is the convergence acceleration of polynomial approximants of f by incorporating information about the growth of f(z) for z → ∞. We consider near polynomial approximation on a compact plane set K, which should be thought of as a circle or a real interval. Our aim is to find sequences (f n ) n of functions which are the product of a polynomial of degree < n and an easy computable second factor and such that (f n ) n converges essentially faster to f on K than the sequence (P n * ) n of best approximating polynomials of degree ≤ n. The resulting method, which we call Reduced Growth method (RG-method) is introduced in Section 2. In Section 5, numerical examples of the RG-method applied to the complex error function and to Bessel functions are given.

Galois theory of Moore-Carlitz-Drinfeld modules

Abstract: In 1896, E. H. Moore showed that the Galois group of the generic vectorial (= additive) q-polynomial of q-degree m is GL(m,q), where m > 0 is any integer and q > 1 is any power of any prime p. We show that, for any integer n > 0, the Galois group of the n-th iterate of the said polynomial is the generalized general linear group GL(m, q, n) consisting of all m by m matrices with invertible determinant over the local ring GF(q)[T]/Tn. For m=1, this was proved by Carlitz in 1938 as part of his explicit class field theory over finite fields. The case of m=1 was further enhanced by Drinfeld in 1974.

Galois cohomology of special orthogonal groups

Abstract: If (A,s) is a central simple algebra of even degree with orthogonal involution, then for the map of Galois cohomology sets from H^1(F, SO(A,s)) to the 2-torsion in the Brauer group of F, we describe fully the image of a given element of H^1(F, SO(A,s)) and prove that this description is correct in two different ways. As an easy consequence, we derive a result of Bartels.

On algebraic properties of selfreciprocal polynomials and of Daubechies filters of low order

Abstract: We show that the generic selfreciprocal polynomial of degree 2n has the Galois group S/sub 2/lS/sub n/. Consequently, "most" selfreciprocal polynomials of degree 2n with coefficients in an algebraic number field have the same Galois group. We use these results to determine algebraic properties of the Daubechies (1988) filters of low order.

Finite neofields and 2-transitive groups

Abstract: Let N be a finite neofield distinct from the Galois field and let G be a group generated by right translations x→x+a of an additive loop of N. We prove that, except for four particular cases, G=SN or G=AN holds.