Generic polynomial

About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.

Classes Doubles, Idéaux de Galois et Résolvantes

Abstract: Results on permutation groups are applyed to Galois theory In particular, to compute rapidly relations among roots of an univariate polynomial

Descent principle in modular Galois theory

Abstract: We propound a descent principle by which previously constructed equations over GF(qn)(X) may be deformed to have incarnations over GF(q)(X) without changing their Galois groups. Currently this is achieved by starting with a vectorial (= additive)q-polynomial ofq-degreem with Galois group GL(m, q) and then, under suitable conditions, enlarging its Galois group to GL(m, qn) by forming its generalized iterate relative to an auxiliary irreducible polynomial of degreen. Elsewhere this was proved under certain conditions by using the classification of finite simple groups, and under some other conditions by using Kantor’s classification of linear groups containing a Singer cycle. Now under different conditions we prove it by using Cameron-Kantor’s classification of two-transitive linear groups.

Algebraic and differential generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier

Abstract: We give a complete answer to the analogue of Grothendieck conjecture on p-curvatures for q-difference equations defined over K(x), where K is any finitely generated extension of Q and q\in K can be either a transcendental or an algebraic number. This generalizes the results in [DV02], proved under the assumption that K is a number field and q an algebraic number. The results also hold for a field K which is a finite extension of a purely transcendental extension k(q) of a perfect field k. If k is a number field and q is a parameter, one can either reduce the equation modulo a finite place of k or specialize the parameter q, or both. In particular for q=1, we obtain a differential equation defined over a number field or in positive characteristic. In \S II, we consider two Galois groups attached to a q-difference module M over K(x): the generic Galois group Gal(M), in the sense of [Kat82]; if char K=0, the generic differential Galois group Gal^D(M), which is a Kolchin differential algebraic group. We deduce an arithmetic description of Gal(M) (resp. Gal^D(M)). In positive characteristic, we prove some devissage. There are many Galois theories for q-difference equations defined over fields such as C, the elliptic functions, or the differential closure of C. In \S III, we show that the Galois D-groupoid [Gra09] of a nonlinear q-difference system generalizes Gal^D(M). In \S IV we give some comparison results between the two generic Galois groups above and the other Galois groups for linear $q$-difference equations in the literature. We compare: the group introduced in [HS08] with the Gal^D(M) and hence with the Galois D-groupoid (cf [Mal09]); Gal(M) and Gal^D(M) to the generic Galois groups of the modules obtained by specialization of q or by reduction in positive characteristic. We relate the dimension of Gal^D(M) to the differential relations among the solutions of M.

GF(2 m ) multiplier using Polynomial Residue Number System

Abstract: This paper studies the polynomial residue representation of Galois field (2m) elements and polynomial residue arithmetic (PRA), according to which a novel approach of performing GF(2m) multiplication using polynomial residue number system (PRNS) is introduced. A channel-serial and a channel-parallel architecture of the PRNS multiplier over GF(2m) are presented. Conclusion is drawn by comparing the synthesis results of these two architectures.

Hopf Galois structures on Kummer extensions of prime power degree

Abstract: Let K be a field of characteristic not p (an odd prime), containing a primitive p n -th root of unity , and let L = K(z) with x p n a the minimal polynomial of z over K: thus L|K is a Kummer extension, with cyclic Galois group G = h i acting on L via (z) = z . T. Kohl, 1998, showed that L|K has p n 1 Hopf Galois structures. In this paper we describe these Hopf Galois structures.