Topic
Generic polynomial
About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.
Papers published on a yearly basis
Papers
More filters
••
TL;DR: In this paper, the set Vf of horizontal critical points of a polynomial function f for the standard Engel structure defined by the 1-forms?3 = dx3? x1dx2 and?4 = dx4? x3dx2, endowed with the sub-Riemannian metric was investigated.
Abstract: We investigate the set Vf of horizontal critical points of a polynomial function f for the standard Engel structure defined by the 1-forms ?3 = dx3 ? x1dx2 and ?4 = dx4 ? x3dx2, endowed with the sub-Riemannian metric $g_{\text {SR}}=d{x_{1}^{2}}+d{x^{2}_{2}}$. For a generic polynomial, we show that the set Γf of points in Vf, where Vf is not transverse to the Engel distribution, does not have a connected component which is contained in a fiber of f. Then, we prove that each trajectory of the horizontal gradient of f approaching the set Vf has a limit.
•
10 Mar 1989
TL;DR: In this paper, the multiplier/summer in the Galois bodies is parametrisable, i.e. it is possible to choose the body CG(2m) in which the polynomial operations are performed, with m being at the most equal to N, N being predetermined by the designer.
Abstract: The multiplier/summer in the Galois bodies is parametrisable, i.e it is possible to chose the Galois body CG(2m) in which the polynomial operations are performed, with m being at the most equal to N, N being predetermined by the designer. It comprises: a decoder (10) organized into N identical elementary cells which receive the generating polynomial G(m:0), and supplying the generating polynomial without its low-weight bit, G(m-1:0) and a polynomial marking the degree of the generating polynomial DG(m-1:0), and a calculating matrix (20), organized into N columns of identical elementary cells, receiving the polynomials A, B and C of the Galois body CG(2m) and supplying a polynomial result P = (A*B)modulo G+C. The invention applies to digital signal processors, for error detecting and correcting coding and decoding systems using BCH or RS codes.
•
01 Jan 2014
TL;DR: In this paper, the Galois closure of a non-Galois extension F/K of a global field was studied and the number of primes lying over an unrami ed place with given residue degree can be given as polynomials in a power of the characteristic of the variety G. This polynomial depend on the length function on the certain subgroups of the Weyl group of G.
Abstract: In this thesis we consider the following question: Given a finite separable non-Galois extension F/K of a global field K, how a prime P of K decomposes in the field F. In the first part, we study the Galois extension M/K where M is the Galois closure of F/K and action of Galois group G of M/K over the set of primes of F lying over a prime P in K. We obtain a one to one correspondence between the double coset space of G with respect to certain subgroups of G (depending on P and F) and the set of primes of F lying over P. Under this correspondence ramification indices and inertia degrees are explicitly determined. Then we investigate the case where G is a finite group of Lie type and F is the intermediate field corresponding to a parabolic subgroup of G. We obtain that the number of primes of F lying over an unrami ed place with given residue degree can be given as polynomials in a power of the characteristic of the variety G. This polynomials depend on the length function on the certain subgroups of the Weyl group of G.
•
TL;DR: In this paper, the existence of generic polynomials for various groups, over fields of positive characteristic, was explicitly established, and the results applied to a broad class of connected linear algebraic groups defined over finite fields satisfying certain conditions on cohomology.
Abstract: We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields satisfying certain conditions on cohomology. In particular, we use our techniques to study constructions for unipotent groups, certain algebraic tori, and certain split semisimple groups. An attractive consequence of our work is the construction of generic polynomials in the optimal number of parameters for all cyclic 2-groups over all fields of positive characteristic. This contrasts with a theorem of Lenstra, which states no cyclic 2-group of order $\ge 8$ has a generic polynomial over $\mathbb{Q}$.
••
TL;DR: In this article, it was shown that every G-Galois field extension L/K has a hyperbolic trace form in the presence of root of unity in a finite group containing a non-abelian Sylow 2-subgroup.
Abstract: Let G be a finite group containing a non-abelian Sylow 2-subgroup. We elementarily show that every G-Galois field extension L/K has a hyperbolic trace form in the presence of root of unity.