Generic polynomial

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

Dihedral Galois groups of even degree polynomials

Abstract: We present a new characterization of dihedral Galois groups of rational irreducible polynomials. It allows us to reduce the problem of deciding whether the Galois group of an even degree polynomial is dihedral, and its computation in the afirmative case, to the case of a quartic or odd degree polynomial, for which algorithms already exist. The characterization and algorithm are extended to permutation groups of order 2n containing an n-cycle.

Rationality Criteria for Galois Extensions

Abstract: Some rationality criteria for finite Galois extensions over C(t) are explained. The first rationality criterion and the second rationality criterion, together with the corresponding examples are contained in the forthcoming lecture notes (27) (see also (23-25)). The rationality criteria in sections 4 and 5, the braid orbit theorem, and the twisted braid orbit theorem, are new. With the last one, the Mathieu group M24 is realized as Galois group over Q. More than 150 years ago Galois attached to every polynomial (over a given field) without double zeros a finite group. In principle he proceeded in the following way: Let K be a field, K an algebraic closure of K and f(X) E K(X) a separable polynomial of degree m with zeros (h, ... , 8m in K. Then the set of polynomials in K(X) := K(X1"" ,Xm), defined by

Lovelock Terms and BRST Cohomology

Abstract: Lovelock terms are polynomial scalar densities in the Riemann curvature tensor that have the remarkable property that their Euler-Lagrange derivatives contain derivatives of the metric of order not higher than two (while generic polynomial scalar densities lead to Euler-Lagrange derivatives with derivatives of the metric of order four). A characteristic feature of Lovelock terms is that their first nonvanishing term in the expansion of the metric around flat space is a total derivative. In this paper, we investigate generalized Lovelock terms defined as polynomial scalar densities in the Riemann curvature tensor and its covariant derivatives (of arbitrarily high but finite order) such that their first nonvanishing term in the expansion of the metric around flat space is a total derivative. This is done by reformulating the problem as a BRST cohomological one and by using cohomological tools. We determine all the generalized Lovelock terms. We find, in fact, that the class of nontrivial generalized Lovelock terms contains only the usual ones. Allowing covariant derivatives of the Riemann tensor does not lead to new structure. Our work provides a novel algebraic understanding of the Lovelock terms in the context of BRST cohomology.

Characterizing Triviality of the Exponent Lattice of a Polynomial Through Galois and Galois-Like Groups

Abstract: The problem of computing the exponent lattice which consists of all the multiplicative relations between the roots of a univariate polynomial has drawn much attention in the field of computer algebra. As is known, almost all irreducible polynomials with integer coefficients have only trivial exponent lattices. However, the algorithms in the literature have difficulty in proving such triviality for a generic polynomial. In this paper, the relations between the Galois group (respectively, the Galois-like groups) and the triviality of the exponent lattice of a polynomial are investigated. The \(\mathbb {Q}\)-trivial pairs, which are at the heart of the relations between the Galois group and the triviality of the exponent lattice of a polynomial, are characterized. An effective algorithm is developed to recognize these pairs. Based on this, a new algorithm is designed and implemented to prove the triviality of the exponent lattice of a generic irreducible polynomial, which considerably improves a state-of-the-art implementation of an algorithm of the same type when the polynomial degree becomes larger. In addition, the concept of the Galois-like groups of a polynomial is introduced. Some properties of the Galois-like groups are proved and, more importantly, a sufficient and necessary condition is given for a polynomial (which is not necessarily irreducible) to have trivial exponent lattice.

Nonlinear differential algorithm to compute all the zeros of a generic polynomial

Abstract: A simple algorithm to compute all the zeros of a generic polynomial is proposed.A simple algorithm to compute all the zeros of a generic polynomial is proposed.