Topic
Generic polynomial
About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.
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TL;DR: In this paper, the authors considered the linear Noether's problem for the fixed field of linear actions on the rational function field and showed that the problem is affirmative for all non-solvable subgroups and the largest and one of the second largest subgroups of GL(4,Q) for the simple group of order 1451520.
Abstract: The linear Noether's problem means the rationality problem for the fixed field of linear actions on the rational function field. This paper deals with a part of our study on the four dimensional linear Noether's problem. Apart from the main part of our study, which will be published in other papers, the results which require complicated calculations by a computer are published here as a separate paper. The problem is affirmative for all of 5 non-solvable subgroups and the largest and one of the second largest subgroups of GL(4,Q). As relevant topics, we remark that PSp(3,2) (the simple group of order 1451520) has a generic polynomial over Q.
5 citations
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TL;DR: Bouc as mentioned in this paper introduced the notion of the b-group, which generalizes the Eulerian functions of a group defined by P. Hall (1936, Quart. Math.7, 134, 151).
5 citations
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TL;DR: In this article, the authors discuss methods for computing the group of polynomials with Galois group and obtain an explicit description of the exceptional numbers, i.e., those for which the specialized polynomial has Galois groups different from the general group.
Abstract: Let $P\in\mathbb Q[t,x]$ be a polynomial in two variables with rational coefficients, and let $G$ be the Galois group of $P$ over the field $\mathbb Q(t)$. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $c$ the specialized polynomial $P(c,x)$ has Galois group isomorphic to $G$ and factors in the same way as $P$. In this paper we discuss methods for computing the group $G$ and obtaining an explicit description of the exceptional numbers $c$, i.e., those for which $P(c,x)$ has Galois group different from $G$ or factors differently from $P$. To illustrate the methods we determine the exceptional specializations of three sample polynomials. In addition, we apply our techniques to prove a new result in arithmetic dynamics.
5 citations
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TL;DR: A simple algorithm to compute all the zeros of a generic polynomial is proposed.
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.
5 citations
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01 Dec 2000
TL;DR: In this article, a polynomial with some parameters which generates cyclic extensions of a given odd prime degree was defined, and proved to be generic in the sense as defined below.
Abstract: Using Cohen's construction of defining polynomials for a cyclic group of odd prime order, we define a polynomial with some parameters which generates cyclic extensions of a given odd prime degree, and prove it to be generic in the sense as defined below.
5 citations