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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 describe the Galois group Gal(F К /F) for arbi-trary ground fields with degree ≤ 6, and apply this description to three interesting related examples.
Abstract: Fix a ground field F. Let be a finite collection of finite degree separable field extensions of F. To study how the fields in К are related to each other, one should construct a joint splitting field F К and describe the Galois group Gal(F К /F) In this paper we describe the general appearance of Gal(F К /F) for arbi-trary F when the F i all have degree ≤ 6. Then we apply this description to three interesting related examples. First and second, we take ground fields the p-adic fields Q 2 and Q 3, and К the collection of all degree ≤ 6 fields. Third, we take ground field the rational number field Q and К the collection of all degree ≤ 6 fields with absolute discriminant of the form 2a3b This paper complements three of our previous papers. The paper [4] roughly speaking considers the Galois theory associated to a single field K of degree ≤ 6, whereas here we consider a finite collection of such K. The paper [5] lists all degree ≤ 6 fields over Q 2 and Q 3 . These tables form the basis of our local examples....
2 citations
12 Sep 2001
TL;DR: In this paper, the concept of neighborhood of a polynomial is analyzed and the square free property for the polynomials in its neighborhood is investigated, based on the idea of using the resultant in order to know the common factors between an empirical poynomial and a generic polynomorph in its neighbourhood.
Abstract: In this paper the concept of neighborhood of a polynomial is analyzed. This concept is spreading into Scientific Computation where data are often uncertain, thus they have a limited accuracy. In this context we give a new approach based on the idea of using resultant in order to know the common factors between an empirical polynomial and a generic polynomial in its neighborhood. Moreover given a polynomial, the Square Free property for the polynomials in its neighborhood is investigated.
2 citations
TL;DR: In this paper, a generic Galois G-extension S/R S / R, where R is a localized polynomial ring over F q F q, and G is an explicit generic polynomial for G in dim F q (A)dim F q (A) parameters, is presented.
2 citations
TL;DR: In this paper, the structure of the relative K 2 group of some truncated polynomial rings over F G with G a finite abelian p-group through calculation of Dennis-Stein symbols was determined.
2 citations
TL;DR: In this paper, an equivariant analog of the Morse formula for a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v was shown.
Abstract: Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula $$ {\rm Ind}^G(v) = \sum_{k = 0}^{n} (-1)^k \chi^G(\partial_{k}^{+}X) $$ which takes its values in A(G). Here IndG(v) denotes the equivariant index of the field v, $\{\partial_{k}^{+}X\}$ the v-induced Morse stratification (see [10]) of the boundary ∂X, and $\chi^G(\partial_{k}^{+}X)$ the class of the (n - k)-manifold $\partial_{k}^{+}X$ in A(G). We examine some applications of this formula to the equivariant real algebraic fields v in compact domains X ⊂ ℝn defined via a generic polynomial inequality. Next, we link the above formula with the equivariant degrees of certain Gauss maps. This link is an equivariant generalization of Gottlieb's formulas ([3, 4]).
2 citations