Generic polynomial

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

Pure Picard-Vessiot extensions with generic properties

Abstract: Given a connected linear algebraic group G over an algebraically closed field C of characteristic 0, we construct a pure Picard-Vessiot extension for G, namely, a Picard-Vessiot extension E ⊃ F, with differential Galois group G, such that E and F are purely differentially transcendental over C. The differential field E is the quotient field of a G-stable proper differential subring R with the property that if F is any differential field with field of constants C and E ⊃ F is a Picard-Vessiot extension with differential Galois group a connected subgroup H of G, then there is a differential homomorphism Φ: R → E such that E is generated over F as a differential field by Φ(R).

Non-linear Group Actions with Polynomial Invariant Rings and a Structure Theorem for Modular Galois Extensions

Abstract: Let $G$ be a finite $p$-group and $k$ a field of characteristic $p>0$. We show that $G$ has a \emph{non-linear} faithful action on a polynomial ring $U$ of dimension $n=\mathrm{log}_p(|G|)$ such that the invariant ring $U^G$ is also polynomial. This contrasts with the case of \emph{linear and graded} group actions with polynomial rings of invariants, where the classical theorem of Chevalley-Shephard-Todd and Serre requires $G$ to be generated by pseudo-reflections. Our result is part of a general theory of "trace surjective $G$-algebras", which, in the case of $p$-groups, coincide with the Galois ring-extensions in the sense of \cite{chr}. We consider the \emph{dehomogenized symmetric algebra} $D_k$, a polynomial ring with non-linear $G$-action, containing $U$ as a retract and we show that $D_k^G$ is a polynomial ring. Thus $U$ turns out to be \emph{universal} in the sense that every trace surjective $G$-algebra can be constructed from $U$ by "forming quotients and extending invariants". As a consequence we obtain a general structure theorem for Galois-extensions with given $p$-group as Galois group and any prescribed commutative $k$-algebra $R$ as invariant ring. This is a generalization of the Artin-Schreier-Witt theory of modular Galois field extensions of degree $p^s$.

Splitting behavior of S n -polynomials

Abstract: We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial \(f(x) \in {\mathbb {Z}}[x]\) with coefficients in a box of side B satisfies: (i) f(x) is irreducible over Open image in new window, with splitting field \(K_{f}/{\mathbb {Q}}\) over Open image in new window having Galois group Sn; (ii) the polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii) f(x) has a prescribed splitting type (mod p) at each prime p in S.

Representation of a real polynomial f(X) as a sum of 2m-th powers of rational functions

Abstract: From Becker’s Satz 2.14 in [B1] it follows that a polynomial f ∈ ℝ[X] admits a representation $$ f = \sum\limits_{{i = 1}}^{\sigma } {\frac{{g_i^{{2m}}}}{{{h^{{2m}}}}}} $$ (1) with gi, h∈ ℝ[X] if and only if f satisfies the following three conditions: (i) 2m divides deg f (ii) 2m divides the order of every real zero of f (iii) f is positive semidefinite Once f satisfies these conditions, the problem arises how to obtain a representation (1) for f. This paper is concerned with that problem.