Permuting the roots of univariate polynomials whose coefficients depend on parameters

Let K be a number field, and let lambda(x,t)\in K[x, t] be irreducible over K(t). Using algebraic geometry and group theory, we study the set of alpha\in K for which the specialized polynomial lambda(x,alpha) is K-reducible. We apply this to show that for any fixed n>=10 and for any number field K, all but finitely many K-specializations of the degree n generalized Laguerre polynomial are K-irreducible and have Galois group S_n. In conjunction with the theory of complex multi...

We investigate some Galois groups of linearized polynomials over fields such as $\mathbb{F}_q(t)$. The space of roots of such a polynomial is a module for its Galois group. We present a realization of the symmetric powers of this module, as a subspace of the splitting field of another linearized polynomial.

Galois/monodromy groups attached to parametric systems of polynomial equations provide a method for detecting the existence of symmetries in solution sets. Beyond the question of existence, one would like to compute formulas for these symmetries, towards the eventual goal of solving the systems more efficiently. We describe and implement one possible approach to this task using numerical homotopy continuation and multivariate rational function interpolation. We describe addit...

In this paper, the changes of representations of a group are used in order to describe its action as algebraic Galois group of an univariate polynomial on the roots of factors of any Lagrange resolvent. By this way, the Galois group of resolvent factors are pre-determinated. In follows, different applications are exposed; in particular, some classical results of algebraic Galois theory.

In this article, we give a numerical algorithm to compute braid groups of curves, hyperplane arrangements, and parameterized system of polynomial equations. Our main result is an algorithm that determines the cross-locus and the generators of the braid group.

The space of monic squarefree polynomials has a stratification according to the multiplicities of the critical points, called the equicritical stratification. Tracking the positions of roots and critical points, there is a map from the fundamental group of a stratum into a braid group. We give a complete determination of this map. It turns out to be characterized by the geometry of the translation surface structure on $\mathbb{CP}^1$ induced by the logarithmic derivative $df/...

In this paper, we propose linear maps over the space of all polynomials $f(x)$ in $\mathbb{F}_q[x]$ that map $0$ to itself, through their evaluation map. Properties of these linear maps throw up interesting connections with permutation polynomials. We study certain properties of these linear maps. We propose to classify permutation polynomials by identifying the generalized eigenspaces of these maps, where the permutation polynomials reside. As it turns out, several classes o...

The problem of computing \emph{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 gr...

We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the theoretical assumption that monodromy actions are generated uniformly, we show that the expected number of homotopy paths tracked by an algorithm following this framework is linear in the number of solutions. We demonstrate that our software implem...

The multivariate Tutte polynomial $\hat Z_M$ of a matroid $M$ is a generalization of the standard two-variable version, obtained by assigning a separate variable $v_e$ to each element $e$ of the ground set $E$. It encodes the full structure of $M$. Let $\bv = \{v_e\}_{e\in E}$, let $K$ be an arbitrary field, and suppose $M$ is connected. We show that $\hat Z_M$ is irreducible over $K(\bv)$, and give three self-contained proofs that the Galois group of $\hat Z_M$ over $K(\bv)$...