Permuting the roots of univariate polynomials whose coefficients depend on parameters

Abstract: We address two interrelated problems concerning permutation of roots of univariate polynomials whose coefficients depend on parameters. First, we compute the Galois group of polynomials

v a r p h i (x) i n m a t h b b C [t_{1}, c d o t s, t_{k}] [x]

over

m a t h b b C (t_{1}, c d o t s, t_{k})

. Provided that the corresponding multivariate polynomial

v a r p h i (x, t_{1}, c d o t s, t_{k})

is generic with respect to its support set

A s u b s e t m a t h b b Z^{k + 1}

, we determine the latter Galois group for any

A

. Second, we determine the Galois group of systems of polynomial equations of the form

p (x, t) = q (t) = 0

where

p

and

q

have prescribed support sets

A_{1} s u b s e t m a t h b b Z^{2}

and

A_{2} s u b s e t 0 i m e s m a t h b b Z

respectively. For each problem, we determine the image of an appropriate braid monodromy map in order to compute the sought Galois group. Among other applications, the present results allow to compute the Galois group of rational functions and give insights on the Galois group of some general enumerative problems; provide methods to compute the kernel of the braid monodromy map associated to

v a r p h i

; connect with Zariski's theorem on the fundamental group of the complement to projective hypersurfaces.

Mathematics Subject Classification ID

Separable extensions, Galois theory (12F10) Braid groups; Artin groups (20F36) Structure of families (Picard-Lefschetz, monodromy, etc.) (14D05)

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