Braid monodromy of univariate fewnomials

Esterov, Alexander; Lang, Lionel

doi:10.2140/gt.2021.25.3053

Cited by 4 publications

(10 citation statements)

References 10 publications

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“…6. To prove Theorem 1.1, we use considerations involving coamoebas, similar to those of [EL21]. In the case k = 1 and |A| = 3, we compute explicitly in Section 2.3 the image under µ br ϕ of a set of generators of π 1 (C \ B, t 0 ).…”

Section: Resultsmentioning

confidence: 99%

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Permuting the roots of univariate polynomials whose coefficients depend on parameters

Esterov¹,

Lang²

2022

Preprint

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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 ϕ) is generic with respect to its support set A ⊂ 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 ⊂ Z 2 and A 2 ⊂ {0} × 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 topics, the present results apply to the computation of the Galois group of some general enumerative problems, to Zariski's theorem on the fundamental group of the complement to projective hypersurfaces and to the determination of the kernel as well as the isomonodromy loci of the braid monodromy maps involved.

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Section: Resultsmentioning

confidence: 99%

“…Later on, we will argue that the image of π br χ in B * Q is the braid group B * A 2 of the universal univariate polynomial supported at A 2 , as studied in [EL21] (here, we see A 2 as the support of the univariate polynomial [q]). We denote P B * N ⊂ B * N the kernel of π br χ so that we have the short exact sequence…”

Section: Resultsmentioning

confidence: 99%

Permuting the roots of univariate polynomials whose coefficients depend on parameters

Esterov¹,

Lang²

2022

Preprint

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“…For a recent study cf. [85] P( 0 ( , L) set theoretically consisting of pairs {( , )| ∈ , ∈ P( 0 ( , L)), ∈ }. In [78] was considered the case = P 2 (resp.…”

Section: Arrangements Of Lines Hyperplanes and Plane Curvesmentioning

confidence: 99%

Complements to ample divisors and Singularities

Libgober¹

2021

Preprint

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“…Theorem 1.1. The Denef-Loeser monodromy conjecture .and moreover its polyhedral version from [13]/ holds true for all non-degenerate hypersurface singularities of four variables.…”

Section: Introductionmentioning

confidence: 99%

“…(2) If such a function b does not exist, then exp.2 i s 0 / is an eigenvalue of the monodromy of f at some point near the origin .and moreover a nearby tropical monodromy eigenvalue of the polyhedron C .f / in the sense of [13]/.…”

Section: Introductionmentioning

confidence: 99%

On the monodromy conjecture for non-degenerate hypersurfaces

2022

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The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations -the Denef-Loeser conjecture on topological zeta functions -is open for surface singularities.We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables.A crucial difference from the known case of three variables is the existence of degenerate singularities arbitrarily close to a non-degenerate one. Thus, even aiming at the study of non-degenerate singularities, we have to go beyond this setting.We develop new tools to deal with such multidimensional phenomena, and conjecture how the proof for non-degenerate singularities of arbitrarily many variables might look like.

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Braid monodromy of univariate fewnomials

Cited by 4 publications

References 10 publications

Permuting the roots of univariate polynomials whose coefficients depend on parameters

Permuting the roots of univariate polynomials whose coefficients depend on parameters

Complements to ample divisors and Singularities

On the monodromy conjecture for non-degenerate hypersurfaces

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