Braid monodromy of univariate fewnomials

Esterov, Alexander¹ ; Lang, Lionel²

¹ Faculty of Mathematics, HSE University, Moscow, Russia
² Department of Electrical Engineering, Mathematics and Science, University of Gävle, Gävle, Sweden

Geometry & topology, Tome 25 (2021) no. 6, pp. 3053-3077.

Résumé

Let ${\mathcal{𝒞}}_d\subset {ℂ}^{d+1}$ be the space of nonsingular, univariate polynomials of degree $d$. The Viète map $\mathcal{𝒱}:{\mathcal{𝒞}}_d\to {Sym}_d\left(ℂ\right)$ sends a polynomial to its unordered set of roots. It is a classical fact that the induced map ${\mathcal{𝒱}}_{\ast }$ at the level of fundamental groups realises an isomorphism between ${\pi }_1\left({\mathcal{𝒞}}_d\right)$ and the Artin braid group $B_d$. For fewnomials, or equivalently for the intersection $\mathcal{𝒞}$ of ${\mathcal{𝒞}}_d$ with a collection of coordinate hyperplanes in ${ℂ}^{d+1}$, the image of the map ${\mathcal{𝒱}}_{\ast }:{\pi }_1\left(\mathcal{𝒞}\right)\to B_d$ is not known in general.

We show that the map ${\mathcal{𝒱}}_{\ast }$ is surjective provided that the support of the corresponding polynomials spans $\mathbb{Z}$ as an affine lattice. If the support spans a strict sublattice of index $b$, we show that the image of ${\mathcal{𝒱}}_{\ast }$ is the expected wreath product of $\mathbb{Z}∕b\mathbb{Z}$ with $B_{d∕b}$. From these results, we derive an application to the computation of the braid monodromy for collections of univariate polynomials depending on a common set of parameters.

Détail

DOI : 10.2140/gt.2021.25.3053

Classification : 20F36, 55R80, 14T05
Keywords: braid group, monodromy, fewnomial, tropical geometry

Bibliographie

[1] V I Arnold, Certain topological invariants of algebraic functions, Trudy Moskov. Mat. Obšč. 21 (1970) 27

[2] E Artin, Theory of braids, Ann. of Math. 48 (1947) 101 | DOI

[3] D Bessis, Zariski theorems and diagrams for braid groups, Invent. Math. 145 (2001) 487 | DOI

[4] D Cheniot, Un théorème du type de Lefschetz, Ann. Inst. Fourier (Grenoble) 25 (1975) 195 | DOI

[5] S D Cohen, The Galois group of a polynomial with two indeterminate coefficients, Pacific J. Math. 90 (1980) 63 | DOI

[6] I Dolgachev, A Libgober, On the fundamental group of the complement to a discriminant variety, from: "Algebraic geometry" (editors A Libgober, P Wagreich), Lecture Notes in Math. 862, Springer (1981) 1 | DOI

[7] A Esterov, Galois theory for general systems of polynomial equations, Compos. Math. 155 (2019) 229 | DOI

[8] A Esterov, L Lang, Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products, preprint (2018)

[9] E Fadell, L Neuwirth, Configuration spaces, Math. Scand. 10 (1962) 111 | DOI

[10] B Farb, D Margalit, A primer on mapping class groups, 49, Princeton Univ. Press (2012)

[11] I M Gelfand, M M Kapranov, A V Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser (1994) | DOI

[12] R P Kent Iv, D Peifer, A geometric and algebraic description of annular braid groups, Int. J. Algebra Comput. 12 (2002) 85 | DOI

[13] L Lang, Monodromy of rational curves on toric surfaces, J. Topol. 13 (2020) 1658 | DOI

[14] A Libgober, On topological complexity of solving polynomial equations of special type, from: "Transactions of the Seventh Army Conference on Applied Mathematics and Computing", ARO Rep. 90, US Army Res. Office (1990) 475