Gale duality for complete intersections
Annales de l'Institut Fourier, Volume 58 (2008) no. 3, pp. 877-891.

We show that every complete intersection defined by Laurent polynomials in an algebraic torus is isomorphic to a complete intersection defined by master functions in the complement of a hyperplane arrangement, and vice versa. We call systems defining such isomorphic schemes Gale dual systems because the exponents of the monomials in the polynomials annihilate the weights of the master functions. We use Gale duality to give a Kouchnirenko theorem for the number of solutions to a system of master functions and to compute some topological invariants of master function complete intersections.

Nous montrons que toute intersection complète définie par des polynômes de Laurent dans un tore algébrique est isomorphe à une intersection complète définie par des « fonctions master » dans le complémentaire d’un arrangement d’hyperplans, et vice versa. On appelle les systèmes définissant de tels schémas isomorphes des systèmes « Gale duaux » car les exposants des monômes apparaissant dans les polynômes annulent les poids des fonctions master. On utilise la dualité de Gale pour donner un théorème de Kouchnirenko sur le nombre de solutions d’un système de fonctions master et pour calculer certains invariants topologiques d’intersections complètes définies par des fonctions master.

DOI: 10.5802/aif.2372
Classification: 14M25, 14P25, 52C35
Keywords: Sparse polynomial system, hyperplane arrangement, master function, fewnomial, complete intersection
Mot clés : système polynomial creux, arrangement d’hyperplan, fonction master, oligonôme, intersection complète
Bihan, Frédéric 1; Sottile, Frank 2

1 Université de Savoie Laboratoire de Mathématiques 73376 Le Bourget-du-Lac Cedex (France)
2 Department of Mathematics Texas A&M University College Station Texas 77843 (USA)
     author = {Bihan, Fr\'ed\'eric and Sottile, Frank},
     title = {Gale duality for complete intersections},
     journal = {Annales de l'Institut Fourier},
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Bihan, Frédéric; Sottile, Frank. Gale duality for complete intersections. Annales de l'Institut Fourier, Volume 58 (2008) no. 3, pp. 877-891. doi : 10.5802/aif.2372.

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