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.
Keywords: Sparse polynomial system, hyperplane arrangement, master function, fewnomial, complete intersection
Mot clés : système polynomial creux, arrangement d’hyperplan, fonction master, oligonôme, intersection complète
@article{AIF_2008__58_3_877_0, author = {Bihan, Fr\'ed\'eric and Sottile, Frank}, title = {Gale duality for complete intersections}, journal = {Annales de l'Institut Fourier}, pages = {877--891}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {58}, number = {3}, year = {2008}, doi = {10.5802/aif.2372}, mrnumber = {2427513}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2372/} }
TY - JOUR AU - Bihan, Frédéric AU - Sottile, Frank TI - Gale duality for complete intersections JO - Annales de l'Institut Fourier PY - 2008 SP - 877 EP - 891 VL - 58 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2372/ DO - 10.5802/aif.2372 LA - en ID - AIF_2008__58_3_877_0 ER -
%0 Journal Article %A Bihan, Frédéric %A Sottile, Frank %T Gale duality for complete intersections %J Annales de l'Institut Fourier %D 2008 %P 877-891 %V 58 %N 3 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2372/ %R 10.5802/aif.2372 %G en %F AIF_2008__58_3_877_0
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. http://archive.numdam.org/articles/10.5802/aif.2372/
[1] Bounds on the number of real solutions to polynomial equations, Int. Math. Res. Not. IMRN (2007) no. 23, pp. 7, Art. ID rnm114 | MR | Zbl
[2] Khovanskii-Rolle continuation for real solutions, 2008 (in preparation)
[3] Newton Polytopes, Usp. Math. Nauk., Volume 1 (1976) no. 3, pp. 201-202 (in Russian)
[4] Polynomial systems with few real zeroes, Math. Z., Volume 253 (2006) no. 2, pp. 361-385 | DOI | MR | Zbl
[5] Polynomial systems supported on circuits and dessins d’enfants, Journal of the London Mathematical Society, Volume 75 (2007) no. 1, pp. 116-132 | DOI | Zbl
[6] Sharpness of fewnomial bounds and the number of components of a fewnomial hypersurface, Algorithms in Algebraic Geometry (IMA), Volume 146, Springer-Verlag (2007), pp. 15-20
[7] New fewnomial upper bounds from Gale dual polynomial systems, Moscow Mathematical Journal, Volume 7 (2007) no. 3, pp. 387-407 | MR
[8] New Betti number bounds for fewnomial hypersurfaces via stratified Morse Theory, 2008 (arXiv:0801.2554)
[9] Intersection theory on toric varieties, Topology, Volume 36 (1997), pp. 335-353 | DOI | MR | Zbl
[10] Newton polyhedra, and the genus of complete intersections, Funktsional. Anal. i Prilozhen., Volume 12 (1978) no. 1, pp. 51-61 | MR | Zbl
[11] First steps in tropical geometry, Idempotent mathematics and mathematical physics (Contemp. Math.), Volume 377, Amer. Math. Soc., Providence, RI, 2005, pp. 289-317 | MR | Zbl
[12] The numerical solution of systems of polynomials, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005 | MR | Zbl
Cited by Sources: