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Discrete Invariants of Generically Inconsistent Systems of Laurent Polynomials

C. R. Math. Rep. Acad. Sci. Canada Vol. 39 (3) 2017, pp. 90-102
Vol.39 (3) 2017
Leonid Monin Details
(Received: 2017-03-17 , Revised: 2017-04-17 )
(Received: 2017-03-17 , Revised: 2017-04-17 )

Leonid Monin,Department of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario, Canada M5S 2E4; e-mail:


Let \( \mathcal{A}_1, \ldots, \mathcal{A}_k \) be finite sets in \( \mathbb{Z}^n \) and let \( Y \subset (\mathbb{C}^*)^n \) be an algebraic variety defined by a system of equations \[f_1 = \ldots = f_k = 0,\] where \( f_1, \ldots, f_k \) are Laurent polynomials with supports in \(\mathcal{A}_1, \ldots, \mathcal{A}_k\). Assuming that \( f_1, \ldots, f_k \) are sufficiently generic, the Newton polyhedron theory computes discrete invariants of \( Y \) in terms of the Newton polyhedra of \( f_1, \ldots, f_k \). It may appear that the generic system with fixed supports \( \mathcal{A}_1, \ldots, \mathcal{A}_k \) is inconsistent. In this paper, we compute discrete invariants of algebraic varieties defined by systems of equations which are generic in the set of consistent system with support in \( \mathcal{A}_1, \ldots, \mathcal{A}_k\) by reducing the question to the Newton polyhedra theory. Unlike the classical situation, not only the Newton polyhedra of \(f_1,\dots,f_k\), but also the supports \(\mathcal{A}_1,\dots,\mathcal{A}_k\) themselves appear in the answers.

Soit \( \mathcal{A}_1, \ldots, \mathcal{A}_k \) un ensemble fini dans \( \mathbb{Z}^n \) et soit \( Y \subset (\mathbb{C}^*)^n \) une variété algébrique définie par un système d’équations \[f_1 = \ldots = f_k = 0,\]\( f_1, \ldots, f_k \) sont les polynômes de Laurent avec support dans \(\mathcal{A}_1, \ldots, \mathcal{A}_k\). Supposant que \( f_1, \ldots, f_k \) soient suffisamment génériques, la théorie du polyèdre de Newton calcule les invariants discrets de \( Y \) en fonction du polyèdre de Newton de \( f_1, \ldots, f_k \). Il peut sembler que le système avec support fixe \( \mathcal{A}_1, \ldots, \mathcal{A}_k \) est inconsistent. Dans ce papier, nous calculons les invariants discrets des variétés algébriques définies par des systèmes d’équations qui sont génériques dans l’ensemble des systèmes cohérents avec support dans \( \mathcal{A}_1, \ldots, \mathcal{A}_k\) en réduisant la question à la théorie du polyèdre de Newton. Contrairement à la situation classique, non seulement le polyèdre de Newton de \(f_1,\dots,f_k\), mais aussi les supports \(\mathcal{A}_1,\dots,\mathcal{A}_k\) eux-mêmes apparaissent dans la solution.

Keywords: Laurent polynomials, Newton polyhedra, generically inconsistent systems, resultants

AMS Subject Classification: Toric varieties; Newton polyhedra 14M25

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