Newton polytopes and irreducible components of complete intersections (Q2810942)

This paper interacts with the theory of Newton polytopes and the geometry of toric varieties. The main result is a generalization of Bernstein and Kushnirenko's theorem. A celebrated theorem of \textit{A.~G.~Kushnirenko} [Invent. Math. 32, 1--31 (1976; Zbl 0328.32007)] and \textit{D.~N.~Bernstein} [Funct. Anal. Appl. 9, 183--185 (1976; Zbl 0328.32001)] states that if \(P_1, \ldots, P_n\) are general functions on the complex torus \((\mathbb{C} ^{*})^n\) with given Newton polytopes \(\Delta_1, \ldots, \Delta_n \subset \mathbb{R}^n\) (respectively), then the number of solutions of the simultaneous equations \(P_1= \cdots = P_n=0\) is equal to \(n!V(\Delta_1, \ldots, \Delta_n)\), where \(V\) is the mixed volume of \(n\) convex bodies \(\Delta_1, \ldots, \Delta_n\).NEWLINENEWLINELet a variety be given in \(({\mathbb{C}^*})^n\) by a generic system of \(k\) equations \(P_1= \cdots = P_k=0\), where \(k \leq n\), with given Newton polytopes \(\Delta_1, \ldots, \Delta_k \subset \mathbb{R}^n\). In the paper under review, the author calculates the number of irreducible components of such a variety. Each component can, in turn, be given by a generic system of equations whose Newton polytopes are determined explicitly. For \(k=n\) the result coincides with Bernstein and Kushnirenko's theorem. For \(k<n\) the proof given in the paper under review is a generalization of the author's earlier proof [Funct. Anal. Appl. 12, 38--46 (1978; Zbl 0406.14035)]. The proof relies on Minkowski's theorem on necessary and sufficient conditions for the mixed volume of \(n\) convex polytopes to be non-zero.NEWLINENEWLINEIt is known that many discrete invariants of a given variety can be determined from the Newton polytopes. The results of this paper enable the author to calculate some discrete invariants for each irreducible component of the variety.