Une nouvelle majoration pour le nombre de solutions d'un système d'équations polynomiales
The main object of the paper under review is a system of equations \[ f_0(\xi )=\dots =f_n(\xi)=0 \] where \(f_i\) are Laurent polynomials in \(n\) variables \(t_1,\dots ,t_n\) with coefficients in \(K[s]\), \(K\) being an algebraically closed field. The authors are interested in the number of isolated solutions \(\xi\in K\times (K^*)^n\) of this system. A classical theorem of \textit{D.~N.~Bernštein} [Funk. Anal. Appl. 9, 1--4 (1975; Zbl 0328.32001)] and \textit{A.~G.~Kušnirenko} [Invent. Math. 32, 1--31 (1976; Zbl 0328.32007)] provides an upper estimate for the number of those isolated points (counted with multiplicities) lying in the torus \((K^*)^{n+1}\) in terms of the mixed volume of \((n+1)\) Newton polytopes corresponding to the \(f_i\)'s regarded as Laurent polynomials in \((n+1)\) variables \(s, t_1, \dots , t_n\). The authors' goal is to improve on this estimate. To this end, they introduce refined combinatorial invariants associated to the system under consideration. For each Laurent polynomial \(f\) as above and each \(v\in {\mathbb P}^1\) they define the \(v\)-adic Newton polytope of \(f\) (lying in \({\mathbb R}^{n+1}\)) and the roof function of this polytope above the usual Newton polytope (lying in \({\mathbb R}^n\)); such a roof function also appears in tropical geometry. Furthermore, for any family of \((n+1)\) concave real functions defined on convex sets in \({\mathbb R}^n\) they define the mixed integral using the operation of sup-convolution (this is a generalization of the mixed volume of convex bodies, more details can be found in their earlier paper [\url{arXiv:math.NT/0406476}]). For a system given by primitive (i.e. not having constant factors in \(K[s]\)) \(f_i\)'s, the main result of the note (Theorem 1.1) states that the number of isolated solutions (counted with multiplicities) does not exceed the sum over all \(v\in {\mathbb P}^1\) of the mixed integrals of the roof functions of the \(v\)-adic Newton polytopes corresponding to the \(f_i\)'s. Moreover, the estimate becomes the equality for generic Laurent polynomials. This result can be extended to nonprimitive Laurent polynomials as well. Moreover, a geometric look at Theorem 1.1 as at a statement on a toric scheme over the projective line allows the authors to generalize it to toric schemes over an arbitrary smooth projective curve (Theorem 3.1) which can be applied to systems of equations over semiabelian varieties. The proofs are contained in a forthcoming paper of the authors, available at [\url{arXiv:0709.3306}].
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