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LSpice
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Since there is no answer yet, and partly in answer to a comment to the OP by Loren, let me add as answer my peeve with the usual formulations of Bernstein–Kushnirenko theorem. The point is the following: starting with fixed finite subsets A1,,An of Zn, the original formulation is a statement about the number of solutions of polynomials f1,,fn such that supp(fi)Ai, and it gives a precise characterization of when this number achieves the maximum. However, in almost every place you look, the statement presented is about polynomials fi such that supp(fi)=Ai.

Of course supp(fi)=Ai is the case people are concerned about in most cases. But the original formulation is clearly stronger (and as far as I remember, has practically the same proof as required for the weaker statement). In addition, the non-degeneracy condition in the stronger (and, let me stress again, original) formulation almost immediately gives a precise characterization for when MV(P1,,Pn)<MV(Q1,,Qn) for PiQi. The first observation of this seems to be due to Maurice Rojas (A convex geometric approach to counting the roots of a polynomial system) in 1994, but it was rediscovered very recently in 2017 (Bihan, Soprunov, Criteria for strict monotonicity of the mixed volume of convex polytopes). The fact that it needed to be "rediscovered" is I think at least partly due to the unawareness of the complete statement of Bernstein–Kushnirenko theorem among its users. In fact I only learned about the complete statement after I started writing the second draft of a book on Bernstein–Kushnirenko theorem and wanted to characterize the strict monotonicy of mixed volume as an application. (To be pedantic — Bernstein–Kushnirenko theorem directly gives the strict monotonicity only for rational polytopes. But the statement about all polytopes then immediately follows from the continuity of mixed volumes with respect to the Hausdorff Measure.)

Also, with respect to the original formulation of non-degeneracy conditions, the set of non-degenerate systems is Zariski open. In the usual formulation, if you are not very careful or unless you allow for some not-very-natural modifications, the set of non-degenerate systems turns out to contain a (nonempty) Zariski open subset, but might not be open itself.

Enough rant; here is the correct statement (see the book mentioned above for a proof): fix an algebraically closed field K.

Given finite subsets Ai of Zn and Laurent polynomials fi with supp(fi)Ai, i=1,,n, the number (counted with multiplicity) of isolated solutions of f1,,fn on (K{0})n is at most MV(conv(A1),,conv(An). This maximum is attained if and only if the following holds:

For every nonzero weighted degree ω on K[x1,,xn], there is no common zero on (K{0})n of the "Ai-supported leading forms" of fi.

The "A-supported leading form" LdA,ω(f) of a Laurent polynomial f supported in a finite (or compact) subset A of Rn is defined as follows: first write Ldω(A) for the "leading ω-face" of A, i.e. the set of all αA such that ω,α=max{ω,α:αA}. If f=αcαxα, then LdA,ω(f):=αLdω(A)cαxα

In particular, if supp(f)Ldω(A)= then LdA,ω(f)=0.

BKK example

Consider e.g. f1=1+x4+x2y4, f2=xy2+x3y3+x6, and A is the set of integral points in the triangle ABC in the picture above (reproduced from the book mentioned above). Even though the Newton polygons of both fi are proper subsets of A, it is immediate to check that the Bernstein–Kushnirenko non-degeneracy condition is satisfied with A1=A2=A, and therefore the number of solutions of f1=f2=0 in (K0)2 is MV(conv(A),conv(A))=2Area(conv(A))=24.

pinaki
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