A refinement of the Bernštein–Kušnirenko estimate

Abstract

A theorem of Kušnirenko and Bernštein shows that the number of isolated roots of a system of polynomials in a torus is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry.

A Poisson formula for the sparse resultant

“…, n, parametrizing the upper envelope of ι(∆ i ). This can be shown by induction on the number of variables n by using (4.15), plus the recursive formulae (4.18) and [PS08,(8.6)]. …”

The Newton Polygon of a Rational Plane Curve

“…The number of solutions of this system can be expressed in combinatorial terms thanks to a result of Philippon and the second author [12,13]. We introduce some combinatorial invariants in order to explain it better.…”

“…As the polynomials in (2.4) are primitive we can apply [13,Thm. 1.2], which shows that the number of solutions is bounded above by the quantity…”

“…In the present text we present an alternative proof of Theorem 1.1 based on the refinement of the Kušnirenko-Bernštein estimate due to Philippon and Sombra [12,13]. Instead of explicitly dealing with the Newton polygon we study its support function h(N(ρ * (T 1 )); ·) : R 2 → R. Set p := char(K), we show that for σ 1 , σ 2 ∈ Z\ pZ, the quantity…”

“…This principle is already illustrated in the example above and explained for the general case in Lemma 3.3. Such factorizations can be computed with gcd operations only, by applying for instance the algorithm in [13,Lem. 6.2].…”