Elimination theory and Newton polytopes

Elimination Theory and Newton Polytopes

Rational Parametrizations, Intersection Theory and Newton Polytopes

The study of the Newton polytope of a parametric hypersurface is currently receiving a lot of attention both because of its computational interest and its connections with Tropical Geometry, Singularity Theory, Intersection Theory and Combinatorics. We introduce the problem and survey the recent progress on it, with emphasis in the case of curves.

Newton Polytopes and Witness Sets

We present two algorithms that compute the Newton polytope of a polynomial f defining a hypersurface H in Cn using numerical computation. The first algorithm assumes that we may only compute values of f—this may occur if f is given as a straightline program, as a determinant, or as an oracle. The second algorithm assumes that H is represented numerically via a witness set. That is, it computes ...

Coercive Polynomials and Their Newton Polytopes

Many interesting properties of polynomials are closely related to the geometry of their Newton polytopes. In this article we analyze the coercivity on Rn of multivariate polynomials f ∈ R[x] in terms of their Newton polytopes. In fact, we introduce the broad class of so-called gem regular polynomials and characterize their coercivity via conditions imposed on the vertex set of their Newton poly...

Dequantization Transform and Generalized Newton Polytopes

For functions defined on Cn or Rn+ we construct a dequantization transform f 7→ f̂ ; this transform is closely related to the Maslov dequantization. If f is a polynomial, then the subdifferential ∂f̂ of f̂ coincides with the Newton polytope of f . For the semiring of polynomials with nonnegative coefficients, the dequantization transform is a homomorphism of this semiring to the idempotent semirin...