Bernstein-Kouchnirenko-Khovanskii with a symmetry

Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we review several applications and discuss implementation and computational aspects.

In this survey I summarize the constructions of toric degenerations obtained from valuations and Gr\"obner theory and describe in which sense they are equivalent. I show how adapted bases can be used to generalize the classical Newton polytope to what is called a $\mathbb B$-Newton polytope. The $\mathbb B$-Newton polytope determines the Newton--Okounkov polytopes of all Khovanskii-finite valuations sharing the adapted standard monomial basis $\mathbb B$.

We introduce a general class of symmetric polynomials that have saturated Newton polytope and their Newton polytope has integer decomposition property. The class covers numerous previously studied symmetric polynomials.

A curve is called nondegenerate if it can be modeled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We show that up to genus 4, every curve is nondegenerate. We also prove that the locus of nondegenerate curves inside the moduli space of curves of fixed genus g > 1 is min(2g+1,3g-3)-dimensional, except in case g=7 where it is 16-dimensional.

The well-known Bernstein-Kushnirenko theorem from the theory of Newton polyhedra relates algebraic geometry and the theory of mixed volumes. Recently the authors have found a far-reaching generalization of this theorem to generic systems of algebraic equations on any quasi-projective variety. In the present note we review these results and their applications to algebraic geometry and convex geometry.

Let C be a smooth complex projective curve of genus g and let X be its second symmetric product. This paper concerns the study of some attempts at extending to X the notion of gonality. In particular, we prove that the degree of irrationality of X is at least g-1 when C is a generic curve, and that the minimum gonality of curves through the generic point of X equals the gonality of C. In order to produce the main results we deal with correspondences on the k-fold symmetric pr...

We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field k of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which completes works of Kushnirenko (Invent. Math., 1976) and Wall (J. Reine Angew. Math., 1999). Given a fixed collection of n convex integral polytopes in R^n, we also give an explicit characterization of systems of n polynomials supported at these p...

In this short note, we determine the Kodaira dimension and some of the plurigenera of (a desingularization of) a symmetric power of a smooth projective variety. We use it to obtain bounds on the genus of curve passing through a fixed number of general points.

A projective non-singular plane algebraic curve of degree d<=4 is called maximally symmetric if it attains the maximum order of the automorphism groups for complex non-singular plane algebraic curves of degree d. For d<=7, all such curves are known. Up to projectivities, they are the Fermat curve for d=5,7, see \cite{kmp1,kmp2}, the Klein quartic for d=4, see \cite{har}, and the Wiman sextic for d=6, see \cite{doi}. In this paper we work on projective plane curves defined ove...

We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in algebraic geometry or convex geometry. Highlights include the following: (1) A completely self-contained proof of an extension of Bernstein's Theorem. Our extension relates volumes of polytopes with the number of connected components of the comp...