Bernstein-Kouchnirenko-Khovanskii with a symmetry

Geometry of sparse systems of polynomial equations (i.e. the ones with prescribed monomials and generic coefficients) is well studied in terms of their Newton polytopes. The results of this study are colloquially known as the Bernstein--Kouchnirenko--Khovanskii toolkit, and unfortunately are not applicable to many important systems, whose coefficients slightly fail to be generic. This for instance happens if some of the equations are obtained from another one by taking part...

We present efficient algorithms for detecting central and mirror symmetry for the case of algebraic curves defined by means of polynomial parametrizations. The algorithms are based on the existence of a linear relationship between two proper polynomial parametrizations of the curve, which leads to a triangular polynomial system (with complex unknowns) that can be solved in a very fast way; in particular, curves parametrized by polynomials of serious degrees can be analyzed in...

The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the Kushnirenko-Bernstein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic rational functions, with explicit genericity conditions. We also show that the var...

In this book we describe an approach through toric geometry to the following problem: "estimate the number (counted with appropriate multiplicity) of isolated solutions of n polynomial equations in n variables over an algebraically closed field k." The outcome of this approach is the number of solutions for "generic" systems in terms of their "Newton polytopes," and an explicit characterization of what makes a system "generic." The pioneering work in this field was done in th...

We show that the Newton polytope of a polynomial has a strong impact on the distribution of its mass and zeros. The basic theme is that Newton polytopes determine allowed and forbidden regions for these distributions. We equip the space of (holomorphic) polynomials of degree $\leq p$ in $m$ complex variables with its usual $SU(m + 1)$-invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polyt...

The first and second most symmetric nonsingular cubic surfaces are x^3+y^3+z^3+t^3=0 and x^2y+y^2z+z^2t+t^2x=0, respectively.

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. Computing monodromy permutations using numerical algebraic geometry gives information about the group, but can only determine it when it is the full symmetric group. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators while the other gives informati...

The Newton polytope $P_f$ of a polynomial $f$ is well known to have a strong impact on its zeros, as in the Kouchnirenko-Bernstein theorem on the number of simultaneous zeros of $m$ polynomials with given Newton polytopes. In this article, we show that $P_f$ also has a strong impact on the distribution of zeros of one or several polynomials. We equip the space of (holomorphic) polynomials of degree $\leq N$ in $m$ complex variables with its usual $SU(m+1)$-invariant Gaussian ...

The Bernshtein-Kushnirenko-Khovanskii theorem provides a generic root count for system of Laurent polynomials in terms of the mixed volume of their Newton polytopes (i.e., the BKK bound). A recent and far-reaching generalization of this theorem is the study of birationally invariant intersection index by Kaveh and Khovanskii. This short note establishes a simple geometric condition on the equality between the BKK bound and the intersection index for a system of vector spaces ...

We define the toric Newton spectrum of a polynomial and we give some applications in singularity theory, combinatorics and mirror symmetry.