Polynomial Equations: Theory and Practice

Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts. The theory is illustrated by many examples using different software packages.Comment: This article will appear as a chapter of a forthcoming book presenting research acitivies conducted in the European Network POEMA. It discusses polynomial equations, with optimization as point of entry. 24 pages, 7 figure

Solving Polynomial Systems via a Stabilized Representation of Quotient Algebras

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to compute the structure of the quotient ring R/I from the null space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous and multi-homogeneous cases are treated. In the presented framework, the concept of a border basis is generalized by relaxing the conditions on the set of basis elements. This allows for algorithms to adapt the choice of basis in order to enhance the numerical stability. We present such an algorithm and show numerical results

Likelihood Equations and Scattering Amplitudes

We relate scattering amplitudes in particle physics to maximum likelihood estimation for discrete models in algebraic statistics. The scattering potential plays the role of the log-likelihood function, and its critical points are solutions to rational function equations. We study the ML degree of low-rank tensor models in statistics, and we revisit physical theories proposed by Arkani-Hamed, Cachazo and their collaborators. Recent advances in numerical algebraic geometry are employed to compute and certify critical points. We also discuss positive models and how to compute their string amplitudes.Comment: 18 page

Toric Eigenvalue Methods for Solving Sparse Polynomial Systems

We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact toric variety $X$. Our starting point is a homogeneous ideal $I$ in the Cox ring of $X$, which gives a global description of this subscheme. It was recently shown that eigenvalue methods for solving this problem lead to robust numerical algorithms for solving (nearly) degenerate sparse polynomial systems. In this work, we give a first description of this strategy for non-reduced, zero-dimensional subschemes of $X$. That is, we allow isolated points with arbitrary multiplicities. Additionally, we investigate the regularity of $I$ to provide the first universal complexity bounds for the approach, as well as sharper bounds for weighted homogeneous, multihomogeneous and unmixed sparse systems, among others. We disprove a recent conjecture regarding the regularity and prove an alternative version. Our contributions are illustrated by several examples.Comment: 41 pages, 7 figure

Solving Equations Using Khovanskii Bases

We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian in its Pl\"ucker embedding. This generalizes established algorithms for toric varieties, and introduces the effective use of Khovanskii bases in computer algebra. We investigate regularity questions and discuss several applications.Comment: 25 pages, 1 figure, 2 table

Tropical Implicitization Revisited

Tropical implicitization means computing the tropicalization of a unirational variety from its parametrization. In the case of a hypersurface, this amounts to finding the Newton polytope of the implicit equation, without computing its coefficients. We present a new implementation of this procedure in Oscar.jl. It solves challenging instances, and can be used for classical implicitization as well. We also develop implicitization in higher codimension via Chow forms, and we pose several open questions.Comment: 18 pages, 3 figure

Gibbs Manifolds

Gibbs manifolds are images of affine spaces of symmetric matrices under the exponential map. They arise in applications such as optimization, statistics and quantum~physics, where they extend the ubiquitous role of toric geometry. The Gibbs variety is the zero locus of all polynomials that vanish on the Gibbs manifold. We compute these polynomials and show that the Gibbs variety is low-dimensional. Our theory is applied to a wide range of scenarios, including matrix pencils and quantum optimal transport.Comment: 22 page

Twisted cohomology and likelihood ideals

A likelihood function on a smooth very affine variety gives rise to a twisted de Rham complex. We show how its top cohomology vector space degenerates to the coordinate ring of the critical points defined by the likelihood equations. We obtain a basis for cohomology from a basis of this coordinate ring. We investigate the dual picture, where twisted cycles correspond to critical points. We show how to expand a twisted cocycle in terms of a basis, and apply our methods to Feynman integrals from physics.Comment: 28 pages, 2 figures, comments are welcom