Cynthia Vinzant

Daniel A. Brake, Jonathan D. Hauenstein, Cynthia Vinzant

Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials with constant coefficients. These algorithms rely on homotopy continuation, monodromy loops, and Cauchy integrals. Several examples are presented which are computed using an implementation that builds on the numerical algebraic geometry software Bertini.

Grigoriy Blekherman, Mario Kummer, Cordian Riener et al.

A quadrature rule of a measure $μ$ on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against $μ$ for all polynomials up to some fixed degree. In this paper, we present a bivariate polynomial whose roots parametrize the nodes of minimal quadrature rules for measures on the real line. We give two symmetric determinantal formulas for this polynomial, which translate the problem of finding the nodes to solving a generalized eigenvalue problem.