From Enumerative Geometry to Solving Systems of Polynomial Equations

An Excursion from Enumerative Geometry to Solving Systems of Polynomial Equations

An Excursion from Enumerative Geometry to Solving Systems of Polynomial Equations with Macaulay

Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a ...

Enumerative Real Algebraic Geometry

Two well-defined classes of structured polynomial systems have been studied from this point of view—sparse systems, where the structure is encoded by the monomials in the polynomials fi—and geometric systems, where the structure comes from geometry. This second class consists of polynomial formulations of enumerative geometric problems, and in this case Question 1.1 is the motivating question o...

ar X iv : a lg - g eo m / 9 70 60 04 v 1 1 0 Ju n 19 97 NUMERICAL SCHUBERT CALCULUS

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gröbner basis for the Plücker ideal of the G...

Numerical Evidence for a Conjecture in Real Algebraic Geometry

Homotopies for polynomial systems provide computational evidence for a challenging instance of a conjecture about whether all solutions are real. The implementation of SAGBI homotopies involves polyhedral continuation, at deformation and cheater's ho-motopy. The numerical diiculties are overcome if we work in the true synthetic spirit of the Schubert calculus by selecting the numerically most f...