Numerical algebraic geometry

Abstract

Numerical algebraic geometry uses numerical data to describe algebraic varieties. It is based on numerical polynomial homotopy continuation, which is an alternative to the classical symbolic approaches of computational algebraic geometry. We present a package, whose primary purpose is to interlink the existing symbolic methods of Macaulay2 and the powerful engine of numerical approximate computations. The core procedures of the package exhibit performance competitive with the other homotopy continuation software. INTRODUCTION. Numerical algebraic geometry [SVW, SW] is a relatively young subarea of computational algebraic geometry that originated as a blend of the well-understood apparatus of classical algebraic geometry over the field of complex numbers and numerical polynomial homotopy continuation methods. Recently steps have been made to extend the reach of numerical algorithms making it possible not only for complex algebraic varieties, but also for schemes, to be represented numerically. What we present here is a description of “stage one” of a comprehensive project that will make the machinery of numerical algebraic geometry available to the wide community of users of Macaulay2 [M2], a dominantly symbolic computer algebra system. Our open-source package [L1, M2] was first released in Macaulay2 distribution version 1.3.1. Stage one has been limited to implementation of algorithms that solve the most basic problem: Given polynomials f1, . . . , fn ∈ C[x1, . . . ,xn] that generate a 0-dimensional ideal I = ( f1, . . . , fn), find numerical approximations of all points of the underlying variety V (I) = {x ∈ Cn | f(x) = 0}. This task is accomplished by using homotopy continuation. To solve a target polynomial system f =( f1, . . . , fn)= 0, one constructs a start polynomial system g=(g1, . . . ,gn) with a similar structure, but readily available solutions. One option for a start system is g = (x f1) 1 − 1, . . . ,x deg( fn) n − 1). Given such an auxiliary system, we consider the homotopy (♠) h= (1− t)g+ γ tf ∈ C[x, t], γ ∈ C∗ . By specializing the values of t to the real interval [0,1], we obtain a collection of paths leading from the known solutions of the start system g = h|t=0 to the unknown solutions of the target system f = h|t=1. The following statement provides the key to making effective numerical computations. Theorem. For all but finitely many values of γ in the homotopy (♠), the homotopy continuation paths have no singularities with a possible exception of the endpoints corresponding to t = 1. Moreover, every solution of the target system, provided there are finitely many, is an endpoint (at t = 1) of some continuation path. 2010 Mathematics Subject Classification. 14Q99; 68N01. NumericalAlgebraicGeometry version 1.3.1. 5 Leykin :::: NumericalAlgebraicGeometry 6