Numerical Algebraic Geometry for Macaulay2

Numerical algebraic geometry uses numerical data to describe algebraic varieties. It is based on numerical polynomial homotopy continuation, which is a technique 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. Numerical algebraic geometry [15, 16] 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 [9], a dominantly symbolic computer algebra system. Our open-source package dubbed NAG4M2 [11] and NumericalAlgebraicGeometry [9] 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, upon solution of which the majority of other prob-) find numerical approximations of all points of the underlying variety V (I) = {x | f (x) = 0}. This task is accomplished by applying the idea of homotopy continuation. To solve a target polynomial system f = (f 1 ,. .. , f n) = 0 construct a start polynomial system g = (g 1 ,. .. , g n) with a " similar structure " (the meaning of this will be explained later), but readily available solutions. Define a homotopy, (1) h = (1 − t)g + γtf ∈ C[x, t], γ ∈ C * , which specialized to the values of t in the real line interval [0, 1] provides a collection of continuation paths leading from the (known) solutions of the