Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety

Abstract

Systems of polynomial equations arise throughout mathematics, engineering, and the sciences. It is therefore a fundamental problem both in mathematics and in application areas to find the solution sets of polynomial systems. The focus of this paper is to compare two fundamentally different approaches to computing and representing the solutions of polynomial systems: numerical homotopy continuation and symbolic computation. Several illustrative examples are considered, using the software packages Bertini and Singular.

Probabilistic Saturations and Alt’s Problem

“…. , dim(Y ) − 1 as defined in (6). This question was primarily motivated by its application to providing a sharp upper bound to Alt's problem, namely g 0 (X, C 8 ) where X = V(f 1 , .…”

A hybrid symbolic-numerical approach to the center-focus problem

“…Following a numerical algebraic geometry approach, solution sets are represented by witness sets that we discuss below. A more detailed comparison of symbolic and numerical approaches is provided in [3].…”

“…, which we call the standard form of system (3). Note that the origin of (4) is a nonhyperbolic singularity at which the associated Jacobian has two purely imaginary eigenvalues λ 1,2 = ±i and λ 3 = −1.…”

What is numerical algebraic geometry?

“…Another approach to compute similar quantities is to use a symbolic approach, e.g., based on Gröbner basis computations over an algebraic number field or over a prime field of characteristic p > 0. There are advantages and disadvantages of each approach, e.g., see [5] for a comparison of approaches to compute irreducible decompositions. A near-term goal is to design hybrid symbolic-numeric methods that utilize advantages of both approaches.…”