A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems

Abstract

Exceptional sets are the sets where the dimension of the fiber of a map is larger than the generic fiber dimension, which we assume is zero. Such situations naturally arise in kinematics, for example, when designing a mechanism that moves when the generic case is rigid. In 2008, Sommese and Wampler showed that one can use fiber products to promote such sets to become irreducible components. We propose an alternative approach using rank constraints on Macaulay matrices. Symbolic computations are used to construct the proper Macaulay matrices, while numerical computations are used to solve the rank-constraint problem. Various exceptional sets are computed, including exceptional RR dyads, lines on surfaces in C 3 , and exceptional planar pentads.

Nonlinear Algebra and Applications

“…Computations are an important cornerstone of nonlinear algebra. Researchers in this field use a collection of different methods ranging from numerical algorithms such as polynomial homotopy continuation [206] to symbolic methods based on Gröbner bases [213] or combine symbolic and numerical algorithms [108]. For carrying symbolic computations out, various computer algebra systems are available, such as Macaulay2 [99], SageMath [224], or Singular [62], to mention a small selection.…”

What is numerical algebraic geometry?

“…Numerical algebraic geometry is concerned with "numerical computations" of objects connected with algebraic sets defined over subfields of the complex numbers. Some examples of such objects include solution sets and irreducible decompositions [80], monodromy groups [56], and exceptional sets of an algebraic map [35,89]. The term "numerical" refers to computations which are potentially inexact, e.g., floating-point arithmetic.…”

Nonlinear algebra and applications

“…Computations are an important cornerstone of nonlinear algebra. Researchers in this field use a collection of different methods ranging from numerical algorithms such as polynomial homotopy continuation [217] to symbolic methods based on Gröbner bases [224] or combine symbolic and numerical algorithms [113]. For carrying out symbolic computations, various computer algebra systems are available, such as Macaulay2 [104], SageMath [235], or Singular [64], to mention a small selection.…”