Branch points of homotopies: Distribution and probability of failure

Jonathan D. Hauenstein, Caroline Hills, Andrew J. Sommese and Charles W. Wampler

Applied Mathematics and Computation, 2025, vol. 493, issue C

Abstract: Homotopy continuation is a standard method used in numerical algebraic geometry for solving multivariate systems of polynomial equations. Techniques such as the so-called gamma trick yield trackable homotopies with probability one. However, since numerical algebraic geometry employs numerical path tracking methods, being close to a branch point may cause concern with finite precision computations. This paper provides a systematic study of branch points of homotopies to elucidate how branch points are distributed and use this information to study the probability of failure when using finite precision. Several examples, including a system arising in kinematics, with various start systems are included to demonstrate the theoretical analysis.

Keywords: Homotopy continuation; Branch points; Ramification points; Path tracking; Probability of failure; Numerical algebraic geometry (search for similar items in EconPapers)
Date: 2025
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0096300324007343
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:493:y:2025:i:c:s0096300324007343

DOI: 10.1016/j.amc.2024.129273

Access Statistics for this article

Applied Mathematics and Computation is currently edited by Theodore Simos

More articles in Applied Mathematics and Computation from Elsevier
Bibliographic data for series maintained by Catherine Liu ().