Ramification points of homotopies: Enumeration and general theory
-
Andrew J. Sommese
,
Caroline Hills
Abstract
Ramification points arise from singularities along solution paths of a homotopy. This paper considers ramification points of homotopies, elucidating the total number of ramification points and providing general theory regarding the properties of the set of ramification points over the same branch point. The general approach utilized in this paper is to view homotopies as lines in the parameter spaces of families of polynomial systems on a projective manifold. With this approach, the number of singularities of systems parameterized by pencils is computed under broad conditions. General conditions are given for when the singularities of the systems parameterized by a line in a space of polynomial systems have multiplicity two. General conditions are also given for there to be at most one singularity in the solution set of any system parameterized by such a line. Several examples are included to demonstrate the theoretical results.
Funding statement: The second author was in part supported by NSF CCF-2331440, Simons Foundation SFM-00005696, and the Robert and Sara Lumpkins Collegiate Professorship. The third author was in part supported by NSF CCF-2331440 and the Robert and Sara Lumpkins Collegiate Professorship. The last author was in part supported in part by the Huisking Foundation, Inc. Collegiate Research Professorship.
Communicated by: M. Joswig
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© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
Artikel in diesem Heft
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
