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PreprintarXiv licencearXiv

Geometry of Newton homotopies: bivariate case

  • Jennifer Buettner
  • Jonathan D. Hauenstein
  • Caroline Hills
  • Hoon Hong
  • Francisco Ponce Carrion
  • Emma L. Schmidt
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Abstract

A standard question in computational real algebraic geometry is to compute all real solutions to a system of polynomial equations with real coefficients. One classical and promising approach is to track along a connected component of a real curve defined by a Newton homotopy, which is dependent upon the selected start point. As the start point varies, different subsets of real solutions may be obtained. This yields a partition of the space of start points into cells, and it is important to understand the structure of this partition in order to develop efficient algorithms based on Newton homotopies. The structure of the boundary of such cells and the number of cells in the corresponding partition are investigated for bivariate systems. Several examples are included to demonstrate the results.

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Cite

BibTeXRIS
BibTeX
@misc{buettner2026geometry,
  title = {{Geometry of Newton homotopies: bivariate case}},
  author = {Jennifer Buettner and Jonathan D. Hauenstein and Caroline Hills and Hoon Hong and Francisco Ponce Carrion and Emma L. Schmidt},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30189},
  eprint = {2609.30189},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30189},
}
RIS
TY  - UNPB
TI  - Geometry of Newton homotopies: bivariate case
AU  - Jennifer Buettner
AU  - Jonathan D. Hauenstein
AU  - Caroline Hills
AU  - Hoon Hong
AU  - Francisco Ponce Carrion
AU  - Emma L. Schmidt
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30189
UR  - https://arxiv.org/abs/2609.30189
AB  - A standard question in computational real algebraic geometry is to compute all real solutions to a system of polynomial equations with real coefficients. One classical and promising approach is to track along a connected component of a real curve defined by a Newton homotopy, which is dependent upon the selected start point. As the start point varies, different subsets of real solutions may be obtained. This yields a partition of the space of start points into cells, and it is important to understand the structure of this partition in order to develop efficient algorithms based on Newton homotopies. The structure of the boundary of such cells and the number of cells in the corresponding partition are investigated for bivariate systems. Several examples are included to demonstrate the results.
ER  -