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PreprintCC BY-SA 4.0arXiv

Resultant of an equivariant polynomial system with respect to a direct product of r symmetric groups

  • Sonagnon Julien Owolabi
  • Ollé Grégroire Kam
  • Ibrahim Nonkane
  • Joel Tossa
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Abstract

In this paper we study the resultant of systems of homogeneous multivariate polynomials which are equivariant under the action of a direct product of symmetric groups. We first treat, in detail, the case of a product of two symmetric groups, and establish a decomposition formula for the resultant of such systems. We then show that this decomposition, together with the underlying combinatorics, extends to an arbitrary (finite) direct product of r symmetric groups. Thanks to these decomposition formulas, we prove that the discriminant of a multivariate homogeneous polynomial invariant under a direct product of r symmetric groups splits into a product of resultants of smaller size that are easier to compute.

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Cite

BibTeXRIS
BibTeX
@misc{owolabi2026resultant,
  title = {{Resultant of an equivariant polynomial system with respect to a direct product of $r$ symmetric groups}},
  author = {Sonagnon Julien Owolabi and Ollé Grégroire Kam and Ibrahim Nonkane and Joel Tossa},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26127},
  eprint = {2609.26127},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26127},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Resultant of an equivariant polynomial system with respect to a direct product of r symmetric groups
AU  - Sonagnon Julien Owolabi
AU  - Ollé Grégroire Kam
AU  - Ibrahim Nonkane
AU  - Joel Tossa
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.26127
UR  - https://arxiv.org/abs/2609.26127
AB  - In this paper we study the resultant of systems of homogeneous multivariate polynomials which are equivariant under the action of a direct product of symmetric groups. We first treat, in detail, the case of a product of two symmetric groups, and establish a decomposition formula for the resultant of such systems. We then show that this decomposition, together with the underlying combinatorics, extends to an arbitrary (finite) direct product of r symmetric groups. Thanks to these decomposition formulas, we prove that the discriminant of a multivariate homogeneous polynomial invariant under a direct product of r symmetric groups splits into a product of resultants of smaller size that are easier to compute.
ER  -