Dimension results for extremal-generic polynomial systems over complete toric varieties - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Journal of Algebra Year : 2024

Dimension results for extremal-generic polynomial systems over complete toric varieties

Matías R Bender
  • Function : Author
  • PersonId : 19333
  • IdHAL : mbender

Abstract

We study polynomial systems with prescribed monomial supports in the Cox rings of toric varieties built from complete polyhedral fans. We present combinatorial formulas for the dimensions of their associated subvarieties under genericity assumptions on the coefficients of the polynomials. Using these formulas, we identify at which degrees generic systems in polytopal algebras form regular sequences. Our motivation comes from sparse elimination theory, where knowing the expected dimension of these subvarieties leads to specialized algorithms and to large speed-ups for solving sparse polynomial systems. As a special case, we classify the degrees at which regular sequences defined by weighted homogeneous polynomials can be found, answering an open question in the Gr\"obner bases literature. We also show that deciding whether a sparse system is generically a regular sequence in a polytopal algebra is hard from the point of view of theoretical computational complexity.

Dates and versions

hal-04102564 , version 1 (22-05-2023)

Licence

Identifiers

Cite

Matías R Bender, Pierre-Jean Spaenlehauer. Dimension results for extremal-generic polynomial systems over complete toric varieties. Journal of Algebra, 2024, 646, pp.156-182. ⟨10.1016/j.jalgebra.2024.01.029⟩. ⟨hal-04102564⟩
64 View
0 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More