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

We study polynomial systems with prescribed monomial supports in the Cox ring of a toric variety built from a complete polyhedral fan. We present combinatorial formulas for the dimension of their associated subvarieties under genericity assumptions on the coefficients of the polynomials. Using these...

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Bibliographic Details
Published in:Journal of algebra Vol. 646; pp. 156 - 182
Main Authors: Bender, Matías, Spaenlehauer, Pierre-Jean
Format: Journal Article
Language:English
Published: Elsevier Inc 15.05.2024
Elsevier
Subjects:
Algebraic Geometry
Computer Science
Mathematics
Sparse polynomial systems
Symbolic Computation
Toric varieties
13P15
68W30
52B20
Toric varieties
Sparse polynomial systems
14M25
ISSN:0021-8693, 1090-266X
Online Access:Get full text
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Summary:We study polynomial systems with prescribed monomial supports in the Cox ring of a toric variety built from a complete polyhedral fan. We present combinatorial formulas for the dimension 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 Gröbner 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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2024.01.029