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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| Published in: | Journal of algebra Vol. 646; pp. 156 - 182 |
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| Abstract | 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. |
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| AbstractList | 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. 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. |
| Author | Spaenlehauer, Pierre-Jean Bender, Matías |
| Author_xml | – sequence: 1 givenname: Matías surname: Bender fullname: Bender, Matías email: matias.bender@inria.fr organization: Inria and CMAP, CNRS, École Polytechnique, Institut Polytechnique de Paris, Palaiseau, France – sequence: 2 givenname: Pierre-Jean orcidid: 0000-0001-7906-0505 surname: Spaenlehauer fullname: Spaenlehauer, Pierre-Jean email: pierre-jean.spaenlehauer@inria.fr organization: Université de Lorraine, Inria, CNRS, France |
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| Snippet | 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... We study polynomial systems with prescribed monomial supports in the Cox rings of toric varieties built from complete polyhedral fans. We present combinatorial... |
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| SubjectTerms | Algebraic Geometry Computer Science Mathematics Sparse polynomial systems Symbolic Computation Toric varieties |
| Title | Dimension results for extremal-generic polynomial systems over complete toric varieties |
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