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PreprintCC BY 4.0arXiv

Combinatorics of degeneracy in vertical systems

  • Carles Checa
  • Nidhi Kaihnsa
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Abstract

We characterize the generic consistency of vertical systems in terms of the iterated circuits of the column matroid of the coefficient matrix. More precisely, we prove that such a system is generically consistent if and only if there exists an iterated circuit such that the dimension of the Minkowski sum of the associated Newton polytopes is equal to the rank of the coefficient matrix. We also give an equivalent characterization in terms of G-spanning flags of the dual matroid. With this, we derive a combinatorial formula for the codimension of the locus of parameter values for which the system has a solution, extending known results on dual defects of toric varieties. When this locus has codimension one, its defining equation yields a generalization of the sparse resultant to vertical systems, which applies also to 𝒜-discriminants. Finally, we characterize when the solution set is contained, for generic parameter values, in a finite union of hyperplanes parallel to a coordinate hyperplane.

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Cite

BibTeXRIS
BibTeX
@misc{checa2026combinatorics,
  title = {{Combinatorics of degeneracy in vertical systems}},
  author = {Carles Checa and Nidhi Kaihnsa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14803},
  eprint = {2610.14803},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14803},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Combinatorics of degeneracy in vertical systems
AU  - Carles Checa
AU  - Nidhi Kaihnsa
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14803
UR  - https://arxiv.org/abs/2610.14803
AB  - We characterize the generic consistency of vertical systems in terms of the iterated circuits of the column matroid of the coefficient matrix. More precisely, we prove that such a system is generically consistent if and only if there exists an iterated circuit such that the dimension of the Minkowski sum of the associated Newton polytopes is equal to the rank of the coefficient matrix. We also give an equivalent characterization in terms of G-spanning flags of the dual matroid. With this, we derive a combinatorial formula for the codimension of the locus of parameter values for which the system has a solution, extending known results on dual defects of toric varieties. When this locus has codimension one, its defining equation yields a generalization of the sparse resultant to vertical systems, which applies also to 𝒜-discriminants. Finally, we characterize when the solution set is contained, for generic parameter values, in a finite union of hyperplanes parallel to a coordinate hyperplane.
ER  -