Combinatorial Reduction of Set Functions and Matroid Permutations through Minor Product Assignment Article Swipe
We introduce an algebraic model, based on the determinantal expansion of the product of two matrices, to test combinatorial reductions of set functions. Each term of the determinantal expansion is deformed through a monomial factor in d indeterminates, whose exponents define a $\mathbb{Z}^{d}$-valued set function. By combining the Grassmann-Plücker relations for the two matrices, we derive a family of sparse polynomials, whose factorisation properties in a Laurent polynomial ring are studied and related to information-theoretic notions. Under a given genericity condition, we prove the equivalence between combinatorial reductions and determinantal expansions with invertible minor products; specifically, a deformation returns a determinantal expansion if and only if it is induced by a diagonal matrix of units in $\mathbb{C}(\mathbf{t})$ acting as a kernel in the original determinant expression. This characterisation supports the definition of a new method for checking and recovering combinatorial reductions for matroid permutations.
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Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2107.01038
- https://arxiv.org/pdf/2107.01038
- OA Status
- green
- Cited By
- 1
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4292423915
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W4292423915Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.48550/arxiv.2107.01038Digital Object Identifier
- Title
-
Combinatorial Reduction of Set Functions and Matroid Permutations through Minor Product AssignmentWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2021Year of publication
- Publication date
-
2021-07-02Full publication date if available
- Authors
-
Mario AngelelliList of authors in order
- Landing page
-
https://arxiv.org/abs/2107.01038Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2107.01038Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/2107.01038Direct OA link when available
- Concepts
-
Mathematics, Matroid, Combinatorics, Monomial, Minor (academic), Invertible matrix, Product (mathematics), Discrete mathematics, Pure mathematics, Geometry, Political science, LawTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
1Total citation count in OpenAlex
- Citations by year (recent)
-
2023: 1Per-year citation counts (last 5 years)
- Related works (count)
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10Other works algorithmically related by OpenAlex
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