Sparse deformations of determinant expansions: hyperdeterminants and complexity of set functions Article Swipe
We introduce an algebraic structure to check the complexity of Z^d-valued set functions, which is interpreted as a combinatorial form of integrability. The algebraic structure is based on the expansion of the determinant of two matrices, one of which has non-vanishing maximal minors. Then, each term of the expansion is deformed through a monomial factor in d indeterminates with exponents defined by the set function. This results in sparse polynomials associated with determinantal constraints: it is proved that, in broad generality, this deformation returns a determinantal expansion if and only if it is integrable, namely, it is induced by a diagonal matrix of monomials acting as a kernel included in the initial determinant expansion. This framework allows extending previous results in the study of signed exponential sums and their applications in statistical physics, tropical geometry, and integrable systems. Furthermore, the hypotheses entailing the integrability of these deformations provide new connections between sparse polynomials, complexity reduction for permutations of subsets, and hyperdeterminants and their factorisation over the ring of Laurent polynomials.
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- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/pdf/2107.01038.pdf
- OA Status
- green
- References
- 13
- Related Works
- 20
- OpenAlex ID
- https://openalex.org/W3198830858
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- OpenAlex ID
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https://openalex.org/W3198830858Canonical identifier for this work in OpenAlex
- Title
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Sparse deformations of determinant expansions: hyperdeterminants and complexity of set functionsWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
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2021Year of publication
- Publication date
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2021-07-02Full publication date if available
- Authors
-
Mario AngelelliList of authors in order
- Landing page
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https://arxiv.org/pdf/2107.01038.pdfPublisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/2107.01038.pdfDirect OA link when available
- Concepts
-
Mathematics, Monomial, Factorization, Laurent polynomial, Diagonal, Laurent series, Integrable system, Algebraic number, Matrix (chemical analysis), Pure mathematics, Algebra over a field, Combinatorics, Mathematical analysis, Geometry, Algorithm, Composite material, Materials scienceTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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13Number of works referenced by this work
- Related works (count)
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20Other works algorithmically related by OpenAlex
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