The signature of a monomial ideal
The irreducible decomposition of a monomial ideal has played an important role in combinatorial commutative algebra, with applications beyond pure mathematics, such as biology. Given a monomial ideal $ I $ of a polynomial ring $ S = k[{\bf x}] $ over a field $ k $ and variables $ {\bf x} = \{x_1, \l...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-09-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241357?viewType=HTML |
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| Summary: | The irreducible decomposition of a monomial ideal has played an important role in combinatorial commutative algebra, with applications beyond pure mathematics, such as biology. Given a monomial ideal $ I $ of a polynomial ring $ S = k[{\bf x}] $ over a field $ k $ and variables $ {\bf x} = \{x_1, \ldots, x_n\} $, its incidence matrix, is the matrix whose rows are indexed by the variables $ {\bf x} $ and whose columns are indexed by its minimal generators. The main contribution of this paper is the introduction of a novel invariant of a monomial ideal $ I $, termed its signature, which could be thought of as a type of canonical form of its incidence matrix, and the proof that two monomial ideals with the same signature have essentially the same irreducible decomposition. |
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| ISSN: | 2473-6988 |