Quasipolynomial behavior via constructibility in multigraded algebra
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911346430312448 |
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| author | Dao, Hailong Miller, Ezra Montaño, Jonathan O'Neill, Christopher Woods, Kevin |
| author_facet | Dao, Hailong Miller, Ezra Montaño, Jonathan O'Neill, Christopher Woods, Kevin |
| contents | Piecewise quasipolynomial growth of Presburger counting functions combines with tame persistent homology module theory to conclude piecewise quasipolynomial behavior of constructible families of finely graded modules over constructible commutative semigroup rings. Functorial preservation of constructibility for families under local cohomology, $\operatorname{Tor}$, and $\operatorname{Ext}$ yield piecewise quasipolynomial, quasilinear, or quasiconstant growth statements for length of local cohomology, $a$-invariants, regularity, depth; length of $\operatorname{Tor}$ and Betti numbers; length of $\operatorname{Ext}$ and Bass numbers; associated primes via $v$-invariants; and extended degrees, including the usual degree, Hilbert-Samuel multiplicity, arithmetic degree, and homological degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18536 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasipolynomial behavior via constructibility in multigraded algebra Dao, Hailong Miller, Ezra Montaño, Jonathan O'Neill, Christopher Woods, Kevin Commutative Algebra Combinatorics Logic Piecewise quasipolynomial growth of Presburger counting functions combines with tame persistent homology module theory to conclude piecewise quasipolynomial behavior of constructible families of finely graded modules over constructible commutative semigroup rings. Functorial preservation of constructibility for families under local cohomology, $\operatorname{Tor}$, and $\operatorname{Ext}$ yield piecewise quasipolynomial, quasilinear, or quasiconstant growth statements for length of local cohomology, $a$-invariants, regularity, depth; length of $\operatorname{Tor}$ and Betti numbers; length of $\operatorname{Ext}$ and Bass numbers; associated primes via $v$-invariants; and extended degrees, including the usual degree, Hilbert-Samuel multiplicity, arithmetic degree, and homological degree. |
| title | Quasipolynomial behavior via constructibility in multigraded algebra |
| topic | Commutative Algebra Combinatorics Logic |
| url | https://arxiv.org/abs/2512.18536 |