Gröbner Bases Native to Term-ordered Commutative Algebras, with Application to the Hodge Algebra of Minors
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917007739322368 |
|---|---|
| author | Grochow, Joshua A. Natarajan, Abhiram |
| author_facet | Grochow, Joshua A. Natarajan, Abhiram |
| contents | Motivated by better understanding the bideterminant (=product of minors) basis on the polynomial ring in $n \times m$ variables, we develop theory \& algorithms for Gröbner bases in not only algebras with straightening law (ASLs or Hodge algebras), but in any commutative algebra over a field that comes equipped with a notion of "monomial" (generalizing the standard monomials of ASLs) and a suitable term order. Rather than treating such an algebra $A$ as a quotient of a polynomial ring and then "lifting" ideals from $A$ to ideals in the polynomial ring, the theory we develop is entirely "native" to $A$ and its given notion of monomial.
When applied to the case of bideterminants, this enables us to package several standard results on bideterminants in a clean way that enables new results. In particular, once the theory is set up, it lets us give an almost-trivial proof of a universal Gröbner basis (in our sense) for the ideal of $t$-minors for any $t$. We note that here it was crucial that theory be native to $A$ and its given monomial structure, as in the standard monomial structure given by bideterminants each $t$-minor is a single variable rather than a sum of $t!$ many terms (in the "ordinary monomial" structure). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11212 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gröbner Bases Native to Term-ordered Commutative Algebras, with Application to the Hodge Algebra of Minors Grochow, Joshua A. Natarajan, Abhiram Commutative Algebra Symbolic Computation Algebraic Geometry Rings and Algebras Motivated by better understanding the bideterminant (=product of minors) basis on the polynomial ring in $n \times m$ variables, we develop theory \& algorithms for Gröbner bases in not only algebras with straightening law (ASLs or Hodge algebras), but in any commutative algebra over a field that comes equipped with a notion of "monomial" (generalizing the standard monomials of ASLs) and a suitable term order. Rather than treating such an algebra $A$ as a quotient of a polynomial ring and then "lifting" ideals from $A$ to ideals in the polynomial ring, the theory we develop is entirely "native" to $A$ and its given notion of monomial. When applied to the case of bideterminants, this enables us to package several standard results on bideterminants in a clean way that enables new results. In particular, once the theory is set up, it lets us give an almost-trivial proof of a universal Gröbner basis (in our sense) for the ideal of $t$-minors for any $t$. We note that here it was crucial that theory be native to $A$ and its given monomial structure, as in the standard monomial structure given by bideterminants each $t$-minor is a single variable rather than a sum of $t!$ many terms (in the "ordinary monomial" structure). |
| title | Gröbner Bases Native to Term-ordered Commutative Algebras, with Application to the Hodge Algebra of Minors |
| topic | Commutative Algebra Symbolic Computation Algebraic Geometry Rings and Algebras |
| url | https://arxiv.org/abs/2510.11212 |