The Gröbner basis for powers of a general linear form in a monomial complete intersection
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918076413378560 |
|---|---|
| author | Kling, Filip Jonsson Lundqvist, Samuel Mohammadi, Fatemeh Orth, Matthias |
| author_facet | Kling, Filip Jonsson Lundqvist, Samuel Mohammadi, Fatemeh Orth, Matthias |
| contents | We study almost complete intersection ideals in a polynomial ring, generated by powers of all the variables together with a power of their sum. Our main result is an explicit description of the reduced Gröbner bases for these ideals under any term order. Our approach is primarily combinatorial, focusing on the structure of the initial ideal. We associate a lattice path to each monomial in the vector space basis of an Artinian monomial complete intersection and introduce a reflection operation on these paths, which enables a key counting argument. As a consequence, we provide a new proof that Artinian monomial complete intersections possess the strong Lefschetz property over fields of characteristic zero. Our results also offer new insights into the longstanding problem of classifying the weak Lefschetz property for such intersections in characteristic $p$. Furthermore, we show that the number of Gröbner basis elements in each degree is connected to several well-known sequences, including the (generalized) Catalan, Motzkin, and Riordan numbers, and connect these numbers to the study of entanglement detection in spin systems within quantum physics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_24028 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Gröbner basis for powers of a general linear form in a monomial complete intersection Kling, Filip Jonsson Lundqvist, Samuel Mohammadi, Fatemeh Orth, Matthias Commutative Algebra Combinatorics Rings and Algebras 13P10, 13E10, 05E40 We study almost complete intersection ideals in a polynomial ring, generated by powers of all the variables together with a power of their sum. Our main result is an explicit description of the reduced Gröbner bases for these ideals under any term order. Our approach is primarily combinatorial, focusing on the structure of the initial ideal. We associate a lattice path to each monomial in the vector space basis of an Artinian monomial complete intersection and introduce a reflection operation on these paths, which enables a key counting argument. As a consequence, we provide a new proof that Artinian monomial complete intersections possess the strong Lefschetz property over fields of characteristic zero. Our results also offer new insights into the longstanding problem of classifying the weak Lefschetz property for such intersections in characteristic $p$. Furthermore, we show that the number of Gröbner basis elements in each degree is connected to several well-known sequences, including the (generalized) Catalan, Motzkin, and Riordan numbers, and connect these numbers to the study of entanglement detection in spin systems within quantum physics. |
| title | The Gröbner basis for powers of a general linear form in a monomial complete intersection |
| topic | Commutative Algebra Combinatorics Rings and Algebras 13P10, 13E10, 05E40 |
| url | https://arxiv.org/abs/2506.24028 |