Matthias Orth: Gröbner bases for the power of a general linear form in an Artinian monomial complete intersection

Abstract.

We study almost complete intersection ideals in a polynomial ring over a field of characteristic zero, generated by powers of all the variables together with a power of their sum. Exploiting the fact that the Hilbert series of the corresponding quotient rings are thin, we determine all reduced Gröbner bases for such ideals. Our approach is primarily combinatorial, focusing on the structure of the initial ideal. To each monomial in the vector space basis of an Artinian monomial complete intersection, we associate a lattice path and introduce a reflection operation on these paths, which enables a key counting argument. As a consequence, our method provides a new proof that Artinian monomial complete intersections have the strong Lefschetz property over fields of characteristic zero.

If time permits, I will talk about several applications of our results in special cases. If the monomial complete intersection is generated by the cubes of the variables, the Gröbner bases have degree structures governed by Motzkin numbers, Riordan numbers, and their convolutions. If the monomial complete intersection is equigenerated and the first power of the variable sum is considered, then we identify generalized Catalan numbers as a subsequence of the sequence of Gröbner basis element counts by degree. These subsequences also appear in quantum physics.

The talk is based on joint work with Filip Jonsson Kling and Samuel Lundqvist (both Stockholm University) and Fatemeh Mohammadi (KU Leuven).