On the degrees of relations on $x_1^{d_1},\dots ,x_n^{d_n},(x_1+\dots +x_n{)}^{d_{n+1}}$ in positive characteristic

Abstract

We give a formula for the smallest degree of a non-Koszul relation on $x_1^{d_1},\dots ,x_n^{d_n},(x_1+\dots +x_n{)}^{d_{n+1}}\in k[x_1,\dots ,x_n]$ (under certain assumptions on $d_1,\dots ,d_{n+1}$) where $k$ is a field of positive characteristic $p$. As an application of our result, we give a formula for the diagonal F-threshold of a diagonal hypersurface. Another application is a characterization, depending on the characteristic $p$ of $k$, of the values of $d_1,\dots ,d_{n+1}$ (satisfying certain assumptions) such that the ring $k[x_1,\dots ,x_{n+1}]/(x_1^{d_1},\dots ,x_{n+1}^{d_{n+1}})$ has the weak Lefschetz property.

References in corpus (2)

• The Lefschetz properties of monomial complete intersections in positive characteristic
• The Weak Lefschetz Property for monomial complete intersections