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

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