On the degrees of relations on x1d1,…,xndn,(x1+…+xn)dn+1 in positive characteristic

We give a formula for the smallest degree of a non-Koszul relation on x1d1,…,xndn,(x1+…+xn)dn+1∈k[x1,…,xn] (under certain assumptions on d1,…,dn+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 d1,…,dn+1 (satisfying certain assumptions) such that the ring k[x1,…,xn+1]/(x1d1,…,xn+1dn+1) has the weak Lefschetz property.