The Weak Lefschetz Property for monomial complete intersections

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Let A= k[x₁,...,xₙ]/(x₁ᵈ,...,xₙᵈ), where k is an infinite field. If k has characteristic zero, then Stanley proved that A has the Weak Lefschetz Property (WLP). Henceforth, k has positive characteristic p. If n=3, then Brenner and Kaid have identified all d, as a function of p, for which A has the WLP. In the present paper, the analogous project is carried out for 4≤ n. If 4≤ n and p=2, then A has the WLP if and only if d=1. If n=4 and p is odd, then we prove that A has the WLP if and only if d=kq+r for integers k,q,d with 1≤ k≤ p-12, r∈q-12,q+12, and q=pᵉ for some non-negative integer e. If 5≤ n, then we prove that A has the WLP if and only if n(d-1)+32≤ p. We first interpret the WLP for the ring k[x₁,..., xₙ]/(x₁ᵈ,..., xₙᵈ) in terms of the degrees of the non-Koszul relations on the elements x₁ᵈ,..., xₙ₋₁ᵈ, (x₁+... +xₙ₋₁)ᵈ in the polynomial ring k[x₁,..., xₙ₋₁]. We then exhibit a sufficient condition for k[x₁,..., xₙ]/(x₁ᵈ,..., xₙᵈ) to have the WLP. This condition is expressed in terms of the non-vanishing in k of determinants of various Toeplitz matrices of binomial coefficients. Frobenius techniques are used to produce relations of low degree on x₁ᵈ,..., xₙ₋₁ᵈ, (x₁+... +xₙ₋₁)ᵈ. From this we obtain a necessary condition for A to have the WLP. We prove that the necessary condition is sufficient by showing that the relevant determinants are non-zero in k.