The Weak Lefschetz Property for monomial complete intersections
Let $A=\pmb k[x_1,...,x_n]/{(x_1^d,...,x_n^d)}$, where $\pmb k$ is an infinite field. If $\pmb k$ has characteristic zero, then Stanley proved that $A$ has the Weak Lefschetz Property (WLP). Henceforth, $\pmb k$ 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 $4\le n$. If $4\le n$ 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 $1\le k\le \frac{p-1}2$, $r\in{\frac{q-1}2,\frac{q+1}2}$, and $q=p^e$ for some non-negative integer $e$. If $5\le n$, then we prove that $A$ has the WLP if and only if $\lfloor\frac{n(d-1)+3}2\rfloor\le p$. We first interpret the WLP for the ring ${{\pmb k}[x_1, ..., x_{n}]}/{(x_1^d, ..., x_{n}^d)}$ in terms of the degrees of the non-Koszul relations on the elements $x_1^d, ..., x_{n-1}^d, (x_1+ ... +x_{n-1})^d$ in the polynomial ring $\pmb k[x_1, ..., x_{n-1}]$. We then exhibit a sufficient condition for ${{\pmb k}[x_1, ..., x_{n}]}/{(x_1^d, ..., x_{n}^d)}$ to have the WLP. This condition is expressed in terms of the non-vanishing in $\pmb k$ of determinants of various Toeplitz matrices of binomial coefficients. Frobenius techniques are used to produce relations of low degree on $x_1^d$, ..., $x_{n-1}^d$, ${(x_1+ ... +x_{n-1})^d}$. 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 $\pmb k$.