On the Hilbert depth of monomial ideals

Let $S=K[x_1,\dots ,x_n]$ be the ring of polynomials over a field $K$. Given two monomial ideals $0\subset I⊊J\subset S$, we present a new method to compute the Hilbert depth of $J\mathrm{/}I$. As an application, we show that if $u\in S$ is a monomial regular of $S\mathrm{/}I$, then $\mathrm{hdepth}(S\mathrm{/}I)\ge \mathrm{hdepth}(S\mathrm{/}(I,u))\ge \mathrm{hdepth}(S\mathrm{/}I)-1.$ Also, we reprove the formula of the Hilbert depth of a squarefree Veronese ideal.