On the Hilbert depth of monomial ideals

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