The sequence of mixed Łojasiewicz exponents associated to pairs of ideals

Abstract

We analyze the sequence ${\mathcal{L}}_J^{\ast }(I)$ of mixed Łojasiewicz exponents attached to any pair $I,J$ of monomial ideals of finite colength of the ring of analytic function germs $({\mathbb{C}}^n,0)\to \mathbb{C}$. In particular, we obtain a combinatorial expression for this sequence when $J$ is diagonal. We also show several relations of ${\mathcal{L}}_J^{\ast }(I)$ with other numerical invariants associated to $I$ and $J$.