, Volume 26, Issue 2, pp 773-798
Date: 31 Jul 2012

Łojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

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Abstract

We give an expression for the Łojasiewicz exponent of a wide class of n-tuples of ideals (I 1,…,I n ) in \(\mathcal {O}_{n}\) using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of Łojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions f:(ℂ n ,0)→(ℂ,0) for which the Łojasiewicz of its gradient map ∇f attains the maximum possible value.

The first author was partially supported by DGICYT Grant MTM2009-08933. The second author was partially supported by DGICYT Grant MTM2009-07291 and CCG08-UAM/ESP-3928.