On real log canonical thresholds

Abstract

We introduce real log canonical threshold and real jumping numbers for real algebraic functions. A real jumping number is a root of the $b$-function up to a sign if its difference with the minimal one is less than 1. The real log canonical threshold, which is the minimal real jumping number, coincides up to a sign with the maximal pole of the distribution defined by the complex power of the absolute value of the function. However, this number may be greater than 1 if the codimension of the real zero locus of the function is greater than 1. So it does not necessarily coincide with the maximal root of the b-function up to a sign, nor with the log canonical threshold of the complexification. In fact, the real jumping numbers can be even disjoint from the non-integral jumping numbers of the complexification.