Maciej Denkowski

Title: On the complex Lojasiewicz inequality with parameter

Abstract: 
A parameter-version of the Lojasiewicz regular separation was first obtained by Lojasiewicz 
and Wachta for compact subanalytic sets. The main feature of their result is that the regular 
separation condition is satisfied with a uniform exponent. From this one may infer that the
Lojasiewicz inequality is satisfied with a uniform exponent in a continuous subanalytic family 
of functions, too. The constant appearing in the inequality usually depends on the parameter. 
In the talk we shall concentrate on the complex counterpart of this result. In particular, we will 
show how a general continuity property (in the sense of currents) of the cycles of zeroes for 
just a continuous family of holomorphic (or c-holomorphic) functions yields a parameter-version 
of the Lojasiewicz inequality with an effective uniform exponent.