Deformations of weighted homogeneous polynomials with line singularities and equimultiplicity

Christophe Eyral, Maria Aparecida Soares Ruas

Published 2016-02-18Version 1

Consider a family $\{f_t\}$ of complex polynomial functions with line singularities, and assume that $f_0$ is weighted homogeneous. We show that if the $1$-st L\^e number and the $1$-st polar number of $f_t$ at $\mathbf{0}$ are independent of $t$, then the family $\{f_t\}$ is equimultiple. In particular, $\{f_t\}$ is equimultiple if the $1$-st polar number of $f_t$ at $\mathbf{0}$ and the local ambient topology of the hypersurface $f_t^{-1}(0)$ at $\mathbf{0}$ are independent of $t$. This partially answers the famous Zariski multiplicity conjecture for this special class of non-isolated singularities.