Construction d'un element remarquable de l'ideal de Bernstein-Sato associe a deux courbes planes analytiques

Let $f_1$ and $f_2$ be two semi-universal deformations of quasi homogeneous polynomials in two variables respectively for the weight vectors ${\rho }_1$ and ${\rho }_2$ such that they satisfy similar conditions to that of semi quasi homogeneous singularities for one weight. By methods inspired by H. Maynadier’s, we give an explicit formula for a Bernstein-Sato polynomial involving two affine forms ${\rho }_i(f_1)s_1+{\rho }_i(f_2)s_2+k$, $i=1,2$. In the particular case $(f_1,f_2)=(x_1^a+x_2^b,x_1^c+x_2^d)$, we calculate the space ${\mathcal{H}}_f$ recently studied by J. Brian\c{c}on, Ph. Maisonobe and M. Merle and we show that it is equal to the zero set of $s_1 s_2 (ab s_1+ ad s_2)(ad s_1+ cd s_2)$.