Geodesic
DEFORMATIONS WITH CONSTANT MILNOR NUMBER AND MULTIPLICITY OF COMPLEX HYPERSURFACES
Glasgow mathematical journal, Tome 46 (2004) no. 1, pp. 121-130

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DOI

We investigate the constancy of the Milnor number of one parameter deformations of holomorphic germs of functions $f:(\C^n,0) \to (\C,0)$ with isolated singularity, in terms of some Newton polyhedra associated to such germs.When the Jacobian ideals $J(\hspace*{1.8pt}f_t) = \left\langle {\partial f_t}/{\partial x_{1}} \ldots ,{\partial f_t}/{\partial x_{n}}\right\rangle $ of a deformation $f_t(x) = f(x)+ \sum_{s=1}^{\ell}\delta_s(t)g_s(x)$ are non-degenerate on some fixed Newton polyhedron $\Gamma_+$, we show that this family has constant Milnor number for small values of $t$, if and only if all germs $g_s$ have non-decreasing $\Gamma$-order with respect to $f$. As a consequence of these results we give a positive answer to Zariski's question for Milnor constant families satisfying a non-degeneracy condition on the Jacobian ideals.
DOI : 10.1017/S0017089503001599
Mots-clés : Primary 32S30, Secondary 32S10 
SAIA, MARCELO JOSÉ; TOMAZELLA, JOÃO NIVALDO. DEFORMATIONS WITH CONSTANT MILNOR NUMBER AND MULTIPLICITY OF COMPLEX HYPERSURFACES. Glasgow mathematical journal, Tome 46 (2004) no. 1, pp. 121-130. doi: 10.1017/S0017089503001599
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     title = {DEFORMATIONS {WITH} {CONSTANT} {MILNOR} {NUMBER} {AND} {MULTIPLICITY} {OF} {COMPLEX} {HYPERSURFACES}},
     journal = {Glasgow mathematical journal},
     pages = {121--130},
     year = {2004},
     volume = {46},
     number = {1},
     doi = {10.1017/S0017089503001599},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089503001599/}
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