Let \(f_0\colon(\mathbb{C}^n,0)\rightarrow \left( \mathbb
{C},0\right)\) be an isolated singularity. We define the number
\begin{equation*}
\mathcal{L}_0(f):=\inf\{\alpha\in\mathbb{R}_+:
\exists _{C > 0}\exists _{r>0}\forall _{\|z\|<r}\|\nabla
f_0(z)\|\geq C\|z\|^{\alpha}
\}
\end{equation*}
and call it the Lojasiewicz exponent of \(f_0.\) In [1],
B. Teissier calculated \(\mathcal{L}_0(f_0)\) in terms of polar
invariants of the singularity \(f_0\) and proved that
\(\mathcal{L}_0(f_0)\) is lower semicontinuous in any
\(\mu\)-constant deformation of the singularity \(f_0.\) A. P\l oski
generalized his result and proved that the Lojasiewicz exponent is
lower semicontinuous in any multiplicity-constant deformation of a
finite holomorphic map germ (see [2]). B. Teissier also
showed that if we do not assume \(\mu\)-constancy, then
\(\mathcal{L}_0(f_0)\) is neither upper or lower semicontinuous (see
[3]). The ``jump phenomena'' of the Lojasiewicz exponent
were rediscovered by some authors (see [4]). The aim of
this talk is to give formulas for jump upwards and downwards of
\(\mathcal{L}_0(f_0)\) in nondegenerate class of curves
singularities in terms of the Newton diagram of \(f_0\). By the jump
downards of \(\mathcal{L}_0(f_0)\) we mean the minimum non-zero
positive difference between the Łojasiewicz exponent of \(f_0\)
and one of its deformations \((f_s).\) We define in analogous way
the jump upwards. We also indicate the deformations, in which the
jumps are attained.
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References- B. Teissier, Variétés polaires I -
Invariant polaires de singularités d'hypersurfaces ,
Invent. Math. 40, 1977, 267-292.
- A. Płoski, Semicontinuity of the Lojasiewicz
exponent , Univ. Iagel. Acta Math. 48, 2011, 103-110.
- B. Teissier, Jacobian Newton polyhedra and
equisingularity , Preceedings R.I.M.S. Conference on
singularities, Kyoto, April 1978, (Publ. R.I.M.S. 1978).
- J. Mc Neal, A. Némethi, The order of
contact of a holomorphic ideal in \(\mathbb{C ^2}\), Math.
Z. 250, 2005, 873-883.
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