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Milnor number of a hypersurface at the origin

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Part of the CMS/CAIMS Books in Mathematics book series (CMS/CAIMS BM,volume 2)

Abstract

The modern theory of applications of Newton polyhedra to affine Bézout problem started from A. Kushnirenko’s work aimed at answering V. I. Arnold’s question on Milnor numbers of generic singularities. In [Kou76] Kushnirenko gave a beautiful formula for a lower bound of the Milnor number at the origin in terms of volumes of the region bounded by the Newton diagram, and showed that the bound is attained in the case that the singularity is Newton non-degenerate. In this chapter, we show that the notion of non-degeneracy at the origin introduced in chapter IX can be used to derive (and generalize) Kushnirenko’s result on Milnor numbers.

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Notes

  1. 1.

    This terminology is taken from [BGM12].

  2. 2.

    The degree of a differentiable map \(\phi : M \rightarrow N\) between oriented differentiable manifolds of the same dimension, where M is compact and N is connected, is the sum of sign of \(df_x\) over all \(x \in \phi ^{-1}(y)\) for a generic \(y \in N\), where \(df_x\) is the derivative map from the tangent space of M at x to the tangent space of N at y, and the sign of \(df_x\) is either 1 or \(-1\) depending on whether \(df_x\) preserves or reverses orientation.

  3. 3.

    “Generic” refers to elements of a nonempty Zariski open (dense) subset of \(\mathcal {L}_0(\mathcal {A})\) in the Zariski topology mentioned in remark IX.7.

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Correspondence to Pinaki Mondal .

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© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

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Mondal, P. (2021). Milnor number of a hypersurface at the origin. In: How Many Zeroes?. CMS/CAIMS Books in Mathematics, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-030-75174-6_11

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