Abstract: In this chapter, we consider the “local” version of the affine Bézout problem, i.e. the problem of estimating the intersection multiplicity of generic hypersurfaces at the origin. This computation is a crucial ingredient of the extension in chapter X of Bernstein’s theorem to the affine space. Recall that the support of a power series \(f = \sum _\alpha c_\alpha x^\alpha \in \mathbbm {k}[[x_1, \ldots , x_n]]\) is \({{\,\mathrm{Supp}\,}}(f):= \{\alpha : c_\alpha \ne 0\}\) and we say that f is supported at \(\mathcal {A}\subset \mathbb {Z}^n\) if \({{\,\mathrm{Supp}\,}}(f) \subset \mathcal {A}\). Now let \(\mathcal {A}_1, \ldots , \mathcal {A}_n\) be (possibly infinite) subsets of \(\mathbb {Z}_{\ge 0}^{n}\). In the case that \(\mathcal {A}_j\) are finite, we saw in chapter VII that within all \(f_j\) supported at \(\mathcal {A}_j\), \(j = 1, \ldots , n\), \([f_1, \ldots , f_n]_{(\mathbbm {k}^*)^n}^{iso}\) takes the maximum value when \(f_1, \ldots , f_n\) are generic. It is possible to talk about “generic” power series supported at \(\mathcal {A}_j\) even if \(\mathcal {A}_j\) is infinite, and it turns out that the intersection multiplicity \([f_1, \ldots , f_n]_{0}\) of \(f_1, \ldots , f_n\) at the origin takes the minimum value when \(f_j\) are generic power series supported at \(\mathcal {A}_j\), \(j = 1, \ldots , n\) (see theorem IX.8 for the precise statement); in this chapter, we compute this minimum and give a Bernstein-Kushnirenko-type characterization of the systems which attain the minimum.
Publication Year: 2021
Publication Date: 2021-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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