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Magic matrix associated with a germ of plane curve and division by the Jacobian ideal. (Matrice magique associée à un germe de courbe plane et division par l’idéal Jacobien.) (French. English summary) Zbl 1135.32028

The authors consider, in the ring of holomorphic functions at the origin of \(\mathbb{C}^2\), the equation \(uf'_x+vf'_y=wf\), for given \(f\) and \(w\). They start the analysis of the equation with the following. Let \(\overline{K}=\bigcup_{d\in \mathbb{N}^*}\mathbb{C}[[x^{1/d}]][1/x]\) be the field of Puiseux series. Suppose \(f=\prod _{i=1}^n (y-a_i)\) is a polynomial in \(\overline{K}[y]\). Let \(\overline{\mathcal{E}}\) be the subspace of \(\overline{K}[y]\) of polynomials of degree strictly smaller than \(n\). One fixes the basis \(\varepsilon_i=\prod_{j\neq i}(y-a_j)\), \(1\leq i\leq n\), of \(\overline{K}\). When \(w\in\overline{\mathcal{E}}\), the column matrices of \(u\) and \(w\) are related via a “magic matrix” determined by the \(a_i\).
Under certain conditions, the authors find the value of a naturally defined valuation on \(\overline{\mathcal{E}}\) of \(u\) and \(v\) and determine their initial term. This analysis recovers, in the case of the germ of a plane curve with an isolated singularity, that \(f^2\) is in the Jacobian ideal of \(f\), which is a particular case of a theorem of H. Skoda and J. Briançon [C. R. Acad. Sci., Paris, Sér. A 278, 949–951 (1974; Zbl 0307.32007)]. As an application, a multiple \(b(s)\) of the Bernstein-Sato polynomial (b-function) of \(f\) is found, and similar functional equation \(b(s)f^s=Pf^{s+1}\), with the property that \(b(s)\) depends only on the topological type of the isolated singularity germ.

MSC:

32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32S10 Invariants of analytic local rings
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14B05 Singularities in algebraic geometry

Citations:

Zbl 0307.32007
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References:

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