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 Mosc. Math. J., 2006, Volume 6, Number 1, Pages 107–117 (Mi mmj238)

Logarithmic vector fields for the discriminants of composite functions

V. V. Goryunov

Department of Mathematical Sciences, University of Liverpool

Abstract: The $K_f$-equivalence is a natural equivalence between map-germs $\varphi\mathbb C^m\mathbb C^n$ which ensures that their compositions $f\circ\varphi$ with a fixed function-germ f on $\mathbb C^n$ are the same up to biholomorphisms of $\mathbb C^m$. We show that the discriminant $\sum$ in the base of a $K_f$-versal deformation of a germ $\varphi$ is Saito's free divisor provided the critical locus of f is Cohen–Macaulay of codimension $m+1$ and all the transversal types of $f$ are $A_k$ singularities. We give an algorithm to construct basic vector fields tangent to $\sum$. This is a generalisation of classical Zakalyukin's algorithm to write out basic fields tangent to the discriminant of an isolated function singularity. The case of symmetric matrix families in two variables is done in detail. For simple singularities, it is directly related to Arnold's convolution of invariants of Weyl groups.

Key words and phrases: Logarithmic vector field, discriminant, composite function, free divisor, matrix singularities.

DOI: https://doi.org/10.17323/1609-4514-2006-6-1-107-117

Full text: http://www.ams.org/.../abst6-1-2006.html
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MSC: Primary 32S05; Secondary 58K20
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Citation: V. V. Goryunov, “Logarithmic vector fields for the discriminants of composite functions”, Mosc. Math. J., 6:1 (2006), 107–117

Citation in format AMSBIB
\Bibitem{Gor06} \by V.~V.~Goryunov \paper Logarithmic vector fields for the discriminants of composite functions \jour Mosc. Math.~J. \yr 2006 \vol 6 \issue 1 \pages 107--117 \mathnet{http://mi.mathnet.ru/mmj238} \crossref{https://doi.org/10.17323/1609-4514-2006-6-1-107-117} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2265950} \zmath{https://zbmath.org/?q=an:1121.58028} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208595700007} 

• http://mi.mathnet.ru/eng/mmj238
• http://mi.mathnet.ru/eng/mmj/v6/i1/p107

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Goryunov V.V., Zakalyukin V.M., “Lagrangian and legendrian varieties and stability of their projections”, Singularities in Geometry and Topology, 2005, 2007, 328–353
2. Miranda-Neto C.B., “A Module-Theoretic Characterization of Algebraic Hypersurfaces”, Can. Math. Bul.-Bul. Can. Math., 61:1 (2018), 166–173