This article is cited in 2 scientific papers (total in 2 papers)
Logarithmic vector fields for the discriminants of composite functions
V. V. Goryunov
Department of Mathematical Sciences, University of Liverpool
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.
MSC: Primary 32S05; Secondary 58K20
Received: February 6, 2006
V. V. Goryunov, “Logarithmic vector fields for the discriminants of composite functions”, Mosc. Math. J., 6:1 (2006), 107–117
Citation in format AMSBIB
\paper Logarithmic vector fields for the discriminants of composite functions
\jour Mosc. Math.~J.
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This publication is cited in the following articles:
Goryunov V.V., Zakalyukin V.M., “Lagrangian and legendrian varieties and stability of their projections”, Singularities in Geometry and Topology, 2005, 2007, 328–353
Miranda-Neto C.B., “A Module-Theoretic Characterization of Algebraic Hypersurfaces”, Can. Math. Bul.-Bul. Can. Math., 61:1 (2018), 166–173
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