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Infinitely near imposed singularities and singularities of polar curves. (English) Zbl 0675.14009
Let \(\xi\) be a complex plane algebroid curve, defined by the equation \(f=0\). This paper is mainly devoted to the study of the singularities of the polar curves of \(\xi\) (defined by \(a\partial f/\partial x+b\partial f/\partial y=0\), a and b being complex numbers). The main results are theorem 8.1 and 11.3.
Theorem 8.1 gives the virtual behaviour of the polar curves and is a precise version of the old claim (Noether) saying that the polar curves have a (n-1)-fold point at each ordinary or infinitely near n-fold point of \(\xi\).
Theorem 11.3 describes the singularities, and in particular the topological types, of general and special polar curves of \(\xi\), assuming \(\xi\) to be unibranched with given characteristic exponents and generic Puiseux coefficients.
The §§ 2 to 6 are devoted to developing a theory of infinitely near imposed singularities which is needed for polar curves. § 12 gives an account of classical work on the subject.
Reviewer: E.Casas-Alvero

MSC:
14H20 Singularities of curves, local rings
14H45 Special algebraic curves and curves of low genus
14B10 Infinitesimal methods in algebraic geometry
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