Factorization of the polar curve and the Newton polygon

Lenarcik, Andrzej; Masternak, Mateusz; Płoski, Arkadiusz

doi:10.2996/kmj/1073670610

Cited by 6 publications

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“…In [13] (see also [14]) the authors considered the pencil f (x, y) − x N = 0. They distinguished one of edges of the Newton diagram of f .…”

Section: Relation With Known Resultsmentioning

confidence: 99%

Ephraim’s Pencils

Gwoździewicz¹

2012

International Mathematics Research Notices

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Let f (x, y) = 0 and l(x, y) = 0 be respectively a singular and a regular analytic curve defined in the neighborhood of the origin of the complex plane. We study the family of analytic curves f (x, y) − t l(x, y) M = 0, where t = 0 is a complex parameter. For all but a finite number of parameters the curves of this family have the same embedded topological type. The exceptional parameters are called special values. We show that the number of special values does not exceed the number of components of the curve f (x, y) = 0 counted without multiplicities. Then we apply this result to estimate the number of critical values at infinity of complex polynomials in two variables.d . A generic curve of this pencil consists of d straight lines crossing at the origin. The curve f t0 (x, y) = 0 is special if and only if t 0 is a critical value of the polynomial H; then f t0 (x, y) = 0 consists of d − 2 single straight lines and one double line. Example 1.3 Let H(Z) = d i=1 (Z − a i ) mi be a complex polynomial with d − 1 nonzero critical values different from H(0). Consider the family of quasihomogeneous polynomials f t (x, y) = d i=1 (x 2 − a i y 3 ) mi − ty 3 deg H . For t = 0 the curve f t (x, y) = 0 consists of d multiple cusps (x 2 − a i y 3 ) mi = 0. For t = H(0) the curve f t (x, y) = 0 consists of deg H −1 cusps and one double line x 2 = 0. If t is a nonzero critical value of H then the curve f t (x, y) = 0 has deg H −2 single cusps and one double cusp. For all other t ∈ C the curve f t (x, y) = 0 decomposes into deg H pairwise different cusps of the form x 2 − b i y 3 = 0 for i = 1, . . . , deg H.This example shows that the bound for the number of special values given in Theorem 1.1 is sharp.

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“…In [13] (see also [14]) the authors considered the pencil f (x, y) − x N = 0. They distinguished one of edges of the Newton diagram of f .…”

Section: Relation With Known Resultsmentioning

confidence: 99%