Special polars and curves with one place at infinity

Meromorphic plane curves

“…The aim of this section is to generalize Abhyankar-Moh Theorem to the families of curves having only one irregular value. When the family is equisingular at infinity, the following result has been proved by R. Ephraim [10]: ii) There is an automorphism σ :…”

“…The family (f λ ) λ∈C is said to be equisingular at infinity if it is equisingular at its points at infinity. Using the theory of polar curves, R. Ephraim proved [10] the following generalization of Abhyankar-Moh Theorem: Assume that the family (f λ ) λ∈C is equisingular at infinity. If int(f x , f y ) = 0, then there is an automorphism σ of C 2 such that f • σ is a coordinate of C 2 .…”

High-school algebra of the theory of dicritical divisors: Atypical fibers for special pencils and polynomials

“…We also provide examples showing that our bound is also sharper than the one of [22,Theorem 1.2]. Notice that Gwoźdiewicz's result is in the same spirit as the following Moh's Theorem [26] as quoted by Ephraim's version [21]. [21])Assume that the complex algebraic curve f (x, y) = 0 has only one branch at infinity.…”

“…Notice that Gwoźdiewicz's result is in the same spirit as the following Moh's Theorem [26] as quoted by Ephraim's version [21]. [21])Assume that the complex algebraic curve f (x, y) = 0 has only one branch at infinity. Then f has no critical values at infinity.…”

Apéry basis and polar invariants of plane curve singularities

“…In general, if M is a smooth germ of a plane curve singularity defined by z e C{X, Y}, then the polar of C with respect to M is the (possibly nonreduced) germ whose defining ideal is generated by the Jacobian /(/, z) = <? (/, z)/d(X, Y) (see [4]). In particular, a general polar P(C) of C is defined by h = J(f,λX + μY) with (λ,μ) general.…”