# Existence des diviseurs dicritiques, d’après S.S. Abhyankar

• ### Vincent Cossart

Université de Versailles Saint-Quentin, Versailles, France
• ### Mickaël Matusinski

Université Bordeaux 1, Talence, France
• ### Guillermo Moreno-Socías

CNRS/UVSQ, Versailles, France

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## Abstract

In this article, there are new proofs of the existence and unicity of dicritical divisors of a pencil of plane curves of $\langle F,G\rangle$ Incidentally, we prove the equivalence between dicritical divisors and Rees valuations. Furthermore, in the case where $G_{\mathrm{red}}$ is regular at the base points of $\langle F,G\rangle$, we have that $F/G$ is residually a polynomial along any dicritical divisor; this reproves geometrically [2, Theorem (7.1)]. As a corollary of the latter proof, we get a generalization of the connectedness theorem of [8].