High-school algebra of the theory of dicritical divisors: atypical fibres for special pencils and polynomials

TL;DR

This paper studies dicritical divisors and polynomials, providing elementary proofs of key results, a constructive method to identify atypical fibers in special pencils, and sharper bounds, including explicit examples reaching these bounds.

Contribution

It offers new elementary proofs, a constructive approach to find atypical fibers, and improved bounds for their number, advancing understanding of dicritical divisors and special pencils.

Findings

Elementary proofs of dicritical divisor results

Constructive method for finding atypical fibers

Sharper bounds on the number of atypical fibers

Abstract

In this work we deal with dicritical divisors, curvettes and polynomials. These objects have been one of the main research interests of S.S. Abhyankar during his last years. In this work we provide some elementary proofs of some S.S. Abhyankar and I. Luengo results for dicriticals in the framework of formal power series. Based on these ideas we give a constructive way to find the atypical fibres of a special pencil and give bounds for its number, which are sharper than the existing ones. Finally, we answer a question of J. Gwo\'zdziewicz finding polynomials that reach his bound.