On hereditarily rational functions

TL;DR

This paper provides an elementary proof of Kollár's theorem on hereditarily rational functions using a strengthened 3ojasiewicz inequality, avoiding resolution of singularities and sharpening the original result.

Contribution

It offers a new, simpler proof of Kollár's theorem that does not depend on desingularization techniques, enhancing the theorem's precision.

Findings

Elementary proof of Kolle1r's theorem

Avoids resolution of singularities

Sharpened version of the theorem

Abstract

In this paper, we give a short proof of a theorem by Koll\'{a}r on hereditarily rational functions. This is an answer to his appeal to find an elementary proof which does not rely so much on resolution of singularities. Our approach does not make use of desingularization techniques. Instead, we apply a stronger version of the \L{}ojasiewicz inequality. Moreover, this allows us to sharpen Koll\'{a}r's theorem.