On hereditarily rational functions

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.