On the Theorems of Biernacki

Abstract

Let $$p(z)$$ be a complex polynomial of degree $$n$$. If we know the location of its zeros, then we can locate the zeros of its $$m$$-th integral. This result has been obtained by Biernacki, and his bound depends only on $$m$$, and not on $$n$$. We found an alternative bound that depends on both $$n$$ and $$m$$, and is better than the previous one in infinitely many cases.