Gert-Martin Greuel, Christoph Lossen, Eugenii Shustin

• We prove that there exists a positive $\alpha$ such thatfor any integer $d\ge 3$ and any topological types $S_1,\dots ,S_n$ of plane curve singularities, satisfying $\mu (S_1)+\cdots +\mu (S_n)\le \alpha d^2$, there exists a reduced irreducible plane curve of degree $d$ with exactly $n$ singular points of types $S_1,\dots ,S_n$, respectively. This estimate is optimal with respect to theexponent of $d$. In particular, we prove that for any topological type $S$ there exists an irreducible polynomial of degree $d\le 14\sqrt{\mu (S)}$ having a singular point of type $S$.