Plane curves of minimal degree with prescribed singularities

Abstract

We prove that there exists a positive α such that for any integer d≥3 and any topological types S 1,…,S n of plane curve singularities, satisfying there exists a reduced irreducible plane curve of degree d with exactly n singular points of types S 1,…,S n , respectively. This estimate is optimal with respect to the exponent of d. In particular, we prove that for any topological type S there exists an irreducible polynomial of degree having a singular point of type S.