Plane curves of minimal degree with prescribed singularities

• We prove that there exists a positive $$\alpha$$ such thatfor any integer $$\mbox{d\ge 3}$$ and any topological types $$\mbox{S_1,\dots,S_n}$$ of plane curve singularities, satisfying $$\mbox{\mu(S_1)+\dots+\mu(S_n)\le\alpha d^2}$$, there exists a reduced irreducible plane curve of degree $$d$$ with exactly $$n$$ singular points of types $$\mbox{S_1,\dots,S_n}$$, 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 $$\mbox{d\le 14\sqrt{\mu(S)}}$$ having a singular point of type $$S$$.

$Rev: 13581$