PLANE-CURVES OF MINIMAL DEGREE WITH PRESCRIBED SINGULARITIES

We prove that there exists a positive alpha such that for any integer d greater than or equal to 3 and any topological types S-1,...,S-n of plane curve singularities, satisfying mu(S-1) + ... + mu(S-n) less tha n or equal to alpha d(2), there exists a reduced irreducible plane cur ve of degree d with exactly n singular points of types S-1,...,S-n, re spectively. 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 d less than or equal to 14 root m u(S) having a singular point of type S.