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No abstract uploaded!

Let$S$denote the class of schlicht functions. D. Bertilsson proved recently that for$f∈S, p<0$and$1<-N<-2|p|+1$the modulus of the$N$th Taylor coefficient of ($f′$)^{$p$}takes its maximal value if$f$is the Koebe function. Here a short proof of a generalisation of this result is presented.

We present an elementary and concrete description of the Hilbert scheme of points on the spectrum of fraction rings$k[X]$_{$U$}of the one-variable polynomial ring over a commutative ring$k$. Our description is based on the computation of the resultant of polynomials in$k[X]$. The present paper generalizes the results of Laksov-Skjelnes [7], where the Hilbert scheme on spectrum of the local ring of a point was described.