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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.

In the present paper we prove, that in the real normed space$X$, having at least three dimensions, the Nordlander’s conjecture about the modulus of convexity of the space$X$is true, i.e. from the validity of Day’s inequality for a fixed real number from the interval (0,2), follows that$X$is an inner product space.

This paper introduces a new efficient algorithm, called MXL_3, for computing Gröbner bases of zero-dimensional ideals. The MXL_3 is based on XL algorithm, mutant strategy, and a new sufficient condition for a set of polynomials to be a Gröbner basis. We present experimental results comparing the behavior of MXL_3 to F_4 on HFE and random generated instances of the MQ problem. In both cases the first implementation of the MXL_3 algorithm succeeds faster and uses less memory than Magma’s implementation of F_4.