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In this contribution we find the solution F : I £ P 2 ðIÞ £ M £ M ! M of the following functional equations Fðt; g; d; Fðg; a; b; a; bÞ; Fðd; a; b; a; bÞÞ ¼ Fðt; a; b; a; bÞ, Fða; a; b; a; bÞ ¼ a, Fðb; a; b; a; bÞ ¼ b. Since we do not insist on any conditions for the solution, the obtained result is greatly general.

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A fast numerical test of multivariate polynomial positiveness with applications Kybernetika, Vol. 54 (2018) A FAST NUMERICAL TEST OF MULTIVARIATE POLYNOMIAL POSITIVENESS WITH APPLICATIONS Petr Augusta and Petra AugustováThe paper presents a simple method to check a positiveness of symmetric multivariate polynomials on the unit multi-circle. The method is based on the sampling polynomials using the fast Fourier transform. The algorithm is described and its possible applications are proposed. One of the aims of the paper is to show that presented algorithm is significantly faster than commonly used method based on the semi-definite programming expression.