About the Blaschke products in polydiscs

We consider multidimensional analogs of Blaschke’s products. The motivation for this consideration was the articles by D. Alpay and A. Iger in connection with the interpolation theory in some functional spaces (Hilbert, Hardy, etc.). We construct the Blaschke multipliers using the Rudin characterization [5] of interior functions in polydics and the Lee-Yang polynomials from the theory of phase transitions in statistical mechanics [1], [2]. As shown by M. Passare and A. Tsikh in [3], the amoeba of the Lee-Yang polynomial adjoins the log-image of a unit polydisc only on the skeleton of the polydisc. The main result of my talk is a theorem about the construction of a multidimensional Blaschke multiplier in odd-dimensional spaces. In such spaces, the Lee-Yang polynomial naturally decomposes into the sum of two polynomials that make up the inner rational function in the polydisc. The description of the permissible denominators of inner rational functions is made by the language of the polar [4] of the real cube $[-1,1]^n$.