The vector equilibrium problems for the logarithmic potential with the Angelesco matrix of interactions are considered. For the case of two arbitrary intervals the solutions in the class of alternating charges and in the class of non-negative measures are researched. It is shown that the solutions are expressed in terms of algebraic functions. It is found that for the problem in the class of charges the genus of the algebraic curve is equal to zero for any location of the intervals. The explicit formulas of the equilibrium charges are found. The explicit formulas of the equilibrium measures are found under certain restrictions on the location of the intervals.

vector equilibrium problem, Angelesco matrix of interaction, extremal measure, logarithmic potential, multiple orthogonal polynomials, spectral curve uniformization

Mathematical problems and theory of numerical methods

Lysov Vladimir Genrikhovich,  v.g.lysov@gmail.com,  KIAM RAS

Tulyakov Dmitriy Nikolaevich,  dntulyakov@gmail.com,  KIAM RAS