On the generalisation of the scalar approach to the weak asymptotics of Hermite–Padé polynomials: some recent achievements

Till this moment, the behaviour of the weak asymptotics of zeros of Hermite–Padé polynomials for the Nikishin system (and some more extensive systems of Markov functions) has been usually investigated in the framework of the vector potential equilibrium problem. In 2017 S. Suetin suggested the new approach to this question dealing with the scalar potential problem with external harmonic field stated on a Riemann surface of genus zero (the Riemann sphere); in 2019 it was generalised for the broader class of functions. This generalisation leads to consideration of the scalar potential problem on a Riemann surface of positive genus with respect to the new type of kernels introduced by E. Chirka in 2018. In my talk I will discuss some recent achievements on the implementation of the scalar approach.