Том 80, № 1 (2025)

The set of roots of any finite system of exponential sums in the space $\mathbb{C}^n$ is called an exponential variety. We define the intersection index of varieties of complementary dimensions, and the ring of classes of numerical equivalence of exponential varieties with operations ‘addition-union’ and ‘multiplication-intersection’. This ring is analogous to the ring of conditions of the torus $(\mathbb{C}\setminus 0)^n$ and is called the ring of conditions of $\mathbb{C}^n$. We provide its description in terms of convex geometry. Namely, we associate an exponential variety with an element of a certain ring generated by convex polytopes in $\mathbb{C}^n$. We call this element the Newtonization of the exponential variety. For example, the Newtonization of an exponential hypersurface is its Newton polytope. The Newtonization map defines an isomorphism of the ring of conditions to the ring generated by convex polytopes in $\mathbb{C}^n$. It follows, in particular, that the intersection index of $n$ exponential hypersurfaces is equal to the mixed pseudo-volume of their Newton polytopes.Bibliography: 32 titles.

The problem of the existence of a limit distribution of the zeros of Hermite–Pade polynomials for a pair of functions forming a Nikishin system is discussed. Two new scalar methods are proposed for the investigation of this problem. The first is based on a potential-theoretic equilibrium problem stated on a two-sheeted Riemann surface and on the use of the Gonchar–Rakhmanov–Stahl ($\operatorname{GRS}$-)method in treating this problem. The second method is based on the existence of a three-sheeted Riemann surface with Nuttall partition into sheets which is associated with a given pair of functions $f$, $f^2$, and it uses only the maximum principle for subharmonic functions. The connection of these methods and the results obtained with Stahl's methods and results of 1987–88 is discussed. Results of numerical experiments are presented.Bibliography: 109 titles.