The Sylvester-Ramanujan system of equations and the complex power moment problem
The author investigates a system of algebraic equations \(\sum_{k=1}^nx_kz_k^j=a_j\), \(0\leq j\leq 2n-1\) with \(2n\) complex unknown \(x_k\), \(z_k\). In the case when \(x_k\), \(z_k\) are nonnegative real the corresponding problem is connected with the power moment problem and had been considered by T. Stieltjes in 1889. In general case a solution \(\{x_k, z_k\}_{k=1}^n\in{\mathbb C}^{2n}\) of the problem is called regular if \(z_k\) are pairwise distinct and all \(x_k\neq 0\). The solvability and the regularity of the problem are investigated in the paper by means of the methods similar to those in the power moment problem. The corresponding complex moment problem is shown to be closely connected with different subjects of analysis such as the theory of binary forms [\textit{B. C. Berndt, S. Bhargava}, ''Ramanujan -- for lowbrows''. Am. Math. Monthly 100, No. 7, 644--656 (1993; Zbl 0795.11001)], quadrature formulas, Hilbert identity, which was used in the solution of the Waring problem, and others.
scientific article; zbMATH DE number 2091536