I had been pointed to Ramanujan's 1912 article Note on a set of simultaneous equations in this answer to my former question about the Solvability of a system of polynomial equations.
While the contents of the paper seemed a bit enigmatic at first reading, I'm now convinced that Ramanujan not only left it to the reader to recover the ideas that had led to the solution but also the motivation to investigate on that special system of polynomial equations.
Questions:
- is anything known about Ramanujan's motivation to work on that problem that somehow doesn't seem to fit his primary research interest; the situation seems analogous to Euler and his characteristic of polyhedra?
- did Ramanujan himself ever refer to the paper in his later work?
- is or was the paper of importance for mathematical research and what are remarkable examples?
- have there been serious attempts to extend the solution method and/or to reduce more general types of systems of polynomial equations to the special case that Ramanujan has solved?
Best Answer
This refers to the third and fourth question in the OP.
Ramanujan's 1912 paper addresses a problem similar to that considered by Sylvester in 1851 [1]. The method to solve the set of algebraic equations $$\sum_{k=1}^{n}x_kz_k^j=a_j,\;\;0\leq j\leq 2n-1$$ in the unknown $x_k$'s and $z_k$'s is different in the two papers, but the final algorithm is similar.
This set of equations shows up in the socalled moment problem, to find a probability measure given its moments. That connection is explored in The Sylvester-Ramanujan System of Equations and The Complex Power Moment Problem by Yuri Lyubich.
[1] J.J. Sylvester, On a remarkable discovery in the theory of canonical forms and of hyperdeterminants, Phil. Magazine 2, 391–410 (1851).