Solution of a problem about symmetric functions
Let \(X\), \(Y\) be independent indeterminates over \(\mathbb{Q}\), the field of rational numbers, and let \(m\) be a positive integer. Let \(N_m= N_m(X, Y)=X^m+ Y^m\) be the Newton symmetric power of order \(m\). The purpose of this paper is to prove the following result: If \(a> b> c\) are positive integers with \((a,b,c)= 1\), then the functions \(N_a\), \(N_b\), \(N_c\) generate the field \(S\) of all symmetric rational functions in \(X\), \(Y\) with rational coefficients, i.e., \(S= \mathbb{Q}(N_a, N_c)\). The authors then go on to prove a conjecture about the minimal degree \(d\) of a polynomial relation satisfied by \(N_a\), \(N_b\), \(N_c\), where by degree of a monomial \(N^i_a N^j_b N^k_c\), they mean \(ai+ bj+ ck\), namely \(d= abc/2\) if \(abc\) is even and \(d= (a- 1)bc/2\) otherwise. The authors supply some geometric applications.
scientific article; zbMATH DE number 2104668