Symmetric polynomials and Hall's theorem
Let A[X,Y] be the polynomial ring in 2n variables \(X=(X_ 1,..,X_ n)\), \(Y=(Y_ 1,...,Y_ n)\) over a commutative unitary ring A and let \({\mathcal P}\) and \({\mathcal D}\) be the following ideals of A[X,Y]: \({\mathcal P}=<P\in A[X,Y]\), \(P(X,s(X))=0\) for all \(s\in S_ n>\); \({\mathcal D}=<\sigma_ k(X)- \sigma_ k(Y)\), \(k=1,...,n>\), where the \(\sigma_ k\) are the k-th elementary symmetric polynomials in n variables. It is shown that \({\mathcal P}={\mathcal D}\) which in consequence yields another proof of the classical result concerning symmetric polynomials. Furthermore by associating to any pair (a,b) of the non-empty set \(\{\) 1,...,n\(\}^ a \)polynomial \(P_{a,b}(X,Y)\) and by investigating the ideal \({\mathcal M}\) generated in \({\mathbb{Z}}[X,Y]\) by these the author derives the well-known matching theorem of Ph. Hall. Finally it is proved that this theorem is equivalent to the statement rad(\({\mathcal M})={\mathcal P}\).
scientific article; zbMATH DE number 4071047