Milne's volume function and vector symmetric polynomials (Q1012153)

Let \(R\) be a commutative ring with identity and \(A=[a_{ij}]\) an \(r\times n\) matrix of indeterminates. The symmetric group \(S_n\) acts on the polynomial ring \(R[a_{ij}\mid 1\leq i\leq r\), \(1\leq j\leq n]\) by permutation of columns of \(A\). The ring of invariants of this action is denoted by \(\text{VSym}_R(A)\) and an element of \(\text{VSym}_R(A)\) is called a vector symmetric polynomial. For \(\alpha=(\alpha_1, \ldots, \alpha_r)\in\mathbb N^r\), set \[ p_\alpha=\sum_{j=1}^n a_{1j}^{\alpha_{1}}\cdots a_{rj}^{\alpha_r}. \] \(p_\alpha\) are called power sums. Define \(e_\alpha\) for \(\alpha\in\mathbb N^r\) by the equation \[ \sum_{\alpha\in\mathbb N^r}e_\alpha t_1^{\alpha_1}\cdots t_r^{\alpha_r} =\prod_{j=1}^n(1+a_{1j}t_1+\cdots+a_{rj}t_r), \] where \(t_1, \ldots, t_r\) are new variables. Both families \(\{p_\alpha\}_{\alpha\in\mathbb N^r}\) and \(\{e_\alpha\}_{\alpha\in\mathbb N^r}\) are known to generate \(\text{VSym}_R(A)\) over \(R\) in case where \(R\) contains the field of rational numbers. \textit{P. Milne} [in: Computational Mathematics and Applications, Academic Press, 89--101 (1992)] introduced a volume function \[ V(u,x_1,\ldots, x_r,A)=\prod_{j=1}^n(u+\prod_{i=1}^r(x_i-a_{ij})), \] where \(u\), \(x_1,\ldots, x_r\) are new variables, in order to consider the number of real roots in a given box of the system \(f_1(x_1,\ldots,x_r)=\cdots=f_k(x_1,\ldots,x_r)=0\) of polynomial equations with real coefficients which has finite complex roots. \(V(u,x_1,\ldots, x_r,A)\) is a vector symmetric polynomial over \(\mathbb Z[u,x_1,\ldots,x_r]\). In this paper, the authors describe \(V(u,x_1,\ldots,x_r,A)\) by the family \(\{\Phi_\alpha(l)\}_{\alpha\in\mathbb N^r,l\in\mathbb N}\) of squarefree vector symmetric polynomials, which is an extension family of \(\{e_\alpha\}_{\alpha\in\mathbb N^r}\) and \(e_\alpha=\Phi_\alpha(|\alpha|)\) for any \(\alpha\in\mathbb N^r\), where \(|\alpha|=\alpha_1+\cdots+\alpha_r\) for \(\alpha=(\alpha_1,\ldots,\alpha_r)\). Relations to power sums \(\{p_\alpha\}_{\alpha\in\mathbb N^r}\) and hypergraphs are also studied.