Transcendent basis for the field of multisymmetric functions

Question

It is known that the field of multisymmetric rational functions (over a field of characteristic $0$), that is, rational function in variables $x_{11}, \ldots, x_{1m}, \ldots, x_{n1}, \ldots, x_{nm}$ that are invariant under the $S_n$ action $g.x_{ij}=x_{g(i)j}$ is rational of transcendent degree $nm$. In the language of geometry $Sym^n(\mathbb{P}^m)$ is birationaly isomorphic to $\mathbb{P}^{nm}$. One example of a transcendent basis is elementary multisymmetric polynomials of multi-indices $(k, 0, \ldots, 0), k=1, \ldots n$ and $(k, 0, \ldots, 1, \ldots, 0), k=0, \ldots n-1$ and the $1$ is in places $2, \ldots, m$ (Discriminants, Resultants, and Multidimensional Determinants, Trm 2.8)

I am interested to know if there is a transcendent basis made entirely from multisymmetric power sum polynomials?

Answer 1

The answer is yes.

Indeed, Remark 7 of the paper

Paul Görlach, Cordian Riener, Tillmann Weißer. Deciding positivity of multisymmetric polynomials. Journal of Symbolic Computation 74 (2016), 603-616. (https://www.sciencedirect.com/science/article/abs/pii/S074771711500098X, see also https://arxiv.org/abs/1409.2707)

says that if we consider degrees arising from different $m$-tuples $(w_1,\ldots,w_m)$ of nonnegative integer weights, each multisymmetric polynomial $f$ can be written as a polynomial expression in the power sums for which the $w$-degree does not exceed the $w$-degree of $f$ for all weights $w$. Thus, for the elementary symmetric functions of the first type we are looking for power sums of multidegrees $(a_1,\ldots,a_m)$ with $$ a_1w_1+\cdots+a_mw_m\le kw_1\le nw_1 $$ for all possible weights, and for the elementary symmetric functions of the second type we are looking for power sums of multidegrees $(a_1,\ldots,a_m)$ with $$ a_1w_1+\cdots+a_mw_m\le kw_1+w_p\le (n-1)w_1+w_p $$ for $p=2,\ldots,m$. Clearly, $$ a_1w_1+\cdots+a_mw_m\le kw_1\le nw_1 $$ for all nonnegative weights implies $a_1\le n$, $a_i=0$ for $i>1$, and, for a given $p$ between $2$ and $m$, $$ a_1w_1+\cdots+a_mw_m\le kw_1+w_p\le (n-1)w_1+w_p $$ for all nonnegative weights implies $a_1\le n-1$, $a_p\le 1$, $a_i=0$ for $i\ne 1,p$. So the constraints from Theorem 2.8' of Gelfand, Kapranov and Zelevinsky are exactly the same as constraints for the requested power sums, giving a required transcendence basis.