Generators of truncated symmetric polynomials (Q326564)

Let \(R = \mathbb{F}[x_1,\ldots,x_n]\) be the polynomial ring over a filed \(\mathbb{F}\) in \(n\) variables. The symmetric group \(\mathfrak{S}_n\) acts on \(R\) by permuting the variables. Denote by \(R^{\mathfrak{S}_n}\) the invariant subring, i.e., the ring of symmetric polynomials. The ideal of truncated symmetric polynomials NEWLINE\[NEWLINEI_{n,d}=(x^{d+1}_1,\ldots, x^{d+1}_n)R\cup R^{\mathfrak{S}_n}.NEWLINE\]NEWLINE in \(R^{S_n}\) was introduced by \textit{A. Adem} and \textit{Z. Reichstein} [Doc. Math., J. DMV 15, 1029--1047 (2010; Zbl 1211.55010)] to present the permutation invariant subring in the cohomology of a finite product of projective spaces.NEWLINENEWLINEBuilding upon their work, the author describes a generating set of the ideal of truncated symmetric polynomials in arbitrary positive characteristic, and offer a conjecture for minimal generators.