Green and Gotzmann theorems for polynomial rings with restricted powers of the variables
Let \(k\) be a field, \(S=k[x_1, \dots, x_n]/(x_1^{a_1}, \dots, x_n^{a_n})\), \(S_d\) the space of homogeneous polynomials of degree \(d\) in \(S\), \(\overline S=k[x_1, \dots, x_n]/(x_1^{a_1}, \dots, x_{n-1}^{a_{n-1}}) \subset S\), \(V\subseteq S_d\) a vector space generated by monomials, \(\overline V\) the vector space generated by the monomials in \(V\) which are not divisible by \(x_n\), and \(V'=\{f/x_n: f\in V\) and \(x_n\mid f\}\). The author obtains as main result in this paper: Let \(V,L\subseteq S_d\) be vector spaces such that \(V\) is strongly stable, \(L\) is generated by an initial lex-segment, and \(\dim V= \dim L\), then (1) \(\dim\overline V\geq\dim \overline L\); (2) \(\dim VS_1\geq\dim LS_1\); (3) Let \(u=\max \{1,a_i-1\mid 1\leq i\leq n,\;a_i< \infty\}\). If \(\dim VS_i= \dim LS_i\) for \(1\leq i\leq u\), then \(\dim VS_j=\dim LS_j\) for all \(j>1\). In this result are contained: (1) a generalized Green's theorem [cf. \textit{M. Green} in: Algebraic curves and projective geometry, Proc. Conf., Trento 1988, Lect. Notes Math. 1389, 76-86 (1989; Zbl 0717.14002)]; (2) a generalized Clements-Lindström theorem [cf. \textit{G. F. Clements} and \textit{B. Lindström}, J. Comb. Theory 7, 230-238 (1969; Zbl 0186.01704); and (3) a generalized Aramova-Herzog-Hibi persistence theorem [cf. \textit{A. Aramova}, \textit{J. Herzog} and \textit{T. Hibi}, J. Algebra 191, No. 1, 174-211 (1997; Zbl 0897.13030)].
scientific article; zbMATH DE number 1308210