Gröbner bases and normal forms in a subring of the power series ring on countably many variables
Let \(K\) be a field and let \(R=Kx_1,x_2,x_3,\dots\) be the ring of power series in countably many variables, with coefficients in \(K\). Denote by \(R'\) the smallest \(K\)-subalgebra of \(R\) that contains all homogeneous elements. The ring \(R'\) is of interest also because it allows a generalization of the notion of generic form in the polynomial ring \(K[x_1,\dots,x_n]\). Ideals generated by generic forms have been the focus of much study, and the investigation of their initial ideals leds to the study of Gröbner bases theory of the ring \(R'\), that is the topic of this paper. The author generalizes the notion of admissible term order to this context and defines the leading power product of an element of \(R'\), and thus the initial ideal \(\text{gr}(I)\) of an arbitrary ideal \(I\subset R'\). It turns out that if \(I\) is locally finitely generated, then \(\text{gr}(I)\) is also locally finitely generated; this implies that \(I\) has a finite truncated Gröbner basis up to any total degree. An example is given of a finitely generated homogeneous ideal which has a non-finitely generated initial ideal with respect to the lexicographic order.
scientific article; zbMATH DE number 1204278