A note on polynomial reduction (Q1314434)

Let \(K\) be a field. A marked polynomial is a polynomial \(f\in K[x_ 1, x_ 2,\dots, x_ n]\) together with a specified leading term \(in(f)\) which can be any of the terms appearing in \(f\). A set \({\mathcal F}\subset K[x_ 1, x_ 2,\dots, x_ n]\) of marked polynomials defines a reduction \(\to_{\mathcal F}\) modulo \({\mathcal F}\) in the usual sense of Gröbner bases theory, and this reduction is noetherian if \({\mathcal F}\) is marked coherently, i.e. if there is an admissible term order on \(K[x_ 1, x_ 2,\dots, x_ n]\) such that for all \(f\in {\mathcal F}\), in \((f)\) is the leading term in \(f\) with respect to this term ordering [see \textit{B. Buchberger}, Multidimensional systems theory, 184-232 (1985; Zbl 0587.13009)]. In this paper the converse of this statement is proved.