On length of generalized fractions \(1/(x_1^n,\dots, x_d^n,1)\) (Q1762794)

Let \(M\) be a finitely generated module over a noetherian local ring \((R,m)\) with \(\dim M= d\). For a system of parameters \((x_{1},x_{2},\dots, x_{d})\) of \(M\) and a positive integer \(n\), consider the generalized fraction \(1/(x_{1}^{n}, x_{2}^{n},\dots, x_{d}^{n},1)\) [concept introduced by \textit{H. Sharp} and \textit{H. Zakeri}, Mathematika 29, 32--41 (1982; Zbl 0497.13006)]. One of the questions on this object is whether its length can be seen as a polynomial in \(n\), for \(n\) large enough. The present paper answers in the negative this question, by indicating a counterexample for any \(d\) \(\geq 3\) . Moreover, the above problem is related with the study of the length of a ring \(R\)/(\(x_{1}^{n}\), \( x_{2}^{n}\),\dots, \(x_{d}^{n}\), \(I\)), where \(I\subset R\) is an arbitrary ideal of \(R\).