Let \(A\) be a commutative ring and \(a_1, \dots, a_d \in A\). In the present paper, the authors study the equation \[ (a_1, \dots, a_n)^n = \bigcap_{\substack{ n_1, \dots, n_d > 0 \\ n_1 + \dots + n_d = n + d - 1 }} (a_1^{n_1}, \dots, a_d^{n_d}) \quad \text{for any \(n > 0\)}. \tag{1} \] The origin of (1) is given by W. Heinzer, L. J. Ratliff jun. and K. Shah [Houston J. Math. 21, No. 1, 29–52 (1995; Zbl 0838.13012)]. They showed that (1) is true whenever \(a_1, \dots, a_d\) is an \(A\)-regular sequence. In particular, an arbitrary system of parameters for a Cohen-Macaulay local ring satisfies (1). The authors prove the converse in a weak sense.
The main theorem of this paper asserts that all the systems of parameters for a Noetherian local ring \(A\) of dimension at least \(2\) satisfy (1) if and only if \(A/H_{\mathfrak m}^0(A)\) is Cohen-Macaulay and \(A\) itself is Buchsbaum. In particular, if \(\text{depth} A > 0\) and all the systems of parameters for \(A\) satisfy (1), then \(A\) is Cohen-Macaulay.
It is a natural question whether \(A\) is Cohen-Macaulay if \(\text{depth} A > 0\) and {some} system of parameters for \(A\) satisfies (1). The authors give a counterexample for this question but the example is not quasi-unmixed.
They also study \(2\)-dimensional generalized Cohen-Macaulay local rings, which are quasi-unmixed, and show that such a ring is Cohen-Macaulay if it is of positive depth and some standard system of parameters satisfies (1).