This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound \(\mu (I^2)\ge 9\) for the number of minimal generators of \(I^2\) with \(\mu (I)\ge 6\). Recently, Gasanova constructed monomial ideals such that \(\mu (I)>\mu (I^n)\) for any positive integer n. In reference to them, we construct a certain class of monomial ideals such that \(\mu (I)>\mu (I^2)>\cdots >\mu (I^n)=(n+1)^2\) for any positive integer n, which provides one of the most unexpected behaviors of the function \(\mu (I^k)\). The monomial ideals also give a peculiar example such that the Cohen–Macaulay type (or the index of irreducibility) of \(R/I^n\) descends.

本文研究了多项式环中多项式理想的最小生成器的数量。对于一个单项式理想我在两个变量，Eliahou，赫尔佐格和SAEM给了一个锋利的下界\（\亩（I ^ 2）\ GE 9 \）为最小的发生器的数量（I ^ 2 \）\与\ （\ mu（I）\ ge 6 \）。最近，加萨诺瓦（Gasanova）构造了单项式理想，使得对于任何正整数n都具有\（\ mu（I）> \ mu（I ^ n）\）。参照它们，我们构造了一类单项式理想，使得\（\ mu（I）> \ mu（I ^ 2）> \ cdots> \ mu（I ^ n）=（n + 1）^ 2 \ ）对于任何正整数n，它提供函数最意外的行为之一\（\ mu（I ^ k）\）。单项式理想还给出了一个特殊的例子，使得\（R / I ^ n \）的Cohen–Macaulay类型（或不可约性指数）下降。