An ideal \(I \subset \mathbb {k}[x_1, \ldots , x_n]\) is said to have linear powers if \(I^k\) has a linear minimal free resolution, for all integers \(k>0\). In this paper, we study the Betti numbers of \(I^k\), for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for \(d=2, 3\) or \(n-1\), and the product of all ideals generated by s variables, for \(s=n-1\) or \(n-2\). We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.

一个理想的\（I \子集\ mathbb {K} [X_1，\ ldots，x_n] \）是说，有如果线性功率\（I-1K-\）具有线性最小自由分辨率，所有整数\（K> 0 \）。在本文中，我们研究具有线性幂的理想I的贝蒂数\（I ^ k \）。我们提供贝蒂数的线性关系，它适用于所有具有线性幂的理想。这对于低尺寸的理想情况特别有用。贝蒂数明确地计算为k的多项式，对于由度为d的所有无平方单项式，对于\（d = 2，3 \）或\（n-1 \）生成的理想，以及所有乘积s产生的理想变量，用于\（s = n-1 \）或\（n-2 \）。对于线性功率的理想，我们还研究了里斯理想的发电机。特别地，我们对由二次元素生成Rees理想的理想感兴趣。这个问题与怀特对类人动物的猜想有关。