Alternate bases are a numeration system that generalizes the Rényi numeration system. It is common in this context to construct examples or counter-examples by specifying the expansions of $1$ in the desired system. While it is easy to show when a system with given expansions of $1$ exists in the Rényi case, the same is not true in the alternate case. In this article, we establish conditions for given words to be the expansions of $1$ in the alternate case. To do so, we use a fixed point theorem on matrices defined from the expansions and obtain the elements of the base from the components of the fixed point. We also obtain a partial result for the uniqueness of such a base. In the latter parts of the article, we use similar techniques to prove the existence of bases with a given sequence of $B$-integers.

翻译：交替基是一种推广了Rényi计数系统的计数系统。在此框架下，通常通过指定所需系统中1的展开式来构造范例或反例。尽管在Rényi情形下，给定1的展开式时系统是否存在的问题易于验证，但在交替情形下却并非如此。本文针对交替情形下给定词能否作为1的展开式建立了判定条件。我们利用由展开式定义的矩阵的不动点定理，从不动点的分量中推导出基元素。同时，本文还得到了此类基唯一性的部分结果。在文章后半部分，我们运用类似技术证明了具有给定B-整数序列的基的存在性。