论文标题
部分可观测时空混沌系统的无模型预测
Integer-valued polynomials on discrete valuation rings of global fields with prescribed lengths of factorizations
论文作者
论文摘要
让$ v $是全球字段$ k $的评估戒指。我们表明，对于所有积极的整数$ k $和$ 1 <n_1 \ leq \ ldots \ leq n_k $，存在$ v $上的整数值多项式，即$ \ text {int}（int}（int}（v）的元素， $ \ text {int}（v）$的不可约元素的因素化，其长度正好为$ n_1，\ ldots，n_k $。实际上，我们显示的是，每个离散的估值域$ v $都具有有限的残留字段，因此$ v $的商字段允许一个估值戒指独立于$ v $，其最大理想是本金或残留物是有限的$ v $。如果$ V $的商字段是任意字段的纯粹先验扩展，则可以满足此属性。在这些情况下，这解决了Cahen，Fontana，Frisch和Glaz提出的一个开放问题。
Let $V$ be a valuation ring of a global field $K$. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $V$, that is, an element of $\text{Int}(V) = \{ f \in K[X] \mid f(V) \subseteq V \}$, which has precisely $k$ essentially different factorizations into irreducible elements of $\text{Int}(V)$ whose lengths are exactly $n_1,\ldots,n_k$. In fact, we show more, namely that the same result holds true for every discrete valuation domain $V$ with finite residue field such that the quotient field of $V$ admits a valuation ring independent of $V$ whose maximal ideal is principal or whose residue field is finite. If the quotient field of $V$ is a purely transcendental extension of an arbitrary field, this property is satisfied. This solves an open problem proposed by Cahen, Fontana, Frisch and Glaz in these cases.
