论文标题
表征绝对不可还原的整数价值多项式在离散估值域上
Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains
论文作者
论文摘要
众所周知，整数值多项式的环是原子，非本金环，提供了不可减至的元素的示例，所有功率都独特地因素（\ emph {\ emph {bection norryreDucibles}）和某些功率具有与小琐事不同的因素不同。 在本文中，我们研究了\ operatorname {int}（r）$ in b $ r $是一个具有有限残基领域的离散估值域，并表明可以明确确定一个norume $ s \ in \ mathbb {n} $降低了$ f^$ f^y $ f^y y $ f^nigand $ f^y y $ f^y y $ f^y y $ f^y y $ f^y n Intorical f.为此，我们建立了$ f $的权力因素与我们关联到$ f $的某个线性地图的内核之间的联系。这种连接从所谓的\ emph {固定的除数内核}来表征绝对不可约性的表征。给定此内核的非平凡元素$ \ boldsymbol {v} $，我们明确构建了$ f^k $的非平凡因素化，但前提是$ k \ ge l $，其中$ l $取决于$ f $以及$ \ boldsymbol {v} $的选择。我们进一步表明，通常无法改善这种界限。此外，我们为$ k $提供其他（较大的）下限，其中仅取决于$ f $的分母的估值和$ r $ $ r $的残留班级字段的大小。
Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (\emph{absolutely irreducibles}) and irreducible elements where some power has a factorization different from the trivial one. In this paper, we study irreducible polynomials $F \in \operatorname{Int}(R)$ where $R$ is a discrete valuation domain with finite residue field and show that it is possible to explicitly determine a number $S\in \mathbb{N}$ that reduces the absolute irreducibility of $F$ to the unique factorization of $F^S$. To this end, we establish a connection between the factors of powers of $F$ and the kernel of a certain linear map that we associate to $F$. This connection yields a characterization of absolute irreducibility in terms of this so-called \emph{fixed divisor kernel}. Given a non-trivial element $\boldsymbol{v}$ of this kernel, we explicitly construct non-trivial factorizations of $F^k$, provided that $k\ge L$, where $L$ depends on $F$ as well as the choice of $\boldsymbol{v}$. We further show that this bound cannot be improved in general. Additionally, we provide other (larger) lower bounds for $k$, one of which only depends on the valuation of the denominator of $F$ and the size of the residue class field of $R$.
