论文标题
多项式上的双降低操作员
Double Lowering Operators on Polynomial
论文作者
论文摘要
最近，莎拉·博克廷·康拉德（Sarah Bockting-Conrad）推出了Tridiagonal对的双降低操作员$ψ$。由$ψ$激励，我们考虑了以下关于多项式的问题。令$ \ mathbb f $表示代数封闭的字段。令$ x $表示不确定的，然后让$ \ mathbb f \ lbrack x \ rbrack $表示由$ x $中的多项式组成的代数，这些代数为$ \ mathbb f $。令$ n $表示一个正整数或$ \ infty $。令$ \ lbrace a_i \ rbrace_ {i = 0}^{n-1} $，$ \ lbrace b_i \ rbrace_ {i = 0}^{n-1} $在$ \ mathbb f $中表示量表\ sum_ {h = 0}^{i-1} b_h $，$ 1 \ leq i \ leq n $。对于$ 0 \ leq i \ leq n $定义多项式$τ_i，η_i\ in \ mathbb f \ lbrack x \ rbrack x \ rbrack $ by $τ_i= \ prod_ = \ prod_ {h = 0}令$ v $表示$ \ mathbb f \ lbrack x \ rbrack $ spand of $ \ lbrace x^i \ rbrace_ {i = 0}^n $。每当$ n \ mathbb fτ_{i-1} $和$ψη_i\ in \ mathbbfη_{i-1-1} $ n \ mathbbfτ_i\ in \ mathbbfτ_i\ in \ mathbbfτ_i\ in \ mathbb fη_{i-1} $时$η_{ - 1} = 0 $。我们在$ \ lbrace a_i \ rbrace_ {i = 0}^{n-1} $，$ \ lbrace b_i \ rbrace_ {i = 0}^{n-1} $上提供了必要和足够的条件，以使其存在非零双重降低映射。有四个解决方案家族，我们将详细描述。
Recently Sarah Bockting-Conrad introduced the double lowering operator $ψ$ for a tridiagonal pair. Motivated by $ψ$ we consider the following problem about polynomials. Let $\mathbb F$ denote an algebraically closed field. Let $x$ denote an indeterminate, and let $\mathbb F\lbrack x \rbrack$ denote the algebra consisting of the polynomials in $x$ that have all coefficients in $\mathbb F$. Let $N$ denote a positive integer or $\infty$. Let $\lbrace a_i\rbrace_{i=0}^{N-1}$, $\lbrace b_i\rbrace_{i=0}^{N-1}$ denote scalars in $\mathbb F$ such that $\sum_{h=0}^{i-1} a_h \not= \sum_{h=0}^{i-1} b_h$ for $1 \leq i \leq N$. For $0 \leq i \leq N$ define polynomials $τ_i, η_i \in \mathbb F\lbrack x \rbrack$ by $τ_i = \prod_{h=0}^{i-1} (x-a_h)$ and $η_i = \prod_{h=0}^{i-1} (x-b_h)$. Let $V$ denote the subspace of $\mathbb F\lbrack x \rbrack$ spanned by $\lbrace x^i\rbrace_{i=0}^N$. An element $ψ\in \operatorname{End}(V)$ is called double lowering whenever $ψτ_i \in \mathbb F τ_{i-1}$ and $ψη_i \in \mathbb F η_{i-1}$ for $0 \leq i \leq N$, where $τ_{-1}=0$ and $η_{-1}=0$. We give necessary and sufficient conditions on $\lbrace a_i\rbrace_{i=0}^{N-1}$, $\lbrace b_i\rbrace_{i=0}^{N-1}$ for there to exist a nonzero double lowering map. There are four families of solutions, which we describe in detail.
