On some resultants formulas of Schur type

TL;DR

This paper derives formulas for resultants of polynomial sequences satisfying certain recurrence relations, generalizing previous results and providing explicit expressions under mild initial conditions.

Contribution

It introduces generalized resultant formulas for polynomial sequences defined by complex recurrence relations, extending prior work to broader cases.

Findings

Derived explicit formulas for resultants of polynomial sequences

Generalized previous results to higher order recurrences

Provided conditions under which formulas hold

Abstract

Let $(r_{A,n}(x))_{n \in \mathbb{N}}$ be a sequence of polynomials with coefficients from a field $K$ satisfying the recurrence relation $r_{A,n}(x)= \sum_{|\alpha|\leq m} t_{\alpha,n}(x)\textbf{r}_{A,n}^\alpha(x)$ of order $d+1 \in \mathbb{N}_{+}$, where $t_{\alpha,n} \in K[x]$, $m \in \mathbb{N}_{+}$ are fixed, $\alpha \in \mathbb{N}^{d+1}$, $|\alpha| = \alpha_0 + \ldots+\alpha_d$ and $\textbf{r}_{A,n}^\alpha(x)=r_{A,n-1}^{\alpha_0}(x)r_{A,n-2}^{\alpha_1}(x)\cdots r_{A,n-d-1}^{\alpha_d}(x).$ We show that under mild assumptions on the initial polynomials $r_{A,0}, \ldots, r_{A,d}$ and the coefficients $t_{\alpha,n}$, we can give the expression for the resultant $\text{Res}(r_{A,n}, r_{A,n-1})$. Our results generalize recent result of Ulas concerning the case $m=1$ and $d=1$.