Orthogonal sequences of polynomials over arbitrary fields
Let \(f\) and \(g\) be polynomials over some field \(F\) with deg \(g<\text{deg } f\). The quotient \(g/f\) is called badly approximable if all the partial quotients of the continued fraction of \(g/f\) have degree 1. For a given \(f\), the order of the set \(\{g\in F[x] \mid \text{deg } g<\text{deg } f\) and \(g/f\) is badly approximable\} is called the orthogonal multiplicity of \(f\). In this paper the author gives a characterization of those polynomials with orthogonal multiplicity one: when \(| F| \neq 2\) then \(f\) has orthogonal multiplicity one if and only if \(f=1\), and when \(| F| =2\) then \(f\) has orthogonal multiplicity one if and only if \(f=X^{e_{0}}(X+1)^{e_{1}}\) where \(\binom{e_{0}+e_{1}}{e_{1}}\equiv 1\bmod 2\). This result generalises two results of \textit{H. Niederreiter} [Monatsh. Math. 103, 269-288 (1987; Zbl 0624.12011)]. The author also proves that if \(| F| \geq d(d+1)/2\) where \(d=\text{deg } f\), then \(f\) has positive orthogonal multiplicity.
scientific article; zbMATH DE number 1146322