Continued Fractions of Quadratic Laurent Series over \(\mathbb{F}_q\): Non-Archimedean Error Equalities, the Quasi-Period, and the Polynomial Pell Equation

Abstract

We transplant to \(\mathbb{F}_q[x]\) the augmented-matrix elementary row operation (ERO) framework for continued fractions of rational numbers, and extend it there from the terminating case to quadratic irrational Laurent series in \(\mathbb{F}_q((1/x))\), whose expansion is infinite. The state of the algorithm is the vector \(v_k=(\sqrt D+m_k,Q_k)^{\!\top}\), one row operation followed by a row swap sends \(v_k\) to \(-\overline{\alpha_{k+1}}\,v_{k+1}\), and we show that the accumulated scalar of the process is exactly \((p_k+q_k\sqrt D)^{-1}\): the machine therefore returns not only the convergents but the associated algebraic integers of \(\mathcal O=\mathbb{F}_q[x][\sqrt D\,]\). Reading the construction at the infinite place gives the exact non-archimedean error equality \(|\alpha-p_k/q_k|=q^{-(\deg q_k+\deg q_{k+1})}\) for the infinite expansion, whence every convergent is a strict best rational approximation for the degree valuation. From the determinant invariant we obtain a norm form, and we recover within the framework the classical facts that the expansion of \(\sqrt D\) is periodic from the first partial quotient onwards with palindromic period, and that the fundamental solution of the polynomial Pell–Abel equation \(P^2-DQ^2\in\mathbb{F}_q^\times\) is the convergent at the end of the quasi-period \(n\) — not of the period \(\ell\), from which \(n\) differs for a substantial proportion of \(D\) in every range we have computed — the regulator being \(R=d+\deg q_{n-1}\), i.e. minus the degree of the accumulated scalar. Periodicity and the distinction between period and quasi-period, unlike the error equality, are genuinely finite-field phenomena. What the construction offers is uniformity of derivation rather than new theorems: all of this is classical and is recovered rather than claimed, including the determination, for \(\deg D=2\), of exactly how often the period exceeds the quasi-period, the proportion being \((q-2)/(q-1)\).