Hyperquadratic power series in \(\mathbb F_3((T^{-1}))\) with partial quotients of degree 1
Let \(m\) be a positive integer, \(\mathbb F_3\) be the finite field of characteristic \(p\) with \(3\) elements and \(\mathbb F(3)\) be the set of power series \(\alpha =\sum_{k\leq k_0} u_kT^k\) where \(k_0\) is integer, \(u_k\in \mathbb F_3\) and \(u_{k_0}\not= 0\). Set \(\eta =(\eta_1,\dots ,\eta_m)\) where \(\eta_1,\dots ,\eta_m\) are non-zero elements from \(\mathbb F_3\) and \(\eta_m=(-1)^{m-1}\). Set also \(\kappa =(k_1,\dots ,k_m)\) where \(k_1,\dots ,k_m\) are integers such that \(k_1\geq 2\) and \(k_{i+1}-k_i\geq 2\) for all \(i=1,\dots ,m-1\). Let us define \(t_{i,n}=k_m(3^n-1)/2+k_i3^n\) for all \(1\leq i\leq m\) and \(n\geq 0\). Set \(\lambda_t=1\) if \(1\leq t\leq t_{1,0}\), \(\lambda_t=(-1)^{mn+i}\) if \(t_{i,n} < t\leq t_{i+1,n}\) for \(1\leq i<m\) and \(\lambda_t=(-1)^{m(n+1)}\) if \(t_{m,n} < t\leq t_{1,n+1}\). Set also \(\mu_t=(-1)^{m(n+1)}\eta_i\) if \( t= t_{i,n}\) or \( t= t_{i,n}+1\), and \(\mu_t=0\) otherwise. Let \(\omega(m,\eta ,\kappa)\in\mathbb F(3)\) be defined by infinite continued fraction expansion \(\omega(m,\eta ,\kappa)=[a_1,\dots ]\) where \(a_n=\lambda_nT+\mu_n\) for all \(n\geq 1\). Then the authors prove that \(\omega(m,\eta ,\kappa)\) is the unique root in \(\mathbb F(3)\) of the quartic equation \[ X=\frac{x_{1+k_m}X^3+(-1)^{m-1}x_{k_m-2}}{y_{1+k_m}X^3+(-1)^{m-1}y_{k_m-2}} \] with \(| \omega(m,\eta ,\kappa)| >1\) where \(\{x_n\}_{n\geq 1}\) and \(\{y_n\}_{n\geq 1}\) are numerators and denominators of partials of \(\omega(m,\eta ,\kappa)\) respectively.
scientific article; zbMATH DE number 6348207