The authors consider a field of formal power series \(\mathbb{K}\) which is constructed as follows:
– let \(K\) be a finite field with \(q\) elements,
– let \(K[x]\) be the ring of polynomials with coefficients in \(K\) with coefficients in\(K\),
– let \(K(x)\) be the quotient field of \(K[x]\),
– define a non-Archimedean valuation \(|\cdot |\) on \(K(x)\) by \[|0|=0\hbox{ and }\left|\frac{a(x)}{b(x)}\right|=q^{\mathrm{deg}(a)-\mathrm{deg}(b)},\]
(\(a(x), b(x)\) are non-zero polynomials in \(K[x], \mathrm{deg}(\cdot )\) denotes the degree of a polynomial)
– then \(\mathbb{K}\) is the completion of \(K(x)\) with respect to \(|\cdot |\) (the absolute value \(|\cdot|\) is uniquely extended, using the same notation).
An element of \(\mathbb{K}\) is called algebraic/transcendental if it is algebraic/transcendental over \(K(x)\). K. Mahler [J. Reine Angew. Math. 166, 137–150 (1932; JFM 58.0207.01)] classified real numbers and separated transcendental reals into disjoint classes \(S\)-, \(T\)- and \(U\)-numbers and P. Bundschuh [J. Reine Angew. Math. 299/300, 411–432 (1978; Zbl 0367.10032)] extended this classification to \(\mathbb{K}\).
Bundschuh also introduced for a transcendental element \(\xi\in\mathbb{K}\) the quantities \(w_n(\xi)\) for \(n\in\mathbb{N}\) and \(\mu(\xi)\) being the smallest integer such that \(w_n(\xi)\) is infinite and \(\mu(\xi)=\infty\) if \(w_n(\xi)\) is always finite. The number \(\mu(\xi)\) is used to define the \(S\)-, \(T\)- and \(U\) numbers. For instance: \(\xi\) is a \(U\)-number if \(w(\xi)=\infty\) and \(\mu(\xi)<\infty\) and a \(U_m\) number if \(w(\xi)=\infty\) and \(\mu(\xi)=m\).
The authors now define continued fractions in the field \(\mathbb{K}\), using the absolute value \(|\cdot |\) and then conclude their section §1 (Introduction) by stating the results:
Theorem 1.1. Let \(\alpha\) be an algebraic formal power series with \(|\alpha|\geq 1,\hbox{ deg}\,(\alpha)=m>1\) and continued fraction expansion \[\alpha=\langle a_0,a_1,a_2,\ldots\rangle .\tag{*}\] Let \(\{r_j\}_{j=0}^{\infty}\) and \(\{s_j\}_{j=0}^{\infty}\) be two infinite sequences of non-negative rational integers satisfying \[0=r_0<s_0<r_1<s_1<r_2<s_2<\cdots \hbox{ and }r_{n+1}-s_n\geq 2.\] Define by \(p_n/q_n\) the \(n\)th convergent of the continued fraction of \((\ast)\) and assume that \[(a)\ \lim_{n\rightarrow\infty}\,\frac{\log{|q_{s_n}|}}{\log{|q_{r_n|}}}=\infty,\quad (b)\ \limsup_{n\rightarrow\infty}\,\frac{\log{|q_{r_{n+1}}|}}{\log{|q_{s_n|}}}<\infty.\] Define \(b_j\in K[x]\ (j=0,1,\ldots)\) by \[b_j=\begin{cases}a_j,&\ r_n\leq j\leq s_n\ (n=0,1,\ldots),\\ \nu_j,&\ s_n<j<r_{n+1} (n=0,1,\ldots),\end{cases}\] where \(\nu_j\in K[x]\) with \(1<|\nu_j|\leq S_1 |a_j|^{S_2}\) are such that \(\sum_{j=s_{n+1}}^{r_{n+1}-1}\,|a_j-\nu_j|\not= 0\) and \(S_1\) and \(S_2\) are fixed positive rational integers. Then the formal power series \(\xi\in \mathbb{K}\) with continued fraction expansion \[\xi=\langle b_0,b_1,b_2,\ldots \rangle\] is a \(U_m\) number.
Theorem 1.2. Let \(L\) be a finite extension of degree \(m\) over \(K(x)\) and \(\alpha_1,\alpha_2,\ldots,\alpha_k\) be in \(L\). Let \(\eta\in\hat{\mathbb{K}}\) be algebraic over \(K(x)\). Assume that \(F(\eta,\alpha_1,\ldots,\alpha_k)=0\), where \(F(y,y_1,\ldots,y_k)\) is a polynomial in \(y,y_1,\ldots,y_k\) over \(K[x]\) with degree at least one in \(y\).
Then: \[\hbox{deg}\,(\eta)\leq dm\] and \[H(\eta)\leq H^mH(\alpha_1)^{\ell_1 m}\cdots H(\alpha_k)^{\ell_k m},\] where \(d\) is the degree of \(F(y,y_1,\ldots,y_k)\) and \(H\) is the maximum of the absolute values of the coefficients of \(F(y,y_1,\ldots,y_k)\).
After this Introduction section, follows §2. Auxiliary results, §3. Proof of Theorem 1.2, §4. Proof of Theorem 1.1 and the References (\(15\) items).
It is quite interesting to see that the authors have succeeded, so many years after Mahler’s classification of transcendental reals, to extend the concept and give really new results for an extension of reals in the setting of an absolute value derived from a non-Archimedean valuation.