Simultaneous approximation in positive characteristic
The authors generalise Diophantine approximation results of Voloch and de Mathan in positive characteristic to higher dimensions. Let \(k\) be field of characteristic \(p>0\) and put \(R=k[X]\), \(K=k(X)\), \({\mathcal K}=k((X^{-1}))\). We endow \({\mathcal K}\) with a non-archimedean absolute value \(|\cdot |\), such that for \(\alpha =\sum_{n=t}^{\infty} a_nX^{-n}\in {\mathcal K}\) with \(a_t\not=0\) we have \(|\alpha |= p^t\). Let \({\mathbf a}=(\alpha_0,\ldots,\alpha_n)\in{\mathcal K}^{n+1}\) be such that its coordinates are linearly independent over \(K\). Further assume that \(\alpha_0,\ldots, \alpha_n\) are algebraic over \(K\) and that \([K(\alpha_1/\alpha_0 ,\ldots, \alpha_n/\alpha_0):K ]=d\). For \({\mathbf P}=(P_0,\ldots,P_n)\in \mathbb{R}^{n+1}\) put \(|{\mathbf P}|:=\max (|P_0|,\ldots,|P_n|)\). Put \(|{\mathbf a}\wedge {\mathbf P}|:= \max_{0\leq i<j\leq n} |\alpha_iP_j-\alpha_iP_j|\) and \(|{\mathbf a}\cdot {\mathbf P}|:= |\alpha_0P_0+\cdots +\alpha_nP_n|\). Let \(e_s({\mathbf a})\) be the supremum of those reals \(e\) such that the inequality \(|{\mathbf a}\wedge {\mathbf P}|\leq |{\mathbf P}|^{-e}\) has solutions \({\mathbf P}\in \mathbb{R}^{n+1}\) with arbitrarily large values of \(|{\mathbf P}|\). By analogues of Dirichlet's theorem and Liouville's theorem, one has \(1/n\leq e_s({\mathbf a})\leq d-1\). Define \(B_s({\mathbf a})\) to be the infimum of all reals \(B\) such that \(|{\mathbf a}\wedge {\mathbf P}|\leq B|{\mathbf P}|^{-e_s(\alpha)}\) has solutions \({\mathbf P}\in \mathbb{R}^{n+1}\) with arbitrarily large values of \(|{\mathbf P}|\); put \(B_s({\mathbf a})=\infty\) if such \(B\) does not exist. Similarly, let \(e_l({\mathbf a})\) be the supremum of those reals \(e\) such that \(|{\mathbf a}\cdot {\mathbf P}|\leq |{\mathbf P}|^{-e}\) has solutions \({\mathbf P}\in \mathbb{R}^{n+1}\) with arbitrarily large values of \(|{\mathbf P}|\). Then \(n\leq e_l(\alpha)\leq d-1\). Let \(B_l({\mathbf a})\) be the infimum of those reals \(B\) such that \(|{\mathbf a}\cdot {\mathbf P}|\leq B|{\mathbf P}|^{-e_l({\mathbf a})}\) has solutions \({\mathbf P}\in \mathbb{R}^{n+1}\) with arbitrarily large values of \(|{\mathbf P}|\) and put \(B_l({\mathbf a})=\infty\) if such \(B\) does not exist. Put \({\mathbf a}^{(q)}=(\alpha_0^q,\ldots,\alpha_n^q)\). In the present paper, the authors consider tuples \({\mathbf a}=(\alpha_0,\ldots,\alpha_n)\in{\mathcal K}^{n+1}\) which satisfy, in addition to the condition that \(\alpha_0,\ldots,\alpha_n\) be linearly independent over \(K\), an equation of the shape \({\mathbf a }=\lambda A{\mathbf a}^{(q)}\) with \(A\in GL_{n+1}(K)\), \(\lambda\in{\mathcal K}\). Their main result states that for such tuples one has \(0< B_s({\mathbf a}), B_l({\mathbf a})<\infty\). Moreover, they deduce explicit upper and lower bounds for \(B_s({\mathbf a})\), \(B_l({\mathbf a})\). This generalises results obtained earlier by \textit{J. Voloch} [Monatsh. Math. 119, 321-325 (1995; Zbl 0827.11039)] and \textit{B. de Mathan} [Acta Arith. 60, 359-370 (1992; Zbl 0763.11048)] for \(n=1\).
Diophantine approximation in positive characteristic
scientific article; zbMATH DE number 1569503