Effective approximation to complex algebraic numbers by algebraic numbers of bounded degree (Q6582259)

Let \(n\) be an integer \(\geq 2\) and \(\xi\) a complex, non-real algebraic number of degree \(d\geq 2\). An elementary, Liouville type result asserts that there exists a positive, effectively computable number \(c_1(\xi ,n)>0\) such that\N\[\N |\xi -\alpha |\geq c_1(\xi ,n)H(\alpha )^{-d/2}\text{ for every }\alpha\not=\xi\text{ that is algebraic of degree }\leq n.\tag{1}\N\]\NHere, \(H(\alpha )\) denotes the height, i.e., the maximum of the absolute values of the coefficients of the minimal polynomial of \(\alpha\). For \(n\geq d-1\), the exponent on \(H(\alpha )\) in (1) can not be improved, more precisely, there is \(c_2(\xi )>0\) such that the inequality \(|\xi -\alpha |\leq c_2(\xi )H(\alpha )^{-d/2}\) is satisfied by infinitely many algebraic numbers \(\alpha\) of degree \(\leq d-1\). For \(n\leq d-2\) the exponent on \(H(\alpha )\) is not optimal anymore. \textit{Y. Bugeaud} and \textit{J.-H. Evertse} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 8, No. 2, 333--368 (2009; Zbl 1176.11031)] proved that there are \(w(\xi ,n )\in\{ (n+1)/2, (n+2)/2\}\) and \(c_3(\xi ,n, \varepsilon )>0\) such that for every \(\varepsilon >0\),\N\[\N|\xi -\alpha |\geq c_3(\xi ,n ,\varepsilon)H(\alpha )^{-w(\xi ,n)-\varepsilon}\N\]\Nfor every \(\alpha\not=\xi\) algebraic of degree \(\leq n\). Their proof depends on Schmidt's Subspace Theorem, therefore they could prove this only with a not effectively computable \(c_3\).\N\NIn the present paper, the authors obtain in certain special cases some effective improvements of (1), with on \(H(\alpha )\) an exponent slightly larger than \(d/2\). More precisely, in certain cases they obtain inequalities\N\[\N|\xi -\alpha |\geq c_4(\xi ,n)H(\alpha )^{-d/2+\kappa (\xi ,n)}\text{ for every }\alpha\not=\xi\text{ that is algebraic of degree }\leq n,\N\]\Nwith \(c_4(\xi ,n)\) and \(\kappa (\xi ,n)\) positive and effectively computable (but very small). In fact, they obtain such improvements in the following cases: \N\N-- \(\xi\) totally complex, \(n=2\), \(d\geq 4\), and \(\mathbb{Q}(\xi )\) not a CM-field if \(d=4\);\N\N-- \(\xi\) totally complex, \(n=3\), \(d\geq 6\), \(\mathbb{Q}(\xi )\) Galois;\N\N-- \(\xi\) totally complex, \(n=4\), \(d\geq 8\), \(\mathbb{Q}(\xi )\) Galois, and for every Galois conjugate \(\sigma (\xi )\) of \(\xi\), the three numbers \(1\), \(\sigma (\xi )+\overline{\sigma (\xi )}\), \(\sigma (\xi )\cdot \overline{\sigma (\xi )}\) are linearly independent over \(\mathbb{Q}\). \N\NThe authors deduce these results from an effective inequality for norm forms. The latter is proved by combining Baker type inequalities with some ingenious combinatorial arguments.