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\(S\)-unit points on analytic hypersurfaces. (English) Zbl 1118.11033
In the present paper, the authors consider points with \(S\)-unit coordinates lying on an analytic hypersurface, i.e., the zero locus of a power series. Let \(K\) be an algebraic number field and \(S\) a finite set of places of \(K\), containing all archimedean places. By a subtorus of \({\mathbf G}_m^n\) we mean an irreducible algebraic subgroup of \({\mathbf G}_m^n\), and by a torus coset in \({\mathbf G}_m^n\) a coset \({\mathbf u}H\) where \(H\) is a subtorus of \({\mathbf G}_m^n\). A special case of a theorem of M. Laurent from 1984 states that if \(\{ {\mathbf x}_h\}_{h=1}^{\infty}\) is a sequence of points in \((K^*)^n\) with \(S\)-unit coordinates and \(f\in K[X_1,\ldots ,X_n]\) is a polynomial such that \(f({\mathbf x}_h)=0\) for \(h=1,2,\ldots\), then \(\{ {\mathbf x}_h\}_{h=1}^{\infty}\) is contained in some finite union \(\bigcup_{i=1}^r {\mathbf u}_iH_i\), where each \({\mathbf u}_iH_i\) is a torus coset of \({\mathbf G}_m^n\) such that \(f\) vanishes identically on \({\mathbf u}_iH_i\).
In the present paper, the authors prove a generalization in which \(f\) is a power series instead of a polynomial. Among other things they prove the following. Let \(K,S\) be as above. For a point \({\mathbf x}=(x_1,\ldots ,x_n)\in K^n\) define \(\| {\mathbf x}\| _v:=\max (| x_1| _v,\ldots ,| x_n| _v)\) for each place \(v\) of \(K\) and \(\hat{h}({\mathbf x})=\sum_{i=1}^n h(x_i)\) where \(h(x)=h(1,x)\) is the absolute logarithmic Weil height of an algebraic number \(x\). Fix a place \(v\) of \(K\) and let \({\mathbf C}_v\) be a completion of an algebraic closure of \(K_v\). View \(K\) as a subfield of \(K_v\). Let \(f({\mathbf X})\in {\mathbf C}_v[[X_1,\ldots ,X_n]]\) be a power series converging in a neighbourhood of \({\mathbf 0}\) of \({\mathbf C}_v^n\). Further, let \(\{ {\mathbf x}_h\}_{h=1}^{\infty}\) be a sequence of points in \((K^*)^n\) with \(S\)-unit coordinates such that \(\log\| {\mathbf x}_h\| _v=O\big(-\hat{h}({\mathbf x}_h)\big)\) as \(h\to\infty\) and such that \(f({\mathbf x}_h)=0\) for \(h=1,2,\ldots\). Then again, \(\{ {\mathbf x}_h\}_{h=1}^{\infty}\) is contained in some finite union \(\bigcup_{i=1}^r {\mathbf u}_iH_i\), where each \({\mathbf u}_iH_i\) is a torus coset of \({\mathbf G}_m^n\) such that \(f\) vanishes identically on \({\mathbf u}_iH_i\).
They deduce this from a more general result which may be viewed as an analogue of a zero lemma of Masser playing an important role in Mahler’s transcendence method. In this general result, \(f({\mathbf X})\) is again a power series in \({\mathbf C}_v[[X_1,\ldots ,X_n]]\) be a power series converging in a neighbourhood of \({\mathbf 0}\) of \({\mathbf C}_v^n\), the points \({\mathbf x}_h\) have “almost \(S\)-unit” coordinates, and the values \(f({\mathbf x}_h)\) need not be zero but lie in \(K\), are “almost \(S\)-integers”, and their heights do not grow to fast as \(h\to\infty\). The conclusion is then that there are finitely many torus cosets \({\mathbf u}_1H_1,\ldots {\mathbf u}_rH_r\) in \({\mathbf G}_m^n\) such that \(\{ {\mathbf x}_h\}_{h=1}^{\infty}\) is contained in \(\bigcup_{i=1}^r {\mathbf u}_iH_i\) and the restriction of \(f\) to \({\mathbf u}_iH_i\) is equal to that of a polynomial. The above results are consequences of the \(p\)-adic Subspace Theorem.

MSC:
11J68 Approximation to algebraic numbers
11J25 Diophantine inequalities
11J81 Transcendence (general theory)
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