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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)
##### Keywords:
S-unit; analytic hypersurface; Subspace Theorem
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##### References:
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