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Zbl 1210.11074
Pila, Jonathan
Counting rational points on a certain exponential-algebraic surface. (English)
[J] Ann. Inst. Fourier 60, No. 2, 489-514 (2010). ISSN 0373-0956; ISSN 1777-5310/e

Let ${\cal X}=\{(x,y,z)\in (0,1)^3:\log x \log y=-\log z\}.$ Let $F\subset{\mathbb R}$ be a number field of degree $f$ over $\mathbb Q$ and let $\varepsilon>0$. The author proves that there exists a constant $c({\cal X},f,\varepsilon)>0 $ such that $$\#{\cal X}(F,T)\leq c({\cal X},f,\varepsilon)(\log T)^{44+\varepsilon}, $$ where ${\cal X}(F,T)$ denotes the set of $F$-rational points of ${\cal X}$ with height $\leq T $.
[Florin Nicolae (Berlin)]

MSC 2000:: *11G50 Heights
03C64 Model theory of ordered structures
14G05 Rationality questions, rational points
11J91 Transcendence theory of other special functions

Keywords: rational points; exponential-algebraic surface

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