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On the algebraic points of a definable set. (English) Zbl 1218.11068
In a previous paper with A. J. Wilkie [“The rational points of a definable set”, Duke Math. J. 133, No. 3, 591–616 (2006; Zbl 1217.11066)], the author proved that if \(X\subset {\mathbb R}^n\) is definable in a given o-minimal structure over \({\mathbb R}\), then for any \(\varepsilon>0\) the counting function \[ N(X\setminus X^{\mathrm{alg}},T)=\# \{ x\in X\setminus X^{\mathrm{alg}}({\mathbb Q}) \mid H(x)\leq T\} \] satisfies \(N(X\setminus X^{\mathrm{alg}},T)=O_{X,\varepsilon} ( T^\epsilon)\) for \(T\geq 1\). Here, the result is adapted to estimating points on \(X\) with coordinates of bounded degrees. Given a positive integer \(k\), define \[ X(k,T)= \{ (x_1,\dots,x_n)\in X \mid [{\mathbb Q}(x_i):{\mathbb Q}]\leq k;\; \max_{1\leq i\leq n} H(x_i)\leq T\}. \] If \(X\subset {\mathbb R}^n\) is definable, \(k\) is a positive integer and \(\varepsilon>0\), then \[ \# \bigl(X\setminus X^{\mathrm{alg}}(k,T)\bigr) =O_{X,k,\epsilon} (T^\varepsilon). \] Further strengthenings of this theorem yield a result for points on \(X\) whose coordinates lie in some finite dimensional \({\mathbb Q}\)-vector subspace of \({\mathbb R}\) and also to a result on a Pfaff curve defined over a fixed number field \(K\subset {\mathbb R}\) of degree \(k\); in the latter case the bound \(O(T^\varepsilon)\) is refined into \(O_{X,k}(\log T) ^{O_X(1)}\) for \(T\rightarrow \infty\).

11G99 Arithmetic algebraic geometry (Diophantine geometry)
03C64 Model theory of ordered structures; o-minimality
11U09 Model theory (number-theoretic aspects)
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