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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 $$T\rightarrow \infty$$.

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