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Rational points on a subanalytic surface. (English) Zbl 1121.11032
This paper is primarily concerned with the distribution of rational points on subanalytic sets in $$\mathbb R^n$$. For the definitions of subanalytic sets, semianalytic sets (a smaller class), and semialgebraic sets, the author refers to E. Bierstone and P. D. Milman [Publ. Math., Inst. Hautes Étud. Sci. 67, 5–42 (1988; Zbl 0674.32002)] and J. Pila [Q. J. Math. 55, 207–223 (2004; Zbl 1111.32004)].
For a subanalytic set $$X\subseteq\mathbb R^n$$, let $$X^a$$ denote the union of all connected semialgebraic subsets of $$X$$ of positive dimension, and let $$X^t=X-X^a$$ be its complement.
For points $$(a_1/b_1,\dots,a_n/b_n)\in\mathbb Q^n$$ with $$a_i,b_i\in\mathbb Z$$, $$b_i>0$$, and $$\gcd(a_i,b_i)=1$$ for all $$i$$, we define a height $$H(a_1/b_1,\dots,a_n/b_n)=\max\{| a_i| ,b_i:1=1,\dots,n\}$$. Note that this height is not the same as the height obtained by embedding into projective space. For sets $$X\subseteq\mathbb R^n$$ and bounds $$B\in\mathbb R$$, we then define $$N(X,B)$$ to be the cardinality of the set $$\{P\in X(\mathbb Q):H(P)\leq B\}$$.
It was conjectured in Pila, op. cit., that if $$X\subseteq\mathbb R^n$$ is a compact subanalytic set then $$N(X,B)\leq c(X,\epsilon)B^\epsilon$$ for all $$\epsilon>0$$ and all $$B\geq1$$.
This paper proves this conjecture in the case $$\dim X=2$$. The proof works by showing that the rational points in question lie on very few intersections of $$X$$ with hypersurfaces of suitable degree. Thus the problem becomes one of obtaining suitably uniform estimates for rational points on such intersections, which are semianalytic curves.
The paper also gives a bound for rational points on algebraic curves. Let $$b,c\geq2$$ be integers and let $$d=\max(b,c)$$. Let $$F\in\mathbb R[x,y]$$ be a polynomial of bidegree $$(b,c)$$ and let $$X$$ be the algebraic curve $$\{P\in\mathbb R^2:F(P)=0\}$$. Then $$N(X,B)\leq (8d^2)^{2d+7}B^{2/d}(\log B)^{2d+4}$$ for all $$B\geq3$$.

##### MSC:
 11D99 Diophantine equations 14G05 Rational points 14P15 Real-analytic and semi-analytic sets 32B20 Semi-analytic sets, subanalytic sets, and generalizations 11G50 Heights
##### Keywords:
subanalytic set; semianalytic set; rational point; height
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