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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
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