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Counting rational points on algebraic varieties. (English) Zbl 1098.14013
Let \(V\) be a geometrically integral subvariety of \(\mathbb{P}^n\), \(n\geq 2\), of dimension \(m\) and degree \(d\geq 2\). The authors conjecture the upper estimate \(N_V(B)=O_{\varepsilon,n,d}(B^{m+\varepsilon})\) to hold for any \(\varepsilon>0\). Here \[ N_V(B):=\text{card}\{{\mathbf a}| {\mathbf a}\in V(\mathbb{Z}),h({\mathbf a})\leq B \} \] is the counting function and \[ h({\mathbf a}):=\max\bigl\{|a_i|\mid 0\leq i\leq n\} \] for \({\mathbf a}\in \mathbb{P}^n(\mathbb{Z})\), \({\mathbf a}=(a_0,a_1,\dots,a_n)\) with h.c.f. \((a_0,a_1, \dots,a_n)=1\). This conjecture is proved to be equivalent to an earlier conjecture of Heath-Brown’s. Therefore it follows from the works of Broberg, Browning, Heath-Brown, and Salberger that the authors’ conjecture holds true if \(m\leq 3\) or if \(V\) is a quadric (that is, if \(d=2)\). In their work reviewed here, the authors prove the conjecture for \(d\geq 6\) and improve on Pila’s estimate \(N_V(B)=O_{\varepsilon,n,d} (B^{m+1/d+\varepsilon}),\varepsilon>0\), for any \(d\geq 3\).
Reviewer: B. Z. Moroz (Bonn)

MSC:
14G05 Rational points
11G35 Varieties over global fields
11D45 Counting solutions of Diophantine equations
11D68 Rational numbers as sums of fractions
11G50 Heights
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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