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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
##### Keywords:
heights; integral algebraic variety
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##### References:
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