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## G. Binyamini - Point counting for foliations over number fields

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BINYAMINI Gal

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Institut Fourier
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### G. Binyamini - Point counting for foliations over number fields

We consider an algebraic \$V\$ variety and its foliation, both defined over a number field. Given a (compact piece of a) leaf \$L\$ of the foliation, and a subvariety \$W\$ of complementary codimension, we give an upper bound for the number of intersections between \$L\$ and \$W\$. The bound depends polynomially on the degree of \$W\$, the logarithmic height of \$W\$, and the logarithmic distance between \$L\$ and the locus of points where leafs of the foliation intersect \$W\$ improperly.

Using this theory we prove the Wilkie conjecture for sets defined using leafs of foliations under a certain assumption about the algebraicity locus. For example, we prove the if none of the leafs contain algebraic curves then the number of algebraic points of degree \$d\$ and log-height \$h\$ on a (compact piece of a) leaf grows polynomially with \$d\$ and \$h\$. This statement and its generalizations have many applications in diophantine geometry following the Pila-Zannier strategy.

I will focus mostly on the proof of the main statement, which uses a combination of differential-algebraic methods related to foliations with some ideas from complex geometry and value distribution theory. If time permits I will briefly discuss the applications to counting algebraic points and diophantine geometry at the end.

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