# An Incidence Theorem in Higher Dimensions

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## Abstract

We prove almost tight bounds on the number of incidences between points and *k*-dimensional varieties of bounded degree in **R**^{d}. Our main tools are the polynomial ham sandwich theorem and induction on both the dimension and the number of points.

### Keywords

Szemerédi–Trotter type incidence theorems Sum-product bounds## Notes

### Acknowledgements

The authors are very grateful to Boris Bukh, Nets Katz, Jordan Ellenberg, and Josh Zahl for helpful discussions and to the Isaac Newton Institute, Cambridge for hospitality while this research was being conducted. We also thank Alex Iosevich, Izabella Łaba, Jiří Matoušek, and János Pach for comments, references, and corrections to an earlier draft of this manuscript. We are thankful for the referee for the careful reading and for the suggestions to improve the readability of the paper. The first author is supported by an NSERC grant and the second author is supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award.

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