# Power-free values of polynomials on symmetric varieties

Browning TD, Gorodnik A. 2017. Power-free values of polynomials on symmetric varieties. Proceedings of the London Mathematical Society. 114(6), 1044–1080.

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Browning, Timothy D

^{IST Austria}^{}; Gorodnik, AlexanderAbstract

Given a symmetric variety Y defined over Q and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y . We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics defined over Q.

Publishing Year

Date Published

2017-06-01

Journal Title

Proceedings of the London Mathematical Society

Acknowledgement

While working on this paper the authors were supported by ERC grants 306457 and 239606, respectively.

Volume

114

Issue

6

Page

1044 - 1080

IST-REx-ID

### Cite this

Browning TD, Gorodnik A. Power-free values of polynomials on symmetric varieties.

*Proceedings of the London Mathematical Society*. 2017;114(6):1044-1080. doi:10.1112/plms.12030Browning, T. D., & Gorodnik, A. (2017). Power-free values of polynomials on symmetric varieties.

*Proceedings of the London Mathematical Society*. Wiley Blackwell. https://doi.org/10.1112/plms.12030Browning, Timothy D, and Alexander Gorodnik. “Power-Free Values of Polynomials on Symmetric Varieties.”

*Proceedings of the London Mathematical Society*. Wiley Blackwell, 2017. https://doi.org/10.1112/plms.12030.T. D. Browning and A. Gorodnik, “Power-free values of polynomials on symmetric varieties,”

*Proceedings of the London Mathematical Society*, vol. 114, no. 6. Wiley Blackwell, pp. 1044–1080, 2017.Browning, Timothy D., and Alexander Gorodnik. “Power-Free Values of Polynomials on Symmetric Varieties.”

*Proceedings of the London Mathematical Society*, vol. 114, no. 6, Wiley Blackwell, 2017, pp. 1044–80, doi:10.1112/plms.12030.**All files available under the following license(s):**

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