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An asymptotic formula for integer points on Markoff-Hurwitz varieties.

Gamburd, Alex and Magee, Michael and Ronan, Ryan (2019) 'An asymptotic formula for integer points on Markoff-Hurwitz varieties.', Annals of Mathematics, 190 (3). 751-809 .

Abstract

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation x21+x22+⋯+x2n=ax1x2⋯xn+k. When n≥4, the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent β that is not in general an integer when n≥4. We give a new interpretation of this exponent of growth in terms of the unique parameter for which there exists a certain conformal measure on projective space.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.4007/annals.2019.190.3.2
Date accepted:16 July 2019
Date deposited:31 July 2019
Date of first online publication:28 October 2019
Date first made open access:01 August 2019

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