Paucity phenomena for polynomial products
Wang V, Xu MW. 2024. Paucity phenomena for polynomial products. Bulletin of the London Mathematical Society.
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Wang, VictorISTA;
Xu, Max Wenqiang
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Abstract
Let P(x)∈Z[x] be a polynomial with at least two distinct complex roots. We prove that the number of solutions (x1,…,xk,y1,…,yk)∈[N]2k to the equation
∏1≤i≤kP(xi)=∏1≤j≤kP(yj)≠0
(for any k≥1 ) is asymptotically k!Nk as N→+∞. This solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums 1N√∑n≤Nf(P(n)) match standard complex Gaussian moments as N→+∞
, where f is the Steinhaus random multiplicative function.
Publishing Year
Date Published
2024-05-31
Journal Title
Bulletin of the London Mathematical Society
Acknowledgement
We thank Oleksiy Klurman, Ilya Shkredov, and Igor Shparlinski for helpful comments on earlier versions of the paper, and thank Yotam Hendel for providing a reference for Lemma 2.1. We also thank the anonymous referee for their generous corrections and comments. The first author has received funding from the European Union's Horizon 2020 Research and Innovation Program under the Marie Skłodowska-Curie Grant Agreement Number: 101034413. The second author is partially supported by the Cuthbert C. Hurd Graduate Fellowship in the Mathematical Sciences, Stanford.
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Cite this
Wang V, Xu MW. Paucity phenomena for polynomial products. Bulletin of the London Mathematical Society. 2024. doi:10.1112/blms.13095
Wang, V., & Xu, M. W. (2024). Paucity phenomena for polynomial products. Bulletin of the London Mathematical Society. London Mathematical Society. https://doi.org/10.1112/blms.13095
Wang, Victor, and Max Wenqiang Xu. “Paucity Phenomena for Polynomial Products.” Bulletin of the London Mathematical Society. London Mathematical Society, 2024. https://doi.org/10.1112/blms.13095.
V. Wang and M. W. Xu, “Paucity phenomena for polynomial products,” Bulletin of the London Mathematical Society. London Mathematical Society, 2024.
Wang V, Xu MW. 2024. Paucity phenomena for polynomial products. Bulletin of the London Mathematical Society.
Wang, Victor, and Max Wenqiang Xu. “Paucity Phenomena for Polynomial Products.” Bulletin of the London Mathematical Society, London Mathematical Society, 2024, doi:10.1112/blms.13095.
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