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Integral solutions to
systems of diagonal equations
Nick Rome and Shuntaro Yamagishi |
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Vol. 340 (2026), No. 1, 179–198
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Abstract |
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We obtain an asymptotic formula for the number of integral solutions to a system of diagonal equations. We obtain an asymptotic formula for the number of solutions with variables restricted to smooth numbers as well. We improve the required number of variables compared to previous results by incorporating recent progress on Waring’s problem and the resolution of the main conjecture in Vinogradov’s mean value theorem. |
Keywords
Hardy–Littlewood method, circle method, diagonal equation,
Waring's problem, exponential sum
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Mathematical Subject Classification
Primary: 11P55
Secondary: 11D45, 11D72, 11P05
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Milestones
Received: 18 September 2024
Revised: 16 September 2025
Accepted: 18 September 2025
Published: 31 October 2025
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Authors |
| Nick Rome | |
| Institute of Analysis and Number
Theory TU Graz Graz Austria |
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| Shuntaro Yamagishi | |
| IST Austria Klosterneuburg Austria |
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| © 2026 MSP (Mathematical Sciences Publishers). |
