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.
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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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