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Combining Igusa's
conjectures on exponential sums and monodromy with
semicontinuity of the minimal exponent
Raf Cluckers and Kien Huu Nguyen |
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Vol. 18 (2024), No. 7, 1275–1303
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DOI: 10.2140/ant.2024.18.1275
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Abstract |
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We combine two of Igusa’s conjectures with recent semicontinuity results by Mustaţă and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in terms of degrees and dimensions only. We provide evidence consisting partly of adaptations of already known results about Igusa’s conjecture on exponential sums, but also some new evidence like for all polynomials in up to variables. We show that, in turn, these bounds imply consequences for Igusa’s (strong) monodromy conjecture. The bounds are related to estimates for major arcs appearing in the circle method for local-global principles. |
Keywords
Igusa's conjectures on monodromy and exponential sums,
local-global principles, circle method, major arcs, minimal
exponent, motivic oscillation index, log canonical
threshold, Igusa's local zeta functions, motivic and
$p$-adic integration, log resolutions
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Mathematical Subject Classification
Primary: 11L07
Secondary: 03C98, 11F23, 11S40, 14E18
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Milestones
Received: 25 February 2022
Revised: 6 April 2023
Accepted: 3 September 2023
Published: 13 June 2024
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Authors |
| Raf Cluckers | ||
| Laboratoire Painlevé Université Lille 1 CNRS - UMR 8524, Cité Scientifique Villeneuve d’Ascq France |
Department of Mathematics KU Leuven Belgium |
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| Kien Huu Nguyen | ||
| Laboratoire de Mathématiques Nicolas
Oresme Université de Caen Normandie CNRS - UMR 6139 Caen France |
Department of Mathematics KU Leuven Belgium |
Thang Long Institute of Mathematics
and Applied Sciences Hanoi Vietnam |
| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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