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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

Vol. 18 (2024), No. 7, 1275–1303
DOI: 10.2140/ant.2024.18.1275

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 4 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.

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
Mathematical Subject Classification
Primary: 11L07
Secondary: 03C98, 11F23, 11S40, 14E18
Received: 25 February 2022
Revised: 6 April 2023
Accepted: 3 September 2023
Published: 13 June 2024
Raf Cluckers
Laboratoire Painlevé
Université Lille 1
CNRS - UMR 8524, Cité Scientifique
Villeneuve d’Ascq
Department of Mathematics
KU Leuven
Kien Huu Nguyen
Laboratoire de Mathématiques Nicolas Oresme
Université de Caen Normandie
CNRS - UMR 6139
Department of Mathematics
KU Leuven
Thang Long Institute of Mathematics and Applied Sciences

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