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Combining Igusa's conjectures on exponential sums and monodromy with semi-continuity of the minimal exponent

Raf Cluckers, Kien Huu Nguyen

Abstract

We combine two of Igusa's conjectures with recent semi-continuity 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.

Combining Igusa's conjectures on exponential sums and monodromy with semi-continuity of the minimal exponent

Abstract

We combine two of Igusa's conjectures with recent semi-continuity 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.

Paper Structure

This paper contains 6 sections, 14 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Generalization to a ring of integers
  3. Link with the monodromy conjecture
  4. Some first evidence
  5. The non-degenerate case
  6. The four variable case

Key Result

Corollary 2.2

For $p$, $f$, $T_0$ and $t_0$ as above, if $T_0$ is furthermore a pole of $(T-1/p)P_{f,p}(T)$, then for infinitely many $m$ and $\xi$ and a constant $c'_p$ independent of $m$, $\xi$.

Theorems & Definitions (35)

  • Conjecture 1
  • Remark 1.2
  • Remark 1.3
  • Corollary 2.2: Igusa2
  • proof
  • Proposition 2.3: Strong Monodromy Conjecture, in a range
  • proof : Proof of Proposition \ref{['prop:mon']}
  • Proposition 2.4
  • proof
  • Remark 2.6
  • ...and 25 more