Igusa’s conjecture for exponential sums

Let $f\in\mathbb{Z}[x_1,\ldots,x_n]$ be nonconstant, and define the normalized exponential sum

E_f(p^m)=p^{-nm}\sum_{x\in(\mathbb{Z}/p^m\mathbb{Z})^n}\exp\!\left(\frac{2\pi i f(x)}{p^m}\right),

where $p$ is prime and $m\geq 1$ . If $\operatorname{lct}(f)$ denotes the log-canonical threshold of the divisor $f=0$ , then there should exist a constant $C_f>0$ such that, uniformly in $p$ and $m$ ,

\left|E_f(p^m)\right|\leq C_f(1+m)^{n-1}p^{-m\operatorname{lct}(f)}.

Equivalently, the decay exponent of the normalized exponential sums should be the log-canonical-threshold exponent, up to the customary polynomial factor in $m$ .

References

Primary source

arXiv

Additional references

Uniform estimate of Lang-Weil type for counting points of schemes over finite rings — arXiv — Kien Huu Nguyen

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed progress

The conjecture remains open in general, while a new paper reports a uniform counting method that advances several predicted bounds.

Igusa’s conjecture predicts uniform decay estimates for exponential sums over varying primes and prime powers. Its full general form remains unsettled.

Known results

Cluckers et al. (2018) proved generalized bounds governed by a log-canonical-threshold exponent and established optimality for important non-rational-singularity cases.
Earlier work proved the modulo $p$ and modulo $p^2$ cases under stated hypotheses.
Special cases include nondegenerate, homogeneous, and quasi-homogeneous exponential sums.
A 2021 result gives exponent $\frac{n-s}{2(d-1)}$ in a degree- $d$ setting, while the expected exponent $\frac{n-s}{d}$ remains open.

October 2026 finite-ring counting advance

Kien Huu Nguyen’s paper reports uniform point-count estimates over $\mathbb{Z}/p^m\mathbb{Z}$ using poles of Igusa local zeta functions, yielding corresponding progress toward the conjectured exponential-sum decay across singularity classes. It does not claim the full conjecture.

Current status (as of October 2026): The full general conjecture remains open; established results cover several singularity classes and special cases, while Nguyen’s finite-ring counting advance is claimed progress and remains unverified.

Sources

No solutions have been posted yet.