PreprintarXiv licencearXiv
On Igusa's conjectures for weighted homogeneous polynomials with isolated singularities
Abstract
In 1991, Denef introduced an explicit formula for Igusa's local zeta functions in terms of embedded resolutions of singularities over p-adic fields with good reduction. In this paper, by slightly generalizing the p-adic McKay correspondence given by the second author in 2017, we extend Denef's result to the setting of ℚ-embedded resolutions of singularities. Consequently, the strong monodromy conjecture with respect to Igusa's local zeta functions holds for weighted homogeneous polynomials with isolated singularities. Moreover, we use our formula to prove Igusa's conjecture for exponential sums in the case of non-degenerate weighted homogeneous polynomials with isolated singularities.
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BibTeX
@misc{nguyen2026igusas,
title = {{On Igusa's conjectures for weighted homogeneous polynomials with isolated singularities}},
author = {Kien Huu Nguyen and Takehiko Yasuda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12222},
eprint = {2610.12222},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12222},
}
RIS
TY - UNPB
TI - On Igusa's conjectures for weighted homogeneous polynomials with isolated singularities
AU - Kien Huu Nguyen
AU - Takehiko Yasuda
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12222
UR - https://arxiv.org/abs/2610.12222
AB - In 1991, Denef introduced an explicit formula for Igusa's local zeta functions in terms of embedded resolutions of singularities over p-adic fields with good reduction. In this paper, by slightly generalizing the p-adic McKay correspondence given by the second author in 2017, we extend Denef's result to the setting of ℚ-embedded resolutions of singularities. Consequently, the strong monodromy conjecture with respect to Igusa's local zeta functions holds for weighted homogeneous polynomials with isolated singularities. Moreover, we use our formula to prove Igusa's conjecture for exponential sums in the case of non-degenerate weighted homogeneous polynomials with isolated singularities.
ER -