Multivariate Igusa theory: Decay rates of exponential sums
| dc.creator | Cluckers, Raf | |
| dc.date | 2003-06-24 | |
| dc.date | 2004-08-10 | |
| dc.date.accessioned | 2026-07-07T04:59:09Z | |
| dc.date.available | 2026-07-07T04:59:09Z | |
| dc.description | We obtain general estimates for exponential integrals of the form \[ E_f(y)=\int_{\mathbb{Z}_{p}^{n}}ψ(\sum_{j=1}^r y_j f_j(x))|dx|, \] where the $f_j$ are restricted power series over $\mathbb{Q}_p$, $y_j\in\mathbb{Q}_p$, and $ψ$ a nontrivial additive character on $\mathbb{Q}_p$. We prove that if $(f_1,...,f_r)$ is a dominant map, then $|E_f(y)| < c|y|^α$ for some $c>0$ and $α<0$, uniform in $y$, where $|y|=\max(|y_i|)_i$. In fact, we obtain similar estimates for a much bigger class of exponential integrals. To prove these estimates we introduce a new method to study exponential sums, namely, we use the theory of $p$-adic subanalytic sets and $p$-adic integration techniques based on $p$-adic cell decomposition. We compare our results to some elementarily obtained explicit bounds for $E_f$ with $f_j$ polynomials. | |
| dc.description | Improved results and presentation | |
| dc.identifier | https://arxiv.org/abs/math/0306351 | |
| dc.identifier | http://arxiv.org/abs/math/0306351 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/67870 | |
| dc.subject | Number Theory | |
| dc.subject | Logic | |
| dc.subject | 11L07, 11U09, 32B20; 11L05, 11S80, 32P05, 32B20, 03C10 | |
| dc.title | Multivariate Igusa theory: Decay rates of exponential sums | |
| dc.type | text |