Effective Hironaka resolution and its Complexity (with appendix on applications in positive characteristic)

TL;DR

This paper provides an a priori complexity estimate for the simplified Hironaka resolution algorithm and demonstrates the existence of canonical desingularization in large characteristic fields, extending prior foundational work.

Contribution

It introduces a complexity bound for the Hironaka algorithm and establishes canonical resolution results over fields with large characteristic.

Findings

Complexity estimate for simplified Hironaka algorithm

Existence of canonical desingularization in large characteristic fields

Extension of resolution techniques to positive characteristic

Abstract

Building upon works of Hironaka, Bierstone-Milman, Villamayor and Wlodarczyk, we give an a priori estimate for the complexity of the simplified Hironaka algorithm. As a consequence of this result, we show that there exists canonical Hironaka embedded desingularization and principalization over fields of large characteristic (relative to the degrees of generating polynomials).