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Nagata's conjecture on a polynomial automorphism in positive characteristic
Abstract
An automorphism of the polynomial ring k[x1,… ,xn] over a field k is said to be tame if it can be obtained by composing affine automorphisms and elementary automorphisms, and wild otherwise. Jung and van der Kulk showed that every automorphism of k[x1,x2] is tame. In 1972, Nagata conjectured that a certain automorphism of k[x1,x2,x3] is wild. In 2003, Shestakov and Umirbaev proved this conjecture for mathopchar k=0. The purpose of this paper is to prove the conjecture for mathopchar k≥ 7. This is the first time that the existence of a wild automorphism has been confirmed in positive characteristic.
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BibTeX
@misc{kuroda2026nagatas,
title = {{Nagata's conjecture on a polynomial automorphism in positive characteristic}},
author = {Shigeru Kuroda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14611},
eprint = {2609.14611},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14611},
}
RIS
TY - UNPB
TI - Nagata's conjecture on a polynomial automorphism in positive characteristic
AU - Shigeru Kuroda
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.14611
UR - https://arxiv.org/abs/2609.14611
AB - An automorphism of the polynomial ring k[x_1,… ,x_n] over a field k is said to be tame if it can be obtained by composing affine automorphisms and elementary automorphisms, and wild otherwise. Jung and van der Kulk showed that every automorphism of k[x_1,x_2] is tame. In 1972, Nagata conjectured that a certain automorphism of k[x_1,x_2,x_3] is wild. In 2003, Shestakov and Umirbaev proved this conjecture for mathopchar k=0. The purpose of this paper is to prove the conjecture for mathopchar k≥ 7. This is the first time that the existence of a wild automorphism has been confirmed in positive characteristic.
ER -