Abstract:A characterization of tame automorphisms of the algebra
A
=
F
[
x
1
,
x
2
,
x
3
]
A=F[x_1,x_2,x_3]
of polynomials in three variables over a field
F
F
of characteristic
0
0
is obtained. In particular, it is proved that the well-known Nagata automorphism is wild. It is also proved that the tame and the wild automorphisms of… Show more
“…(17) affirmatively. On the other hand, if the answer is no for some coordinate q of P 3 , then we would obtain a new proof of the Nagata conjecture without using the previous results of Shestakov and Umirbaev (9)(10)(11).…”
Section: Some Open Problemsmentioning
confidence: 99%
“…This condition can be effectively determined by an algorithm motivated by ideas of Shestakov and Umirbaev (9)(10)(11). By the algorithm, we are able to prove that all wild coordinates of K[z] [x, y] (1, 2) are also wild coordinates of P 3 ϭ K[x, y, z], hence we obtain many wild coordinates of P 3 .…”
Section: Main Theoremmentioning
confidence: 99%
“…Then the proof of theorem 1 in ref. 9 shows that ( f, g, h) is either elementarily reducible, or admits a reduction of one of the types I-IV.…”
Section: Theorem 2 a Triple {F G H} ʚ P 2 Is Tame If And Only If Imentioning
confidence: 99%
“…The elementary reducibility of ( f, g, h) is algorithmically recognizable by Theorem 1, if (f, g, h) admits a reduction of one of types I-IV, then the proof is similar to the proof of theorem 3 in ref. 9.…”
Section: Theorem 3 Let F G H ʦ P 2 Such That K[f G H] ϭ P 2 Thmentioning
confidence: 99%
“…Nagata (8) conjectured in 1972 that there exist wild automorphisms in Aut K P 3 . The famous Nagata conjecture was recently proved by Shestakov and Umirbaev (9)(10)(11). However, the following strong version of the Nagata conjecture remains open.…”
It is proved that there exist wild coordinates in the polynomial algebra in three variables over a field of characteristic zero. This result implies the famous Nagata conjecture.
“…(17) affirmatively. On the other hand, if the answer is no for some coordinate q of P 3 , then we would obtain a new proof of the Nagata conjecture without using the previous results of Shestakov and Umirbaev (9)(10)(11).…”
Section: Some Open Problemsmentioning
confidence: 99%
“…This condition can be effectively determined by an algorithm motivated by ideas of Shestakov and Umirbaev (9)(10)(11). By the algorithm, we are able to prove that all wild coordinates of K[z] [x, y] (1, 2) are also wild coordinates of P 3 ϭ K[x, y, z], hence we obtain many wild coordinates of P 3 .…”
Section: Main Theoremmentioning
confidence: 99%
“…Then the proof of theorem 1 in ref. 9 shows that ( f, g, h) is either elementarily reducible, or admits a reduction of one of the types I-IV.…”
Section: Theorem 2 a Triple {F G H} ʚ P 2 Is Tame If And Only If Imentioning
confidence: 99%
“…The elementary reducibility of ( f, g, h) is algorithmically recognizable by Theorem 1, if (f, g, h) admits a reduction of one of types I-IV, then the proof is similar to the proof of theorem 3 in ref. 9.…”
Section: Theorem 3 Let F G H ʦ P 2 Such That K[f G H] ϭ P 2 Thmentioning
confidence: 99%
“…Nagata (8) conjectured in 1972 that there exist wild automorphisms in Aut K P 3 . The famous Nagata conjecture was recently proved by Shestakov and Umirbaev (9)(10)(11). However, the following strong version of the Nagata conjecture remains open.…”
It is proved that there exist wild coordinates in the polynomial algebra in three variables over a field of characteristic zero. This result implies the famous Nagata conjecture.
Abstract. In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.