Abstract
Let k be a field of characteristic p > 0. We discuss the automorphisms of the polynomial ring k[x1,…,xn] of order p, or equivalently the Z/pZ-actions on the affine space \({\textbf {A}}_{k}^{n}\). When n = 2, such an automorphism is known to be a conjugate of an automorphism fixing a variable. It is an open question whether the same holds when n ≥ 3. In this paper, (1) we give a negative answer to this question when n = 3. In fact, we show that every Ga-action on \({\textbf {A}}_{k}^{3}\) of rank three yields an automorphism of k[x1,x2,x3] of order p which is not a conjugate of an automorphism fixing a variable. We give a family of such automorphisms of k[x1,x2,x3] by constructing a family of rank three Ga-actions on \({\textbf {A}}_{k}^{3}\). (2) For the automorphisms of k[x1,x2,x3] induced by this family of Ga-actions, we show that the invariant ring is isomorphic to k[x1,x2,x3] if and only if the plinth ideal is principal, under some mild assumptions. (3) We study the Nagata type automorphism of R[x1,x2], where R is a UFD of characteristic p > 0. This type of automorphism is of order p. We give a necessary and sufficient condition for the invariant ring to be isomorphic to R[x1,x2]. This condition is equivalent to the condition that the plinth ideal is principal.
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Notes
The author announced this theorem, together with a negative answer to Question 1.1, on the occasion of the 13th meeting of Affine Algebraic Geometry at Osaka on March 5, 2015 (see [18]).
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This work is partly supported by JSPS KAKENHI Grant Numbers 15K04826 and 18K03219.
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Kuroda, S. Polynomial Automorphisms of Characteristic Order and Their Invariant Rings. Transformation Groups (2022). https://doi.org/10.1007/s00031-022-09764-2
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DOI: https://doi.org/10.1007/s00031-022-09764-2