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Pascal finite polynomial automorphisms. (English) Zbl 1433.14053
Pascal finite automorphisms of the polynomial algebra \(K[X]=K[X_1,\ldots,X_n]\) over a field \(K\) were introduced in the previous paper by the authors [J. Algebra Appl. 16, No. 8, Article ID 1750141, 13 p. (2017; Zbl 1371.14066)]. For a polynomial map \(F:K^n\to K^n\) the authors define an endomorphism \(\sigma_F:K[X]\to K[X]\) by \(\sigma_F(P)=P\circ F\) and a \(\sigma_F\)-derivation \(\Delta_F\) of \(K[X]\) by \(\Delta_F(P)=\sigma_F(P)-P\), \(P\in K[X]\). The polynomial \(F\) is Pascal finite, if there exists an integer \(m\) such that \(\Delta_F^m(X)=0\) and hence the polynomial map \(X\to F(X)\) is invertible. In characteristic 0, a Pascal finite automorphism is the exponential map of a locally nilpotent derivation. Since Pascal finite automorphisms are defined in any characteristic, they constitute a generalization of exponential automorphisms to positive characteristic.
In the paper under review the authors establish several properties of Pascal finite automorphisms. For example, the Pascal finite property is stable under taking powers but not under composition. The authors show that any triangulable polynomial automorphism is Pascal finite. Also, in characteristic 0, Hubbers’ classification describes up to conjugation all cubic polynomial maps in dimension 4. It has turned out that seven of the eight conjugation classes are Pascal finite. Other interesting properties and examples are established as well, some of them inspired by the Jacobian conjecture. In particular, the authors formulate a generalization of the exponential generators conjecture to arbitrary characteristic.
MSC:
14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
14R15 Jacobian problem
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14E07 Birational automorphisms, Cremona group and generalizations
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