# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Adamus, E.; Bogdan, P.; Crespo, T.; Hajto, Z., An effective study of polynomial maps, J. Algebra Appl., 16, 8, 13, (2017) · Zbl 1371.14066 [2] de Bondt, M., Quasi-translations and counterexamples to the homogeneous dependence problem, Proc. Amer. Math. Soc., 134, 2849-2856, (2006) · Zbl 1107.14054 [3] de Bondt, M.; Yan, D., Some remarks on the Jacobian conjecture and Drużkowski mappings, J. Algebra, 384, 267-275, (2013) · Zbl 1290.14036 [4] van den Essen, A., Polynomial Automorphisms and the Jacobian Conjecture, 190, (2000), Birkhäuser Verlag · Zbl 0962.14037 [5] van den Essen, A., The sixtieth anniversary of the Jacobian Conjecture: A new approach, Annales Polonici Mathematici, 76, 77-87, (2001) · Zbl 0982.14035 [6] Furter, J-Ph., On the degree of iterates of automorphisms of the affine plane, Manuscripta Mathematica, 98, 183-193, (1999) · Zbl 0944.32017 [7] Furter, J-Ph.; Maubach, S., Locally finite polynomial endomorphisms, J. Pure Appl. Algebra, 211, 445-458, (2007) · Zbl 1127.14054 [8] Gorni, G.; Zampieri, G., On cubic-linear polynomial mappings, Indagationes Mathematicae, 8, 471-492, (1997) · Zbl 0904.58005 [9] Keller, O. H., Ganze Cremona-Transformationen, Monatsh. Math. Phys., 47, 299-306, (1939) · JFM 65.0713.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.