×

A characterization of Keller maps. (English) Zbl 1267.14077

Let \(k\) be a field of characteristic \(0\) and \(\phi \) be a \(k\)-endomorphism of \(k[x_{1},\ldots ,x_{n}].\) Let \(F=(\phi (x_{1}),\ldots ,\phi (x_{n})):k^{n}\rightarrow k^{n}\) be the polynomial mapping associated to \( \phi\). If the Jacobian \(\text{Jac}F\) of \(F\) is a constant \(\neq 0\) (i.e., \(\text{Jac}F\in k\setminus \{0\}\)) then we say \(\phi \) satisfies the Jacobian condition. The main result is that \(\phi \) satisfies the Jacobian condition if and only if \(\phi \) maps irreducible polynomials to square-free polynomials. Then the famous Jacobian Conjecture can be reformulated: if \(k\) -endomorphism \(\phi \) of \(k[x_{1},\dots ,x_{n}]\) maps irreducible polynomials to square-free polynomials then \(\phi \) maps irreducible polynomials to irreducible ones (because then, by Bakalarski result, \(\phi \) is an automorphism).

MSC:

14R15 Jacobian problem
13N15 Derivations and commutative rings
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bakalarski, S., Jacobian problem for factorial varieties, Univ. Iagel. Acta Math., XLIV, 31-34 (2006) · Zbl 1128.14044
[2] Ehrenfeucht, A., Kryterium absolutnej nierozkładalności wielomianów (in Polish), Prace Mat., 2, 167-169 (1956) · Zbl 0074.25505
[3] van den Essen, A., Polynomial Automorphisms and the Jacobian Conjecture (2000), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0962.14037
[4] van den Essen, A.; Nowicki, A.; Tyc, A., Generalizations of a lemma of Freudenburg, J. Pure Appl. Algebra, 177, 43-47 (2003) · Zbl 1040.13005
[5] van den Essen, A.; Shpilrain, V., Some combinatorial questions about polynomial mappings, J. Pure Appl. Algebra, 119, 47-52 (1997) · Zbl 0899.13009
[6] Freudenburg, G., A note on the kernel of a locally nilpotent derivation, Proc. Amer. Math. Soc., 124, 27-29 (1996) · Zbl 0857.13005
[7] Jelonek, Z., A solution of the problem of van den Essen and Shpilrain, J. Pure Appl. Algebra, 137, 49-55 (1999) · Zbl 0929.13014
[8] Jędrzejewicz, P., A characterizetion of one-element \(p\)-bases of rings of constants, Bull. Pol. Acad. Sci. Math., 59, 19-26 (2011) · Zbl 1216.13017
[9] Jędrzejewicz, P., A note on rings of constants of derivations in integral domains, Colloq. Math., 122, 241-245 (2011) · Zbl 1221.13040
[10] P. Jędrzejewicz, Jacobian conditions for \(p\)-bases, Comm. Algebra (2012), in press (http://dx.doi.org/10.1080/00927872.2011.587213). · Zbl 1254.13028
[11] Keller, O.-H., Ganze Cremona-Transformationen, Monatsh. Math. Phys., 47, 299-306 (1939) · JFM 65.0713.02
[12] A. Nowicki, Polynomial derivations and their rings of constants, Nicolaus Copernicus University, Toruń, 1994. · Zbl 1236.13023
[13] Nowicki, A., Rings and fields of constants for derivations in characteristic zero, J. Pure Appl. Algebra, 96, 47-55 (1994) · Zbl 0811.12003
[14] Schinzel, A., Selected Topics on Polynomials (1982), University of Michigan Press: University of Michigan Press Ann Arbor, Mich · Zbl 0487.12002
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.