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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The irreducibility of symmetric Yagzhev maps


Author: Sławomir Bakalarski
Journal: Proc. Amer. Math. Soc. 138 (2010), 2279-2281
MSC (2000): Primary 14R15, 12E05
Published electronically: March 10, 2010
MathSciNet review: 2607856
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Abstract: Let $ F:\mathbb{C}^n \rightarrow \mathbb{C}^n$ be a polynomial mapping in Yagzhev form, i.e.

$\displaystyle F(x_1,\ldots,x_n)=(x_1+H_1(x_1,\ldots,x_n),\ldots,x_n+H_n(x_1,\ldots,x_n)),$

where $ H_i$ are homogeneous polynomials of degree 3. We show that if $ \mathrm{Jac}(F) \in \mathbb{C}^*$ and the Jacobian matrix of $ F$ is symmetric, then the polynomials $ x_i+H_i(x_1,\ldots,x_n)$ are irreducible as elements of the ring $ \mathbb{C}[x_1,\ldots,x_n]$.


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Additional Information

Sławomir Bakalarski
Affiliation: Institute of Computer Science, Jagiellonian University, Prof. Stanisława Łojasiewicza 6, 30-348 Kraków, Poland
Email: Slawomir.Bakalarski@uj.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10109-9
PII: S 0002-9939(10)10109-9
Received by editor(s): March 15, 2009
Received by editor(s) in revised form: July 8, 2009
Published electronically: March 10, 2010
Communicated by: Martin Lorenz
Article copyright: © Copyright 2010 American Mathematical Society