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

 

Theorems of W. W. Stothers and the Jacobian Conjecture in two variables


Author: Edward Formanek
Journal: Proc. Amer. Math. Soc. 139 (2011), 1137-1140
MSC (2010): Primary 14R15, 11C08
Published electronically: August 30, 2010
MathSciNet review: 2748408
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Abstract: Differential equations of the form $ r p(z)q'(z) - sp'(z)q(z) = \gamma p(z)$ or $ rp(z)q'(z) - sp'(z)q(z) = \gamma$, where $ \gamma$ is a nonzero complex number and $ r,s$ are positive integers, have arisen in attempts to solve the two-variable Jacobian Conjecture. Solutions of such equations, in which $ p(z)$ and $ q(z)$ are monic complex polynomials of positive degrees $ r$ and $ s$, give rise to extra-special pairs of polynomials in the sense of W. W. Stothers. Stothers showed that, modulo automorphisms of $ \mathbb{C}[z]$, there are only finitely many extra-special pairs of a given degree $ n$. This implies that, modulo automorphisms of $ \mathbb{C}[z]$, there are only finitely many solutions of the above differential equations in which $ p(z)$ and $ q(z)$ are monic polynomials of given degrees $ r$ and $ s$.


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Edward Formanek
Affiliation: 52 Living Edens Court, Las Vegas, Nevada 89148
Email: formanek@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10523-3
PII: S 0002-9939(2010)10523-3
Keywords: Jacobian Conjecture, polynomial abc-theorem
Received by editor(s): October 2, 2007
Received by editor(s) in revised form: April 28, 2009
Published electronically: August 30, 2010
Communicated by: Ted Chinburg
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.