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The Jacobian Conjecture as a problem in combinatorics

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arxiv math/0511214 v2 pith:JDRIX7D3 submitted 2005-11-08 math.CO math.ACmath.AG

The Jacobian Conjecture as a problem in combinatorics

classification math.CO math.ACmath.AG
keywords conjecturejacobiancasesymmetricgivehomogeneousalgebrabeen
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The Jacobian Conjecture has been reduced to the symmetric homogeneous case. In this paper we give an inversion formula for the symmetric case and relate it to a combinatoric structure called the Grossman-Larson Algebra. We use these tools to prove the symmetric Jacobian Conjecture for the case $F=X-H$ with $H$ homogeneous and $JH^{3}=0$. Other special results are also derived. We pose a combinatorial statement which would give a complete proof the Jacobian Conjecture.

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