Geodesic
Toward a combinatorial proof of the Jacobian conjecture!
The electronic journal of combinatorics, The Zeilberger Festschrift volume, Tome 18 (2011) no. 2
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The Jacobian conjecture [Keller, Monatsh. Math. Phys., 1939] gives rise to a problem in combinatorial linear algebra: Is the vector space generated by rooted trees spanned by forest shuffle vectors? In order to make headway on this problem we must study the algebraic and combinatorial properties of rooted trees. We prove three theorems about the vector space generated by binary rooted trees: Shuffle vectors of fixed length forests are linearly independent, shuffle vectors of nondegenerate forests relative to a fixed tree are linearly independent, and shuffle vectors of sufficient length forests are linearly independent. These results are proved using the acyclic digraph method for establishing that a coefficient matrix has full rank [Singer, The Electronic Journal of Combinatorics, 2009]. We also provide an infinite class of counterexamples to demonstrate the need for sufficient length in the third theorem.
DOI : 10.37236/2023
Classification : 05E05, 05C05, 14R15, 15A03
Mots-clés : binary rooted trees, acyclic digraph method 
@article{10_37236_2023,
     author = {Dan Singer},
     title = {Toward a combinatorial proof of the {Jacobian} conjecture!},
     journal = {The electronic journal of combinatorics},
     year = {2011},
     volume = {18},
     number = {2},
     doi = {10.37236/2023},
     zbl = {1243.05241},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/2023/}
}
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Dan Singer. Toward a combinatorial proof of the Jacobian conjecture!. The electronic journal of combinatorics, The Zeilberger Festschrift volume, Tome 18 (2011) no. 2. doi: 10.37236/2023

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