Toward a combinatorial proof of the Jacobian conjecture?

1986 ◽  
pp. 370-380 ◽  
Author(s):  
Doron Zeilberger
Keyword(s):  
Jacobian Conjecture ◽  
Combinatorial Proof

scholarly journals Toward a Combinatorial Proof of the Jacobian Conjecture!

10.37236/2023 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Dan Singer
Keyword(s):  
Vector Space ◽  
Full Rank ◽  
Coefficient Matrix ◽  
Rooted Trees ◽  
Jacobian Conjecture ◽  
Combinatorial Proof ◽  
Combinatorial Properties ◽  
Linearly Independent ◽  
Acyclic Digraph ◽  
Fixed Tree

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.

Ont he monodromy representation of polynomial maps in n variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 361-367
Author(s):  
A. Némethi ◽  
I. Sigray
Keyword(s):  
Cohomology Group ◽  
Jacobian Conjecture ◽  
Natural Basis ◽  
Generic Fiber ◽  
Monodromy Representation ◽  
Polynomial Maps ◽  
Polynomial Map

For a   non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As  applications,  we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case  n=2 is treated.

scholarly journals On a tropical version of the Jacobian conjecture

2020 ◽  
Author(s):  
Dima Grigoriev ◽  
Danylo Radchenko
Keyword(s):  
Jacobian Conjecture

scholarly journals A combinatorial proof of Cayley's theorem on Pfaffians

1966 ◽  
Vol 1 (2) ◽  
pp. 224-232 ◽  
Author(s):  
John H. Halton
Keyword(s):  
Combinatorial Proof

scholarly journals COMBINATORIAL KNOT FLOER HOMOLOGY AND DOUBLE BRANCHED COVERS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350014
Author(s):  
FATEMEH DOUROUDIAN
Keyword(s):  
Floer Homology ◽  
Combinatorial Proof ◽  
Branched Covers ◽  
Knot Floer Homology ◽  
Branched Cover ◽  
Heegaard Diagram

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

scholarly journals Polynomial Retracts and the Jacobian Conjecture

1999 ◽  
Vol 352 (1) ◽  
pp. 477-484 ◽  
Author(s):  
Vladimir Shpilrain ◽  
Jie-Tai Yu
Keyword(s):  
Jacobian Conjecture

Polynomial remainders and plane automorphisms

2003 ◽  
Vol 68 (1) ◽  
pp. 73-79
Author(s):  
Takis Sakkalis
Keyword(s):  
Jacobian Conjecture ◽  
Real Polynomial ◽  
Polynomial Automorphisms ◽  
Polynomial Map

This note relates polynomial remainders with polynomial automorphisms of the plane. It also formulates a conjecture, equivalent to the famous Jacobian Conjecture. The latter provides an algorithm for checking when a polynomial map is an automorphism. In addition, a criterion is presented for a real polynomial map to be bijective.

scholarly journals Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Liuquan Wang
Keyword(s):  
Analogous Result ◽  
Triple Product ◽  
Combinatorial Proof ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Pentagonal Number Theorem ◽  
Nonnegative Coefficients ◽  
Jacobi's Triple Product Identity

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

scholarly journals Combinatorial Constructions of Weight Bases: The Gelfand-Tsetlin Basis

10.37236/305 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Patricia Hersh ◽  
Cristian Lenart
Keyword(s):  
Lie Algebras ◽  
Simple Algorithm ◽  
Irreducible Representations ◽  
Combinatorial Proof ◽  
Young Tableaux ◽  
Constructive Approach ◽  
Function Identity ◽  
New Interpretation ◽  
Combinatorial Constructions ◽  
Extremal Properties

This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of $\mathfrak{ sl}$$_n$, the celebrated Gelfand-Tsetlin basis is the only such basis known. Using the setup of supporting graphs developed by Donnelly, we present a new interpretation and a simple combinatorial proof of the Gelfand-Tsetlin formulas based on a rational function identity (all the known proofs use more sophisticated algebraic tools). A constructive approach to the Gelfand-Tsetlin formulas is then given, based on a simple algorithm for solving certain equations on the lattice of semistandard Young tableaux. This algorithm also implies certain extremal properties of the Gelfand-Tsetlin basis.

scholarly journals Combinatorial Proof of a Curious $q$-Binomial Coefficient Identity

10.37236/462 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Victor J. W. Guo ◽  
Jiang Zeng
Keyword(s):  
Binomial Coefficient ◽  
Combinatorial Proof ◽  
Binomial Coefficient Identity

Using the Algorithm Z developed by Zeilberger, we give a combinatorial proof of the following $q$-binomial coefficient identity $$ \sum_{k=0}^m(-1)^{m-k}{m\brack k}{n+k\brack a}(-xq^a;q)_{n+k-a}q^{{k+1\choose 2}-mk+{a\choose 2}} $$ $$=\sum_{k=0}^n{n\brack k}{m+k\brack a}x^{m+k-a}q^{mn+{k\choose 2}}, $$ which was obtained by Hou and Zeng [European J. Combin. 28 (2007), 214–227].

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