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.

Kruskal's theorem for matroids

1968 ◽  
Vol 64 (1) ◽  
pp. 3-4 ◽  
Author(s):  
D. J. A. Welsh
Keyword(s):  
Vector Space ◽  
Finite Subset ◽  
Spanning Tree ◽  
General Theorem ◽  
Independent Set ◽  
Easy Method ◽  
Maximal Weight ◽  
Linearly Independent ◽  
Kruskal’S Theorem ◽  
Special Case

AbstractKruskal's theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.

scholarly journals Isomorphisms on countable vector spaces with recursive operations

1974 ◽  
Vol 18 (2) ◽  
pp. 230-235 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Vector Space ◽  
Independent Set ◽  
Vector Spaces ◽  
Recursive Structure ◽  
Recursively Enumerable ◽  
Linearly Independent ◽  
Vector Addition

Terminology and notation may be found in Dekker [1] and [2]. Briefly, we fix a recursively enumerable (r.e.) field F with recursive structure, and let Ū be the vector space over F consisting of ultimately vanishing countable sequences of elements of F with the usual definitions of vector addition and multiplication by a scalar. A subspace V of Ū is called an α-space if V has a basis B which is contained in some r.e. linearly independent set S.

On Diophantine approximations of the solutions of q-functional equations

2002 ◽  
Vol 132 (3) ◽  
pp. 639-659 ◽  
Author(s):  
Tapani Matala-aho
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Linear Independence ◽  
Number Fields ◽  
Growth Conditions ◽  
Linear Forms ◽  
Dimension Estimate ◽  
Linearly Independent ◽  
Diophantine Approximations ◽  
Image Position

Given a sequence of linear forms in m ≥ 2 complex or p-adic numbers α1, …,αm ∈ Kv with appropriate growth conditions, Nesterenko proved a lower bound for the dimension d of the vector space Kα1 + ··· + Kαm over K, when K = Q and v is the infinite place. We shall generalize Nesterenko's dimension estimate over number fields K with appropriate places v, if the lower bound condition for |Rn| is replaced by the determinant condition. For the q-series approximations also a linear independence measure is given for the d linearly independent numbers. As an application we prove that the initial values F(t), F(qt), …, F(qm−1t) of the linear homogeneous q-functional equation where N = N(q, t), Pi = Pi(q, t) ∈ K[q, t] (i = 1, …, m), generate a vector space of dimension d ≥ 2 over K under some conditions for the coefficient polynomials, the solution F(t) and t, q ∈ K*.

scholarly journals A Combinatorial Proof of Postnikov's Identity and a Generalized Enumeration of Labeled Trees

10.37236/1890 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Seunghyun Seo
Keyword(s):  
Rooted Trees ◽  
Binary Trees ◽  
Combinatorial Proof ◽  
Labeled Trees ◽  
Plane Trees

In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled $k$-ary trees, rooted labeled trees, and labeled plane trees.

Finite Unions of Quasi-Independent Sets

1984 ◽  
Vol 27 (4) ◽  
pp. 490-493 ◽  
Author(s):  
David Grow ◽  
William C. Whicher
Keyword(s):  
Vector Space ◽  
Independent Sets ◽  
Linearly Independent

AbstractThe analogue of Horn's theorem characterizing finite unions of linearly independent sets in a vector space is shown to fail in the group of integers.

Combinations of Orthogonal Spectra to Estimate Component Spectra in Multicomponent Mixtures

1987 ◽  
Vol 41 (2) ◽  
pp. 227-234 ◽  
Author(s):  
H. Bruce Friedrich ◽  
Jung-Pin Yu
Keyword(s):  
Factor Analysis ◽  
Vector Space ◽  
Dimensional Vector ◽  
Multicomponent Mixtures ◽  
Dimensional Vector Space ◽  
Number Of Components ◽  
Linearly Independent

A factor analysis of a set of spectra of multicomponent mixtures yields n orthogonal spectra where n is the number of components in the mixtures. Combinations of these orthogonal spectra, done so as to eliminate the contributions to the spectra from one of the components, yield a set of n – 1 linearly independent abstract spectra. Repetition of this process is used to estimate the spectrum of each of the components without identifying the range of the n-dimensional vector space in which the solution must fall. Several techniques for estimating the appropriate combinations of orthogonal spectra are compared.

scholarly journals A General Basis Theorem

1959 ◽  
Vol 11 (3) ◽  
pp. 139-141 ◽  
Author(s):  
A. P. Robertson ◽  
J. D. Weston
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Upper Bound ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Basis Theorem ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Linearly Independent ◽  
General Basis

The well-known “basis theorem” of elementary algebra states that in a finite-dimensional vector space, any two bases have the same number of elements; or, equivalently, that a vector space is n-dimensional if it has a basis consisting of n vectors (where the dimension of a vector space is defined to be the least upper bound of the numbers k for which there exist k linearty independent vectors, and a basis is defined to be a maximal set of linearly independent vectors). This theorem has an analogue in the theory of groups : if an Abelian group has a finite maximal set of independent non-cyclic elements, the number of elements in one such set being n, then no set of independent non-cyclic elements can have more than n members.

scholarly journals Relation between saturated and normal operators

BIBECHANA ◽  
2013 ◽  
Vol 10 ◽  
pp. 115-117
Author(s):  
Nagendra Pd Sah
Keyword(s):  
Vector Space ◽  
Finite Dimensional ◽  
Linearly Independent ◽  
Linear Maps ◽  
Normal Operators ◽  
The Ideal

A vector space X with algebra of all linear maps ∑(X) from X into itself and the ideal of all finite dimensional linear maps with dual (Conjugate) transformation T* to T from X' to itself form a relation in terms of relatively regular and linearly independent which is sufficient for mentioned title. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.9342   BIBECHANA 10 (2014) 115-117

scholarly journals On Catalan Trees and the Jacobian Conjecture

10.37236/1546 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Dan Singer
Keyword(s):  
Equivalence Classes ◽  
Jacobian Conjecture ◽  
Complex Numbers ◽  
Total Degree ◽  
Combinatorial Properties ◽  
Algebraic Result ◽  
Ring Of Polynomials

New combinatorial properties of Catalan trees are established and used to prove a number of algebraic results related to the Jacobian conjecture. Let $F=(x_1+H_1,x_2+H_2,\dots,x_n+H_n)$ be a system of $n$ polynomials in $C[x_1,x_2,\dots,x_n]$, the ring of polynomials in the variables $x_1,x_2, \dots, x_n$ over the field of complex numbers. Let $H=(H_1,H_2,\dots,H_n)$. Our principal algebraic result is that if the Jacobian of $F$ is equal to 1, the polynomials $H_i$ are each homogeneous of total degree 2, and $({{\partial H_i}\over {\partial x_j}})^3=0$, then $H\circ H\circ H=0$ and $F$ has an inverse of the form $G=(G_1,G_2,\dots,G_n)$, where each $G_i$ is a polynomial of total degree $\le6$. We prove this by showing that the sum of weights of Catalan trees over certain equivalence classes is equal to zero. We also show that if all of the polynomials $H_i$ are homogeneous of the same total degree $d\ge2$ and $({{\partial H_i}\over {\partial x_j}})^2=0$, then $H\circ H=0$ and the inverse of $F$ is $G=(x_1-H_1,\dots,x_n-H_n)$.

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