Narrow Results

We prove that generating series for colored modular graphs satisfy some systems of partial differential equations generalizing Burgers or heat equations. The solution is obtained by genus expansion of the generating function. The initial term of this expansion is the corresponding generating function for trees. For this term the system of differential equations is equivalent to the inversion problem for the gradient mapping defined by the initial condition. This enables to state the Jacobian conjecture in the language of generating functions. The use of generating functions provides rather short and natural proofs of resent results of Zhao and of the well-known Bass–Connell–Wright tree inversion formula.

In this paper, we study a so-called Condition C1 on square matrices with complex coefficients and a weaker Condition C2. For Druzkowski maps Condition C2 is equivalent to the Jacobian conjecture. We show that these conditions satisfy many good properties and in particular are satisfied by a dense subset of the set of square matrices of a given rank $r$. Based on this, we propose a heuristic argument for the truth of the Jacobian conjecture. We propose some new equivalent formulations and some generalizations of the Jacobian conjecture, and some approaches (including computer algebra and numerical methods) toward resolving it. We show that some of these equivalent formulations are automatically satisfied by generic Druzkowski matrices. Applications and experimental results are included.

The structure of the group Aut(ℂⁿ) of biholomorphisms of ℂⁿ is largely unknown if n > 1. In stark contrast Aut(ℂ) is rather small, consisting of the non-constant affine linear maps. The description of Aut(ℂ) follows from the observation that an injective holomorphic function f : ℂ → ℂ satisfying f(0) = 0 and f′(0) = 1 must be the identity. These considerations suggest that similar characterizations of the identity might be useful in understanding the structure of Aut(ℂⁿ). Using geometric methods we prove that an injective holomorphic map f : ℂⁿ → ℂⁿ is the identity I if and only if the power series at 0 of f - I has no terms of order ≤ 2n + 1 and the function |det Df(z)| |z|²ⁿ |f(z)|^-2n is subharmonic throughout ℂⁿ.

The well-known Dixmier conjecture [5] asks if every algebra endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism.

We bring a hopefully easier to solve conjecture, called the γ, δ conjecture, and show that it is equivalent to the Dixmier conjecture.

In the group generated by automorphisms and anti-automorphisms of A₁, all the involutions belong to one conjugacy class, hence:

• Every involutive endomorphism from (A₁, γ) to (A₁, δ) is an automorphism (γ and δ are two involutions on A₁).

• Given an endomorphism f of A₁ (not necessarily an involutive endomorphism), if one of f(X), f(Y) is symmetric or skew-symmetric (with respect to any involution on A₁), then f is an automorphism.

In this note, we investigate Jacobian conjecture through investigation of automorphisms of polynomial rings in characteristic $p$. Making use of the technique of inverse limits, we show that under Jacobian condition for a given homomorphism $\phi$ of the polynomial ring k[$x_1,\dots ,x_n$], if $\phi$ preserves the maximal ideals, then $\phi$ is an automorphism.

A non-associative ring which contains a well-known associative ring or Lie ring is interesting. In this paper, a method to construct a Lie admissible non-associative ring is given; a class of simple non-associative algebras is obtained; all the derivations of the non-associative simple algebra defined in this paper are determined; and finally, a solid algebra is defined.

We define a degree stable Lie algebra. Since the special type Lie algebra S⁺(2) is degree stable, we find the automorphism group Aut_Lie(S⁺(2)) of the Lie algebra S⁺(2) and prove the Jacobian conjecture of the Lie algebra S⁺(2).

Jędrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well.

In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducibility properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over ℂ and hence any field of characteristic zero) are irreducible.

Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman.

We classify all polynomial maps of the form $H=(u(x,y,z),v(x,y,z),h(x,y,z))$ in the case when the Jacobian matrix of $H$ is nilpotent and the highest degree of $z$ in $v$ is no more than 1. In addition, we generalize the structure of polynomial maps $H$ to $H=(H_1(x_1,x_2,\cdots ,x_n),b_3{x}_3+\cdots +b_n{x}_n+H_2^{(0)}(x_2),H_3(x_1,x_2),\dots ,H_n(x_1,x_2))$.

In this paper we consider the Jacobian conjecture in the view of hypercomplex analysis. For a polynomial mapping P(w) = (p1(w),p2(w)): ℂ² → ℂ² we discuss the left holomorphic quaternionic function f(z1, z2, z3) = p1 (w) + jp2 (w) where w = (x0 + x1 i, x2 + x3,i) and z1 = x1- x0i, z2 = x2 - x0i, z3 = x3 - x0i. Then we give a new approach to the Jacobian conjecture by the use of argument principle for quaternionic holomorphic functions.