A weight-homogenous condition to the real Jacobian conjecture in

It is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

Combinatorial QFT and the Jacobian Conjecture

The Jacobian Conjecture states that any complex n-dimensional locally invertible polynomial system is globally invertible with polynomial inverse. In 1982, Bass et al. proved an important reduction theorem stating that the conjecture is true for any degree of the polynomial system if it is true in degree three. This degree reduction is obtained with the price of increasing the dimension n. We show in this chapter a result concerning partial elimination of variables, which implies a reduction of the generic case to the quadratic one. The price to pay is the introduction of a supplementary parameter 0≤n′≤n, parameter which represents the dimension of a linear subspace where some particular conditions on the system must hold. We exhibit a proof, in a QFT formulation, using the intermediate field method exposed in Chapter 3.

An extension to the planar Markus–Yamabe Jacobian conjecture

We extend the planar Markus–Yamabe Jacobian conjecture to differential systems having Jacobian matrix with eigenvalues with negative or zero real parts.

Analytic Automorphisms and Transitivity of Analytic Mappings

In this paper, we investigate analytic automorphisms of complex topological vector spaces and their applications to linear and nonlinear transitive operators. We constructed some examples of polynomial automorphisms that show that a natural analogue of the Jacobian Conjecture for infinite dimensional spaces is not true. Also, we prove that any separable Fréchet space supports a transitive analytic operator that is not a polynomial. We found some connections of analytic automorphisms and algebraic bases of symmetric polynomials and applications to hypercyclicity of composition operators.

Analytic Automorphisms and Transitivity of Analytic Mappings

In this paper we investigate analytic automorphisms of complex topological vector spaces and their applications to linear and nonlinear transitive operators. We constructed some examples of polynomial automorphisms which show that a natural analogue of the Jacobian Conjecture for infinite dimensional spaces is not true. Also, we prove that any separable Fréchet space supports a transitive analytic operator which is not a polynomial. We found some connections of analytic automorphisms and algebraic bases of symmetric polynomials and applications to hypercyclisity of composition operators.