Related papers: Symmetric Jacobians

Jacobian conjectures (that nonsingular implies invertible) for rational everywhere defined maps of real n-space to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The associated…

We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a…

The purpose of this note is to give an affirmative answer to a conjecture appearing in [Integral Transforms Spec. Funct. 26 (2015) 90-95].

Using the local bijectivity of Keller maps, we give a proof of two-dimensional Jacobian conjecture.

In this paper we study the orthogonality conditions satisfied by Jacobi polynomials $P_n^{(\alpha,\beta)}$ when the parameters $\alpha$ and $\beta$ are not necessarily $>-1$. We establish orthogonality on a generic closed contour on a…

It is shown that every polynomial function $P : \mathbb{C}^2\longrightarrow \mathbb{C}$ with irreducible fibres of same a genus is a coordinate. In consequence, there does not exist counterexamples F = (P,Q) to the Jacobian conjecture such…

Let $ K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a {\it retraction}, or {\it projection}) $\varphi:…

For given polynomial map $F:\C^2\to\C^2$ with nonvanishing jacobian we associate a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

Let $\cal{A}$ be the algebra of quaternions $\mathbb{H}$ or octonions $\mathbb{O}$. In this manuscript a new proof is given, based on ideas of Cauchy and D' Alembert, of the fact that an ordinary polynomial $f(t) \in {\cal{A}}\, [t]$ has a…

In this paper, we discuss the associated family of harmonic maps $\mathcal{F}: M \rightarrow G/K$ from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type which are either algebraic or totally symmetric. These…

We study Jacobi matrices on trees with one end at inifinity. We show that the defect indices cannot be greater than 1 and give criteria for essential selfadjointness. We construct certain polynomials associated with matrices, which mimic…

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…

The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that…

We prove an unexpected general relation between the Jacobian syzygies of a projective hypersurface $V\subset \mathbb{P}^n$ with only isolated singularities and the nature of its singularities. This allows to establish a new method for the…

Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input…

We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over the complex numbers all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When…

Using the author's inversion formula for automorphisms of the Weyl algebras with polynomial coefficients and the bound on its degree a slightly shorter (algebraic) proof is given of the result of A. Belov-Kanel and M. Kontsevich that the…

We prove that the Logarithm of the Jacobian of a sense preserving harmonic mappings between surfaces is superharmonic, provided that the Gaussian curvature of the image domain is non-negative.

We consider non-singular and Jacobian maps whose components are polynomial in the variable y. We prove that if a map has y-degree one, then it is the composition of a triangular map and a quasi-triangular map. We also prove that…

The asymptotic variety of a counterexample of Pinchuk type to the strong real Jacobian conjecture is explicitly described by low degree polynomials.