Related papers: Some arithmetic aspects of polynomial maps
We introduce endomorphisms of special jacobians and show that they satisfy polynomial equations with all integer roots which we compute. The eigen-abelian varieties for these endomorphisms are generalizations of Prym-Tjurin varieties and…
Given a finite field $\F_q$ and $n\in \N^*$, one could try to compute all polynomial endomorphisms $\F_q^n\lp \F_q^n$ up to a certain degree with a specific property. We consider the case $n=3$. If the degree is low (like 2,3, or 4) and the…
We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.
The groups of automorphisms of the Jacobian algebras are found.
We investigate Selmer groups of Jacobians of curves that admit an action of a non-trivial group of automorphisms, and give applications to the study of the parity of Selmer ranks. Under the Shafarevich--Tate conjecture, we give an…
If a symmetric multilinear algebra is weakly nil, then it is Engel. This result may be regarded as an infinite-dimensional analogue of the well-known Jacobian theorem, which states that if a polynomial mapping has a polynomial inverse, then…
The famous Jacobian conjecture asks if a morphism $f:K[x,y]\to K[x,y]$ having an invertible Jacobian is invertible ($K$ is a characteristic zero field). We show that if one of the following three equivalent conditions is satisfied, then $f$…
In this paper, we investigate the existence of fixed-point-free automorphisms for finite-dimensional Lie algebras. By a result of Jacobson, a Lie algebra admitting a fixed-point-free automorphism is solvable. We prove that such a Lie…
A particular case of the Jacobian conjecture is considered and for small dimensional cases a computational approach is offered
The main result of this paper is the following version of the real Jacobian conjecture: "Let $F=(p,q):\R^2\to\R^2$ be a polynomial map with nowhere zero Jacobian determinant. If the degree of $p$ is less than or equal to $4$, then $F$ is…
We study affine Jacobi structures on an affine bundle $\pi:A\to M$, i.e. Jacobi brackets that close on affine functions. We prove that there is a one-to-one correspondence between affine Jacobi structures on $A$ and Lie algebroid structures…
We introduce a theory of finite polynomial cohomology with coefficients in this paper. We prove several basic properties and introduce an Abel-Jacobi map with coefficients. As applications, we use such a cohomology theory to study…
A quantum integrable model is considered which describes a quantization of affine hyper-elliptic Jacobian. This model is shown to possess the property of duality: a dual model with inverse Planck constant exists such that the…
The direct or algorithmic approach for the Jacobian problem, consisting of the direct construction of the inverse polynomials is proposed. The so called principle and derived Jacobi conditions are proposed and discussed. The algorithmic…
We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture, and describe completely the solution set.
We present algorithms which, given a genus 2 curve $C$ defined over a finite field and a quartic CM field $K$, determine whether the endomorphism ring of the Jacobian $J$ of $C$ is the full ring of integers in $K$. In particular, we present…
We investigate the decomposition of Jacobians of superelliptic curves based on their automorphisms. For curve with equation $y^n=f(x^m)$ we provide an necessary and sufficient condition in terms of $m$ and $n$ for the decomposition of the…
We study Jacobi pairs in details and obtained some properties. We also study the natural Poisson algebra structure $(\PP,[...,...],...)$ on the space $\PP:=\C[y]((x^{-\frac1N}))$ for some sufficient large $N$, and introduce some…
This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic…
Let $\mathcal{C}$ be a smooth, projective, genus $g\geq 2$ curve, defined over $\mathbb{C}$. Then $\mathcal{C}$ has \emph{many automorphisms} if its corresponding moduli point $p \in \mathcal{M}_g$ has a neighborhood $U$ in the complex…