2020/04/07 by Tuyen Trung Truong, Truong, Tuyen Trung
Computer Science · Engineering · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Complex Variables (math.CV) #Control and Dynamics of Mobile Robots #Dynamical Systems (math.DS) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2004.03309
openalex publication_date 2020/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For 2 vectors x,y∈ ℝm, we use the notation x * y =(x1y1,… ,xmym), and if x=y we also use the notation x2=x*x and define by induction xk=x*(xk-1). We use for the usual inner product on ℝm. For A an m× m matrix with coefficients in ℝ, we can assign a map FA(x)=x+(Ax)3:~ℝm→ ℝm. A matrix A is Druzkowski iff det(JFA(x))=1 for all x∈ ℝm. Recently, Jiang Liu posted a preprint on arXiv asserting a proof of the Jacobian conjecture, by showing the properness of FA(x) when A is Druzkowski, via some inequalities in the real numbers. In the proof, indeed Liu asserted the properness of FA(x) under more general conditions on A, see the main body of this paper for more detail. Inspired by this preprint, we research in this paper on the question of to what extend the above maps FA(x) (even for matrices A which are not Druzkowski) can be proper. We obtain various necessary conditions and sufficient conditions for both properness and non-properness properties. A complete characterisation of the properness, in terms of the existence of non-zero solutions to a system of polynomial equations of degree at most 3, in the case where A has corank 1, is obtained. Extending this, we propose a new conjecture, and discuss some applications to the (real) Jacobian conjecture. We also consider the properness of more general maps x± (Ax)k or x± A(xk). By a result of Druzkowski, our results can be applied to all polynomial self-mappings of ℂm or ℝm.