Throughout this paper, k will denote a commutative ring containing the rational numbers Q, and k = k[x{, . . . , xn] will be the polynomial ring over k . If ƒ : k —• k is a polynomial map (i.e., a fc-algebra homomorphism), then ƒ is a polynomial automorphism provided there is an inverse ƒ " which is also a polynomial map. Very little is known about the group of polynomial automorphisms, and ind...

This paper describes a comparative study of performance between the estimated image Jacobian that come from taking into account the geometry epipolar of a system of two cameras, and the well known analytic image Jacobian that is utilized for most applications in visual servoing. Image Based Visual Servoing architecture is used for controlling a 3 d.o.f. articular system using two cameras in eye...

This paper presents a new method for calculating operational-space inertia matrices, and other related quantities, for branched kinematic trees. It is based on the exploitation of branch-induced sparsity in the joint-space inertia matrix and the task Jacobian. Detailed cost figures are given for the new method, and its efficacy is demonstrated by means of a realistic example based on the ASIMO ...

Let F = (F1, F2, · · ·Fn) be an n-tuple of formal power series in n variables of the form F (z) = z + O(|z|). It is known that, there exists a unique formal differential operator A(z) = ∑n i=1 ai(z) ∂ ∂zi such that F (z) = exp(A)z as formal series. In this article, we show the Jacobian J(F ) and the Jacobian matrix J(F ) of F can also be given by some exponential formulas. Namely, J(F ) = exp(A...

Let z = (z1, · · · , zn) and ∆ = ∑n i=1 ∂ 2 ∂z i the Laplace operator. The main goal of the paper is to show that the wellknown Jacobian conjecture without any additional conditions is equivalent to the following what we call vanishing conjecture: for any homogeneous polynomial P (z) of degree d = 4, if ∆P(z) = 0 for all m ≥ 1, then ∆P(z) = 0 when m >> 0, or equivalently, ∆P(z) = 0 when m > 3 2...

We give a proof of the two dimensional Jacobian conjecture. We also prove that if (F, G) is a Jacobian pair with deg y F ≥ 1, then F is a monic polynomial of y up to a scalar.

We show that the iterated images of a Jacobian pair f : C 2 → C 2 stabilize; that is, all the sets f k (C 2) are equal for k sufficiently large. More generally, let X be a closed algebraic subset of C N , and f : X → X an open polynomial map with X − f (X) a finite set. We show that the sets f k (X) stabilize, and for any cofinite subset Ω ⊆ X with f (Ω) ⊆ Ω, the sets f k (Ω) stabilize. We appl...

Let φ be a polynomial over K (a field of characteristic 0) such that the Hessian of φ is a nonzero constant. Let φ̄ be the formal Legendre Transform of φ. Then φ̄ is well-defined as a formal power series over K. The Hessian Conjecture introduced here claims that φ̄ is actually a polynomial. This conjecture is shown to be true when K = R and the Hessian matrix of φ is either positive or negative de...