SOME REMARKS ON THE JACOBIAN CONJECTURE AND POLYNOMIAL ENDOMORPHISMS

In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization which states that under some conditions, a polynomial endomorphism with r homogeneous parts of positive degree does not have r times the same image point on a line through the origin, in case its Jacobian determinant does not vanish anywhere on that line. As a consequence, a Keller map of degree r does not take the same values on r > 1 collinear points, provided r is a unit in the base field. Next, we show that for invertible maps x + H of degree d such that ker JH has n - r independent vectors over the base field, in particular for invertible power linear maps x + (Ax)(*d) with rk A = r, the degree of the inverse of x + H is at most d(r).