A conjecture in relation to Loewner's conjecture

Let f be a smooth function of two variables x, y and for each positive integer n, let d(n) f be a symmetric tensor field of type (0,n) defined by d(n) f: = Sigma 1(=0)(n) ((n)(i)) (partial derivative(n-i)(x)partial derivative(i)(y)f)dx(n-i)dy(i) and D-dnf a finitely many-valued one-dimensional distribution obtained from d(n) f: for example, D-d1f is the one-dimensional distribution defined by the gradient vector field of f; D-d2f consists of two one-dimensional distributions obtained from one-dimensional eigenspaces of Hessian of f. In the present paper, we shall study the behavior of D-dnf around its isolated singularity in ways which appear in [1]-[4]. In particular, we shall introduce and study a conjecture which asserts that the index of an isolated singularity with respect to D-dnf is not more than one.