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:=∑^n_{i=0} n \atopwithdelims() i \big( ∂
n-i_x ∂^i_y f\big) dx
n-i dy^i and ˜{\cal D}_{d^n f} a finitely many-valued one-dimensional distribution obtained from d^n f: for example, ˜{\cal D}_{d^1 f} is the one-dimensional distribution defined by the gradient vector field of f; ˜{\cal D}_{d^2 f} 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 ˜{\cal D}_{d^n f} 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 ˜{\cal D}_{d^n f} is not more than one.
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