Thom's conjecture on singularities of gradient vector fields (Q1198475)

A partial answer to the following Thom conjecture is given: let \(f(x)\) be a germ of real analytic function at the origin \(0\in\mathbb{R}^ n\) and let \(X=\text{grad} f(x)\) be the gradient vector field of \(f(x)\). If an integral curve \(g(t)\) of \(X\) tends to the origin \(0\in\mathbb{R}^ n\), then there exists a unique tangential direction \(\lim_{t\to\infty} g(t)/| g(t)|\). The author proves that if \(f(x)=P_ k(x)+P_{k+1}(x)+\dots\), \(P_ k\neq 0\), where \(P_ j(x)\) are homogeneous polynomial of degree \(i\) and \(\dim Sp(P_ k)\leq 1\) (where \(Sp(P_ k)=\{x=(x_ 1,\dots,x_ n)\in\mathbb{R}^ n\): \(x_ i{\partial P_ k \over \partial x_ j} = x_ j{\partial P_ k \over \partial x_ i}\), \(i,j=1,\dots,n\}\), is a cone algebraic set containing 0), then the Thom's conjecture holds. As a corollary he shows that the Thom's conjecture holds in the two- dimensional case.