On Tangent Cones of Analytic Sets and Łojasiewicz Exponents

For an analytic set V in K n , K = R or C , which contains the origin 0 ∈ K n , the geometric tangent cone of V at 0 is the set of vectors in K n which are the limits of secant lines passing through the origin and non-zero sequences in V that converge to the origin; the algebraic tangent cone of V at 0 is the algebraic set defined by the ideal generated by the initial forms of all analytic functions f whose germs at 0 are in the ring of germs of analytic functions in K n about 0 which vanish on the germ of V at 0. In this paper, we give some characterization for the geometric tangent cone and compare these two tangent cones of V at 0. In particular, we characterize the geometric tangent cone of V at 0 via the so-called Ł ojasiewicz exponent of an analytic map germ along a line and compute this number in some special cases.