On Tangent Cones of Analytic Sets and $$\L $$ojasiewicz Exponents

Abstract

For an analytic set V in $${\mathbb {K}}^n,$$$$\mathbb K={\mathbb {R}}$$ or $${\mathbb {C}}$$, which contains the origin $$0\in \mathbb K^n$$, the geometric tangent cone of V at 0 is the set of vectors in $${\mathbb {K}}^n$$ 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 $${\mathbb {K}}^{n}$$ 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 $$\L $$ojasiewicz exponent of an analytic map germ along a line and compute this number in some special cases.