In this paper we study directional second-order tangent sets of real and complex analytic sets. For an analytic set $X\subseteq\mathbb{ K}^n$ and a nonzero tangent direction $u\in T_0X$, we compare the geometric second-order tangent set $T^2_{ 0,u}X$, defined via second-order expansions of analytic arcs, with the algebraic second-order tangent set $T^{ 2,a}_{ 0,u}X$, defined by initial forms of the defining equations. We prove the general inclusion $T^2_{ 0,u}X\subseteq T^{ 2,a}_{ 0,u}X$ and construct explicit real and complex analytic examples showing that the inclusion is strict. We introduce a second-jet formulation along fixed tangent directions and show that $T^2_{ 0,u}X=T^{ 2,a}_{ 0,u}X$ if and only if the natural second-jet map from analytic arcs in $X$ to jets on the tangent cone $C_0X$ is surjective. This surjectivity is established for smooth analytic germs, homogeneous analytic cones, hypersurfaces with nondegenerate tangent directions, and nondegenerate analytic complete intersections. As an application, we derive second-order necessary and sufficient optimality conditions for $C^2$ optimization problems on analytic sets.
Categories: math.AG math.CV math.OC