In this paper we study homogeneous polynomials and vector subspaces of a polynomial ring that are divisible by a large power of a linear form. We compute their cactus and border cactus ranks. We show that for d ≥ 5, the component of the cactus variety κ14(ν d (P 6 )) other than the secant variety σ14(ν d (P 6 )) consists of degree d polynomials divisible by (d − 3)-th power of a linear form. For d ≥ 6 we present an algorithm for deciding whether a point in the cactus variety κ14(ν d (P 6 )) belongs to the secant variety σ14(ν d (P 6 )). Analogously, we show that for d ≥ 5, the component of the Grassmann cactus variety κ8,3(ν d (P 4 )) other than the Grassmann secant variety σ8,3(ν d (P 4 )) consists of subspaces divisible by (d − 2)-th power of a linear form. Finally, for d ≥ 5 we present an algorithm for deciding whether a point in the Grassmann cactus variety κ8,3(ν d (P 4 )) belongs to the Grassmann secant variety σ8,3(ν d (P 4 )).
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