Ranks with Respect to a Projective Variety and a Cost-Function

Let X⊂Pr be an integral and non-degenerate variety. A “cost-function” (for the Zariski topology, the semialgebraic one, or the Euclidean one) is a semicontinuous function w:=[1,+∞)∪+∞ such that w(a)=1 for a non-empty open subset of X. For any q∈Pr, the rank rX,w(q) of q with respect to (X,w) is the minimum of all ∑a∈Sw(a), where S is a finite subset of X spanning q. We have rX,w(q)+∞ for all q. We discuss this definition and classify extremal cases of pairs (X,q). We give upper bounds for all rX,w(q) (twice the generic rank) not depending on w. This notion is the generalization of the case in which the cost-function w is the constant function 1. In this case, the rank is a well-studied notion that covers the tensor rank of tensors of arbitrary formats (PARAFAC or CP decomposition) and the additive decomposition of forms. We also adapt to cost-functions the rank 1 decomposition of real tensors in which we allow pairs of complex conjugate rank 1 tensors