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...

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Published in:AppliedMath 2022-09, Vol.2 (3), p.457-465
Main Author: Ballico, Edoardo
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Language:English
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real rank
semialgebraic function
tensor rank
typical rank
X-rank
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description 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)
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tensor rank
typical rank
X-rank
title Ranks with Respect to a Projective Variety and a Cost-Function
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