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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Bibliographic Details
Published in:AppliedMath 2022-09, Vol.2 (3), p.457-465
Main Author: Ballico, Edoardo
Format: Article
Language:English
Subjects:
real rank
semialgebraic function
tensor rank
typical rank
X-rank
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Summary: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)
ISSN:2673-9909
2673-9909
DOI:10.3390/appliedmath2030026