2023/05/31 by Arthur Bik, Jan Draisma, Bik, Arthur +5 · 2 citations
Engineering · #Algebraic Geometry (math.AG) #Elasticity and Material Modeling #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2305.19866
openalex publication_date 2023/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank ≤ r locus is not Zariski closed, and understanding the closure (the locus with "border rank" ≤ r) is an important problem. We make two contributions in this direction: we prove a de-bordering result, which bounds border rank as a function of rank; and we show that the limits required to realize a point of border rank ≤ r do not become increasingly complicated as the dimension of the vector space increases. We prove both results for a fairly general class of ranks. We deduce our theorems on ranks from foundational results on GL-varieties, which are infinite dimensional algebraic varieties on which the infinite general linear group acts. For example, an important result concerns the existence of curves on GL-varieties.