تقرير
Uniformity for limits of tensors
| العنوان: | Uniformity for limits of tensors |
|---|---|
| المؤلفون: | Bik, Arthur, Draisma, Jan, Eggermont, Rob, Snowden, Andrew |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Algebraic Geometry, Mathematics - Representation Theory |
| الوصف: | 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 $\le r$ locus is not Zariski closed, and understanding the closure (the locus with "border rank" $\le r$) 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 $\le r$ 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 $\mathbf{GL}$-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 $\mathbf{GL}$-varieties. Comment: 25 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2305.19866 |
| رقم الأكسشن: | edsarx.2305.19866 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |