论文标题
最低边界等级的简明张量
Concise tensors of minimal border rank
论文作者
论文摘要
我们确定在$ c^m \ otimes c^m \ otimes c^m $中定义最小边界排名的简洁张量的定义方程式，当$ m = 5 $和一组简明的最小边框等级$ 1 _*$ - 通用量张量 - 当$ m = 5,6 $时。我们借助两个最近的发展解决了代数复杂性理论的经典问题：布齐斯卡·布奇斯基（Buczyńska-Buczyński）定义的111个方程式以及Jelisiejew-Sisivic在通勤矩阵上的结果。我们引入了一种新的代数张量，即111个代数，并利用它来增强弗里德兰的正常形式，以满足Strassen方程的$ 1 $定型张量。我们使用111个代数来表征野生最小边界等级张量，并将其分类为$ c^5 \ otimes c^5 \ otimes c^5 $。
We determine defining equations for the set of concise tensors of minimal border rank in $C^m\otimes C^m\otimes C^m$ when $m=5$ and the set of concise minimal border rank $1_*$-generic tensors when $m=5,6$. We solve this classical problem in algebraic complexity theory with the aid of two recent developments: the 111-equations defined by Buczyńska-Buczyński and results of Jelisiejew-Šivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland's normal form for $1$-degenerate tensors satisfying Strassen's equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in $C^5\otimes C^5\otimes C^5$.
