Invariant Theory, Tensors and Computational Complexity

[摘要] The main problem addressed in this dissertation is the problem of giving strong upper bounds on the degree of generators for invariant rings. In the cases of matrix invariants and matrix semi-invariants, we give polynomial upper bounds. An exciting consequence of these bounds is a polynomial time algorithm for rational identity testing. We use an approach inspired by ideas from Popov and Derksen to reduce the problem to finding invariants that define the null cone. The theory of blow-ups of matrix spaces and non-commutative rank is crucial in finding invariants that define the null cone. We also give a polynomial time algorithm for deciding if the orbit closures of two points intersect for matrix invariants and semi-invariants. In addition, we give some applications for proving lower bounds on the border rank of tensors.

[发布机构] University of Michigan

[效力级别] tensor rank

[关键词] degree bounds for invariant rings;tensor rank;non-commutative circuits;Mathematics;Science;Mathematics