Determination Problems for Orbit Closures and Matrix Groups
summary
Gives a concrete algorithm: a polynomial-space procedure that decides, given an algebraic variety and a number s, whether the variety equals the orbit closure of some point under an s-generated commutative algebraic matrix group. Uses Jordan decomposition, module theory and arithmetic lattices, reduces semi‑simple parts to guessing generators nondeterministically, and checks membership via real‑closed (or complex algebraic) field tests. Also shows the cyclic case outputs a generator in the cyclic case and notes s is effectively bounded by dimension D, but leaves noncommutative groups open.
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chapters (12)
- 0:00Introduction
- 1:05Orbit problems in matrix groups
- 3:34Algebraic invariants in loops
- 5:20Fibonacci numbers on algebraic sets
- 8:34Loop synthesis and orbit closure
- 11:48Cyclic group determination
- 13:37Multiple matrix generators
- 16:11Problem statements and complexity
- 19:01Commutativity and future work
- 20:05Mathematical structure
- 22:54Polynomial space procedures
- 23:20Q&A
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