Edmonds' problem and the membership problem for orbit semigroups of quiver representations

A central problem in algebraic complexity, posed by J. Edmonds [16], asks to decide if the span of a given l-tuple W=(W1,…,Wl) N×N complex matrices contains non-singular matrix. In this paper, we provide quiver invariant theoretic approach problem. Viewing W as representation l-Kronecker Kl, Edmonds' can be rephrased asking there exists semi-invariant on space (CN×N)l weight (1,?1) that does not vanish at W. other words, is belongs orbit semigroup Let Q an arbitrary acyclic and Q. We study membership for semi-group focusing so-called W-saturated weights. first show any ?, checking ? done deterministic polynomial time. Next, let (Q,R) bound with algebra A=KQ/?R? assume satisfies relations R. A/AnnA(W) tame then W-saturated. Our results systematic way producing families tuples which solved effectively.