A Special Case of Rational Identity Testing and the Brešar-Klep Theorem

Abstract

We explore a special case of rational identity testing and algorithmic versions of two theorems on noncommutative polynomials, namely, Amitsur's theorem [S.A Amitsur, 1966] and the Brešar-Klep theorem [Brešar and Klep, 2008] when the input polynomial is given by an algebraic branching program (ABP). Let f be a degree-d n-variate noncommutative polynomial in the free ring Q<x_1,x_2,...,x_n> over rationals.  
1) We consider the following special case of rational identity testing: Given a noncommutative ABP as white-box, whose edge labels are linear forms or inverses of linear forms, we show a deterministic polynomial-time algorithm to decide if the rational function computed by it is equivalent to zero in the free skew field Q<(X)>. Given black-box access to the ABP, we give a deterministic quasi-polynomial time algorithm for this problem.
2) Amitsur's theorem implies that if a noncommutative polynomial f is nonzero on k x k matrices then, in fact, f(M_1,M_2,...,M_n) is invertible for some matrix tuple (M_1,M_2,...,M_n) in (M_k(ℚ))^n. While a randomized polynomial time algorithm to find such (M_1,M_2,...,M_n) given black-box access to f is simple, we obtain a deterministic s^{O(log d)} time algorithm for the problem with black-box access to f, where s is the minimum ABP size for f and d is the degree of f.
3) The Brešar-Klep Theorem states that the span of the range of any noncommutative polynomial f on k x k matrices over Q is one of the following: zero, scalar multiples of I_k, trace-zero matrices in M_k(Q), or all of M_k(Q). We obtain a deterministic polynomial-time algorithm to decide which case occurs, given white-box access to an ABP for f. We also give a deterministic s^{O(log d)} time algorithm given black-box access to an ABP of size s for f. Our algorithms work when k >= d. 
Our techniques are based on some automata theory combined with known techniques for noncommutative ABP identity testing [Ran Raz and Amir Shpilka, 2005; Michael A. Forbes and Amir Shpilka, 2013].