A Special Case of Rational Identity Testing and the Brešar-Klep Theorem
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].
Rational identity testing
ABP with inverses
Brešar-Klep Theorem
Invertible image
Amitsur’s theorem
Theory of computation
Theory of computation~Algebraic complexity theory
10:1-10:14
Regular Paper
We thank the reviewers of MFCS 2020 for their invaluable feedback.
V.
Arvind
V. Arvind
Institute of Mathematical Sciences (HBNI), Chennai, India
Abhranil
Chatterjee
Abhranil Chatterjee
Institute of Mathematical Sciences (HBNI), Chennai, India
Rajit
Datta
Rajit Datta
Chennai Mathematical Institute, India
Partha
Mukhopadhyay
Partha Mukhopadhyay
Chennai Mathematical Institute, India
10.4230/LIPIcs.MFCS.2020.10
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V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay
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