License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2020.10
URN: urn:nbn:de:0030-drops-126807
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12680/
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Arvind, V. ; Chatterjee, Abhranil ; Datta, Rajit ; Mukhopadhyay, Partha

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

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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].

BibTeX - Entry

@InProceedings{arvind_et_al:LIPIcs:2020:12680,
  author =	{V. Arvind and Abhranil Chatterjee and Rajit Datta and Partha Mukhopadhyay},
  title =	{{A Special Case of Rational Identity Testing and the Bre{\v{s}}ar-Klep Theorem}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Javier Esparza and Daniel Kr{\'a}ľ},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12680},
  URN =		{urn:nbn:de:0030-drops-126807},
  doi =		{10.4230/LIPIcs.MFCS.2020.10},
  annote =	{Keywords: Rational identity testing, ABP with inverses, Bre{\v{s}}ar-Klep Theorem, Invertible image, Amitsur’s theorem}
}

Keywords: Rational identity testing, ABP with inverses, Brešar-Klep Theorem, Invertible image, Amitsur’s theorem
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020


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