Determinant Equivalence Test over Finite Fields and over Q
The determinant polynomial Det_n(x) of degree n is the determinant of a n x n matrix of formal variables. A polynomial f is equivalent to Det_n(x) over a field F if there exists a A in GL(n^2,F) such that f = Det_n(A * x). Determinant equivalence test over F is the following algorithmic task: Given black-box access to a f in F[x], check if f is equivalent to Det_n(x) over F, and if so then output a transformation matrix A in GL(n^2,F). In (Kayal, STOC 2012), a randomized polynomial time determinant equivalence test was given over F = C. But, to our knowledge, the complexity of the problem over finite fields and over Q was not well understood.
In this work, we give a randomized poly(n,log |F|) time determinant equivalence test over finite fields F (under mild restrictions on the characteristic and size of F). Over Q, we give an efficient randomized reduction from factoring square-free integers to determinant equivalence test for quadratic forms (i.e. the n=2 case), assuming GRH. This shows that designing a polynomial-time determinant equivalence test over Q is a challenging task. Nevertheless, we show that determinant equivalence test over Q is decidable: For bounded n, there is a randomized polynomial-time determinant equivalence test over Q with access to an oracle for integer factoring. Moreover, for any n, there is a randomized polynomial-time algorithm that takes input black-box access to a f in Q[x] and if f is equivalent to Det_n over Q then it returns a A in GL(n^2,L) such that f = Det_n(A * x), where L is an extension field of Q and [L : Q] <= n.
The above algorithms over finite fields and over Q are obtained by giving a polynomial-time randomized reduction from determinant equivalence test to another problem, namely the full matrix algebra isomorphism problem. We also show a reduction in the converse direction which is efficient if n is bounded. These reductions, which hold over any F (under mild restrictions on the characteristic and size of F), establish a close connection between the complexity of the two problems. This then leads to our results via applications of known results on the full algebra isomorphism problem over finite fields (Rónyai, STOC 1987 and Rónyai, J. Symb. Comput. 1990) and over Q (Ivanyos {et al}., Journal of Algebra 2012 and Babai {et al}., Mathematics of Computation 1990).
Determinant equivalence test
full matrix algebra isomorphism
Lie algebra
Theory of computation~Algebraic complexity theory
62:1-62:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://eccc.weizmann.ac.il/report/2019/042/.
We would like to thank Youming Qiao for pointing us to the module decomposition algorithm in [Alexander L. Chistov et al., 1997]. NG would like to thank Vineet Nair for discussions on the structure of the Lie algebra of Det. We thank him for sharing his proof of Theorem 10. We also thank anonymous reviewers for their comments.
Ankit
Garg
Ankit Garg
Microsoft Research India, Bangalore, India
Nikhil
Gupta
Nikhil Gupta
Department of Computer Science and Automation, Indian Institute of Science, India
Neeraj
Kayal
Neeraj Kayal
Microsoft Research India, Bangalore, India
Chandan
Saha
Chandan Saha
Department of Computer Science and Automation, Indian Institute of Science, India
10.4230/LIPIcs.ICALP.2019.62
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Ankit Garg, Nikhil Gupta, Neeraj Kayal, and Chandan Saha
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