Garg, A and Gupta, N and Kayal, N and Saha, C (2019) Determinant equivalence test over finite fields and over q. In: 46th International Colloquium on Automata, Languages, and Programming, 7 July 2019 - 12 July 2019, Patras.
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The determinant polynomial Detn(x) of degree n is the determinant of a n × n matrix of formal variables. A polynomial f is equivalent to Detn(x) over a field F if there exists a A ∈ GL(n2, F) such that f = Detn(A · x). Determinant equivalence test over F is the following algorithmic task: Given black-box access to a f ∈ F[x], check if f is equivalent to Detn(x) over F, and if so then output a transformation matrix A ∈ GL(n2, 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 ∈ Q[x] and if f is equivalent to Detn over Q then it returns a A ∈ GL(n2, Ł) such that f = Detn(A · x), where Ł is an extension field of Q and [Ł: 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).
| Item Type: | Conference Paper |
|---|---|
| Publication: | Leibniz International Proceedings in Informatics, LIPIcs |
| Publisher: | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
| Additional Information: | The copyright for this article belongs to Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
| Keywords: | Automata theory; Equivalence classes; Linear transformations; Mixtures; Number theory; Polynomial approximation; Set theory, Equivalence tests; Formal variables; Full matrix algebra isomorphism; Isomorphism problems; Lie Algebra; Polynomial-time algorithms; Randomized reductions; Transformation matrices, Matrix algebra |
| Department/Centre: | Division of Electrical Sciences > Computer Science & Automation |
| Date Deposited: | 23 Dec 2022 09:39 |
| Last Modified: | 23 Dec 2022 09:39 |
| URI: | https://eprints.iisc.ac.in/id/eprint/78512 |
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