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Generic complexity of the Diophantine problem
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Alexander Rybalov
Published/Copyright:
May 2, 2013
Abstract.
The generic-case approach to algorithmic problems was suggested by Myasnikov, Kapovich, Schupp and Shpilrain in 2003. This approach studies the behavior of an algorithm on “most” or “typical” inputs. The remaining inputs form the so-called black hole of the algorithm. In the present paper we consider Hilbert's tenth problem and use arithmetic circuits for the representation of Diophantine equations. We prove that this Diophantine problem is generically hard in the following sense. For every generic polynomial algorithm deciding this problem, there exists a polynomial algorithm for random generation of inputs from the black hole.
Received: 2012-08-16
Published Online: 2013-05-02
Published in Print: 2013-05-01
© 2013 by Walter de Gruyter Berlin Boston
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Articles in the same Issue
- Masthead
- Shortlex automaticity and geodesic regularity in Artin groups
- Generic complexity of the Diophantine problem
- Secrecy without one-way functions
- Constructing a pseudo-free family of finite computational groups under the general integer factoring intractability assumption
- A new algorithm to find apartments in coset geometries
- Faithful representations of limit groups II
- Public key exchange using matrices over group rings
