Generic complexity of the Diophantine problem

Abstract

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.