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The Constraint Satisfaction Problem Dichotomy Conjecture of Feder and Vardi (1999) has in the last 10 years been profitably reformulated as a conjecture about the set SP fin (A) of subalgebras of finite Cartesian powers of a finite universal algebra A. One particular strategy, advanced by Dalmau in his doctoral thesis (2000), has confirmed the conjecture for a certain class of finite algebras A which, among other things, have the property that the number of subalgebras of A n is bounded by an exponential polynomial. In this paper we characterize the finite algebras A with this property, which we call having few subpowers, and develop a representation theory for the subpowers of algebras having few subpowers. Our characterization shows that algebras having few subpowers are the finite members of a newly discovered and surprisingly robust Maltsev class defined by the existence of a special term we call an edge term. We also prove some tight connections between the asymptotic behavior of the number of subalgebras of A n and some related functions on the one hand, and some standard algebraic properties of A on the other hand. The theory developed here was applied to the Constraint Satisfaction Problem Dichotomy Conjecture, completing Dalmau's strategy.

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