Complexity measures and hierarchies for the evaluation of integers, polynomials, and n-linear forms

The difficulty of evaluating integers and polynomials has been studied in various frameworks ranging from the addition-chain approach [5] to integer evaluation to recent efforts aimed at generating polynomials that are hard to evaluate [2,8,10]. Here we consider the classes of integers and polynomials that can be evaluated within given complexity bounds and prove the existence of proper hierarchies of complexity classes. The framework in which our problems are cast is general enough to allow any finite set of binary operations rather than just addition, subtraction, multiplication, and division. The motivation for studying complexity classes rather than specific integers or polynomials is analogous to why complexity classes are studied in automata-based complexity: (i) the immense difficulty associated with computing the complexity of a specific integer or polynomial; (ii) the important insight obtained from discovering the structure of the complexity classes.