Addition Requirements for Rational Functions

A notion of rank or independence for arbitrary sets of rational functions is developed, which bounds from below the number of additions and subtractions required of all straight-line algorithms which compute those functions. This permits a uniform derivation of the best lower bounds known for a number of familiar sets of rational functions. The result is proved without the use of substitution arguments. This not only provides an interesting contrast to standard approaches for arithmetic lower bounds, but also allows the algebraic setting to be somewhat generalized. Keywords: additions, algorithms, analysis of algorithms, arithmetic complexity, computational complexity, dimensionality, lower bounds, matrix multiplication, optimality, polynomials, rational functions.