Consider four basic arithmetic operators +, −, *, and /, and an arbitrary rational expression of finitely many variables from a given field K (assumed to be algebraically closed) and these operators. Such an expression is computable in finite time; an interesting question is to compute the theoretical lower bound on the number of operations needed to evaluate the expression. Various special forms of such expressions as linear forms and bilinear forms have been studied, but not much is known for arbitrary rational expressions. The purpose of this paper is to obtain some new insight into this problem of algebraic complexity based on synthetic and algebraic geometry. The author has considered linear forms and computational directed acyclic graphs and has developed some interesting theoretical results. He has also studied the relationship of the problem with classical invariant theory.

The paper is rigorous, detailed, and extremely theoretical in the sense that the results cannot readily be used in developing faster algorithms to evaluate expressions. It is fairly self-contained, although readers will need some logic background to go through the proofs. The paper is apparently aimed at pure mathematicians.