AlgebraAufwand

AlgebraAufwand is a proposed metric for quantifying the computational effort required to perform symbolic algebra tasks on a given input. It is intended to capture the inherent difficulty of operations such as simplification, expansion, factorization, substitution, and solving within a unified framework.

Formalization and scope: In a basic model, the AlgebraAufwand A(P) of a problem P is defined as the minimum number of primitive algebraic operations needed to transform the input into the desired output. Primitive operations typically include addition, subtraction, multiplication, division, exponentiation, and basic substitutions. More advanced tasks such as polynomial gcd, factorization, or the computation of Gröbner bases can be incorporated as single steps or as cost-weighted steps, depending on the chosen cost model. The metric can be considered static (worst-case for a fixed input size) or dynamic (instance-specific).

Computation and interpretation: Exactly determining A(P) is generally difficult, so researchers rely on proxy measures to estimate it. Common proxies include the number of nonzero terms, the total degree, sparsity of the expression, the size of intermediate expressions, and known algorithmic costs for specific tasks. For example, expanding a high-degree binomial or multinomial typically increases the estimated AlgebraAufwand due to rapid growth in term count, whereas solving a sparse linear system may have lower estimated cost under efficient methods.

Relation to theory and practice: The concept aligns with algebraic complexity theory and the cost models used in computer algebra systems. It serves as a heuristic to compare algorithms, benchmark software, and teach about problem difficulty. Limitations include dependence on the chosen set of primitive operations and the fact that many problems admit multiple, incomparable decompositions of effort.