Algorithms for Algebraic Curves

1.1. Languages. Let Σ be an alphabet i.e. a finite set of symbols. The set of finite words on Σ is denoted Σ∗. A language is a subset of Σ∗. For a language L ⊂ Σ∗ the associated decision problem is as follows. On input a word w ∈ Σ∗ we want to decide if w belongs to L or not. Examples : even integers, prime integers, etc. For a function f : Σ∗ → Σ∗, the associated functional problem is as follows. On input a word w ∈ Σ∗ we want to evaluate f(w). Examples : multiplying, factoring, etc.