Computing periods

Abstract

Computer algebra systems (CAS) brought algebraic numbers $\overline{Q}$ accessible to end-users.Still, there are uncountable transcendentals C$\backslash$Q.Kontsevich and Zagier introduced periods to fill in this gap: a period is the difference between the volumes of two algebraic sets. Periods are receiving increasing interest in algebraic model theory as they have finite descriptions, i.e., the polynomials' coefficients, and include all algebraic reals as well as some transcendentals. Recent research has located their worst-case complexity in low levels of the Grzegorczyk Hierarchy. Meanwhile, regular floating-point arithmetic incurs rounding errors that accumulate over time and hamper reliable computations. Interval arithmetic deals with an interval [a

b] where the actual value reside instead of a single floating-point approximation to keep track of the error bounds. The error bound, however, may blow up beyond use. On the other hand, Exact Real Computation is an approximation of infinite computations of Type-2 machines by finite computations with arbitrary precision, i.e., given an output precision n, to produce a dyadic approximation a/$2^n$. In this background, this thesis specify a general problem of computing periods in sense of Exact Real Computation

then, introduces, analyzes, and evaluates a rigorous algorithms for rigorously computing periods up to error $2^{-n}$. We discuss pros and cons of traditional discrete computation and Exact Real Computation with our previous algorithms. Furthermore, we show the general problem of computing periods up to error $2^{-n}$ is \classNP-hard.