Calcium: computing in exact real and complex fields

Calcium is a C library for real and complex numbers in a form suitable for exact algebraic and symbolic computation. Numbers are represented as elements of fields $\mathbb{Q}(a_1,\ldots,a_n)$ where the extensions numbers $a_k$ may be algebraic or transcendental. The system combines efficient field operations with automatic discovery and certification of algebraic relations, resulting in a practical computational model of $\mathbb{R}$ and $\mathbb{C}$ in which equality is rigorously decidable for a large class of numbers.

💡 Research Summary

Calcium is a C library designed to perform exact arithmetic on real and complex numbers by representing each number as an element of a finitely generated field ℚ(a₁,…,aₙ). The extensions aₖ may be algebraic (defined by minimal polynomials and isolating balls) or transcendental (π, γ, e^z, log z, etc.), and are introduced symbolically rather than numerically. The core mathematical framework treats the field as the fraction field of the polynomial ring ℚ