Towards faster real algebraic numbers

This paper presents a new encoding scheme for real algebraic number manipulations which enhances current Axiom’s real closure. Algebraic manipulations are performed using different instantiations of sub-resultant-like algorithms instead of Euclidean-like algorithms. We use these algorithms to compute polynomial gcds and Bezout relations, to compute the roots and the signs of algebraic numbers. This allows us to work in the ring of real algebraic integers instead of the field of real algebraic numbers avoiding many denominators.