Ideal Theory and Gröbner Bases and Affine Algebras [HB 47]

A new algorithm by Allan Steel (to be published) solves a suite of fundamental problems associated with arbitrary algebraic function fields. These fields may be constructed as chains of algebraic extensions (using polynomial quotient rings, affine algebras, or standard algebraic function fields) and transcendental extensions (using rational function fields). The fields may have any characteristic and the algebraic extensions may be inseparable (which can happen in small characteristic).

The problems now solved include decomposition of multivariate ideals and factorization of polynomials over such fields.

New features:

• Primary decomposition and radical computation is now fully supported for ideals over arbitrary algebraic function fields of any characteristic (including non-perfect fields).
• Primary decomposition and radical computation is now supported for ideals over finite fields of arbitrary dimension.
• Factorization of polynomials is now fully supported over arbitrary algebraic function fields of any characteristic.
• New fast modular algorithm for GCD of polynomials over arbitrary algebraic function fields.