Generalized normal forms and polynomial system solving

This paper describes a new method for computing the normal form of a polynomial modulo a zero-dimensional ideal I. We give a detailed description of the algorithm, a proof of its correctness, and finally experimentations on classical benchmark polynomial systems. The method that we propose can be thought as an extension of both the Gröbner basis method and the Macaulay construction. We have weaken the monomial ordering requirement for bases computations, which allows us to construct new type of representations for the quotient algebra. This approach yields more freedom in the linear algebra steps involved, which allows us to take into account numerical criteria while performing the symbolic steps. This is a new feature for a symbolic algorithm, which has a huge impact on the practical efficiency.