Structures of precision losses in computing approximate Gröbner bases

Pivoting in Extended Rings for Computing Approximate Gröbner Bases

It is well known that in the computation of Gröbner bases arbitrarily small perturbations in the coefficients of polynomials may lead to a completely different staircase, even if the solutions of the polynomial system change continuously. This phenomenon is called artificial discontinuity in Kondratyev’s Ph.D. thesis. We show how such phenomenon may be detected and even “repaired” by using a ne...

Computing inhomogeneous Gröbner bases

In this paper we describe how an idea centered on the concept of self-saturation allows several improvements in the computation of Gröbner bases via Buchberger’s Algorithm. In a nutshell, the idea is to extend the advantages of computing with homogeneous polynomials or vectors to the general case. When the input data are not homogeneous, we use a technique described in Section 2: the main tool ...

FGb: A Library for Computing Gröbner Bases

FGb is a high-performance, portable, C library for computing Gröbner bases over the integers and over finite fields. FGb provides high quality implementations of state-of-the-art algorithms (F4 and F5) for computing Gröbner bases. Currently, it is one of the best implementation of these algorithms, in terms of both speed and robustness. For instance, FGb has been used to break several cryptosys...

Modular algorithms for computing Gröbner bases

Intermediate coefficient swell is a well-known difficulty with Buchberger’s algorithm for computing Gröbner bases over the rational numbers. p-Adic and modular methods have been successful in limiting intermediate coefficient growth in other computations, and in particular in the Euclidian algorithm for computing the greatest common divisor (GCD) of polynomials in one variable. In this paper we...

Computing Gröbner bases for quasi-homogeneous systems