Numerical computation of Gröbner bases

Approximate Gröbner Bases and Overdetermined Algebraic Systems

We discuss computation of Gröbner bases using approximate arithmetic for coefficients. We show how certain considerations of tolerance, corresponding roughly to accuracy and precision from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. We provide examples of solving overdetermined systems of polynomial equations. As a secondary feature we...

Numerical Computation of Gröbner Bases for Zero-dimensional Polynomial Ideals

It is well known that in the computation of Gröbner bases an arbitrarily small perturbation in the coefficients of polynomials may lead to a completely different staircase even if the roots of the polynomials change continuously. This phenomenon is called pseudo singularity in this paper. We show how such phenomenon may be detected and even “repaired” by adding a new variable and a binomial rel...

On Gröbner Bases Computation on a Supercomputer Using REDUCE

Solving finite algebraic systems using numeric Gröbner bases and eigenvalues

Systems of algebraic equations with finitely many solutions arise in many areas of applied mathematics. Among these are motion planning, robotics, computer−aided design, and graphics. We will discuss the design and implementation of a hybrid symbolic− numeric method, and a Mathematica implementation thereof, that finds all solutions to an algebraic system. It makes use of numeric Gröbner bases ...

Efficient decoding of (binary) cyclic codes above the correction capacity of the code using Gröbner bases

This paper revisits the topic of decoding cyclic codes with Gröbner bases. We introduce new algebraic systems, for which the Gröbner basis computation is easier. We show that formal decoding formulas are too huge to be useful, and that the most efficient technique seems to be to recompute a Gröbner basis for each word (online decoding). We use new Gröbner basis algorithms and “trace preprocessi...