Online Gröbner Basis [OGB]

Abstract

1 What OGB does OGB calculates in the polynomical ring Q[x1, x2, ..., xn], i.e the set of all polynomials in n variables with rational coefficients. It does all Gröbner Basis calculations using lexicographical ordering, defined as follows: A given term T1 = c1x 1 x a2 2 ...xn n is lexicographically greater than another term T2 = c2x b1 1 x2 2 ...xn n (and so we write T1 > T2) iff the first non-zero term in the vector (a1 − b1, a2 − b2, ..., an − bn) is positive. Here cj ∈ Q, i.e. cj = m/n with m, n ∈ Z. OGB calculates the Gröbner Basis, the minimal Gröbner Basis (with the property that the leading monomial in polynomial i is not a factor of the leading monomial in polynomial j for any i 6= j), or the reduced Gröbner Basis (with the property that the leading monomial in polynomial i is not a factor of any monomial in polynomial j for j 6= i). 2 ...yet another Gröbner Basis calculator??? This software implements known algorithms that have been implemented many other places. However the focus of OGB is on 1. Applicability to solving systems of equations: This is why the lexicographic ordering is chosen, and the reduced basis is calculated. It is known that if there are a sufficient number of equations, and if the system is solvable, the reduced basis under lexicographical ordering is always triangular and moreover the last polynomial is univariate. 2. Pedagogy: OGB is written to educate both students and academics who are not mathematicians but use mathematics (scientists, engineers,...)