On the Complexity of a Gröbner Basis Algorithm

While the computation of Gröbner bases is known to be an expspace-complete problem, the generic behaviour of algorithms for their computation is much better. We study generic properties of Gröbner bases and analyse precisely the best algorithm currently known, F5. 1. Gröbner Bases Gröbner bases are a fundamental tool in computational algebra. They provide a multivariate generalization of Euclidean division and Euclid’s algorithm for the gcd, as well as a generalization of Gaussian elimination to higher degrees. A very clear introduction is given in [3]; in this section we recall the basic definitions and properties. 1.1. Definitions. We consider polynomials in K[x] = K[x1, . . . , xn], where K is a field. The first step is to define a generalization of the univariate degree. Definition 1. A monomial ordering is a total order on the set of monomials xα that is compatible with the product and such that 1 is the smallest monomial. A monomial ordering can be given by a nonsingular real matrix A: the vectors of exponents are multiplied by the matrix and the resulting vectors are compared lexicographically. A technical condition encodes that 1 is minimal. Basic examples of orderings are: the lexicographic order, with identity matrix; the total degree order, also called grevlex, whose matrix has a first line of 1’s above an antidiagonal matrix of −1’s; the elimination orders whose matrix decompose diagonally into blocks of grevlex matrices. An order is said to refine the degree when the corresponding matrix has a fist line of 1’s. A polynomial can be expanded as a sum of terms, each term being a monomial times a coefficient. The leading term LT(p) of a polynomial p is then defined with respect to any monomial ordering. The next step is to define an analogue of the Euclidean division. This process is called reduction, it depends on a given monomial ordering. Given a polynomial f and a set of polynomials B = {f1, . . . , fs}, it returns a polynomial r such that f = a1f1 + · · ·+ asfs + r, where ai ∈ K[x] for i = 1, . . . , s, and the leading monomials of the fi’s do not divide that of r. One says that f reduces to r by B. Definition 2. A Gröbner basis of an ideal I ⊂ K[x] for a given monomial ordering is a finite set B ⊂ I such that any f ∈ I reduces to 0 by B. The basis is called reduced when the fi’s all have leading coefficient 1 and when none of the fi’s involves a monomial which reduces by B \ {fi}. 86 On the Complexity of a Gröbner Basis Algorithm An important consequence of the Hilbert basis theorem is the existence of Gröbner bases, thus of finite sets of generators, for all polynomial ideals. For a given monomial ordering, any ideal has a single reduced Gröbner basis.