Groebnerbases

Groebner bases are a central tool in computational algebraic geometry and computer algebra for studying ideals in polynomial rings over a field. Let k be a field and consider the polynomial ring k[x1, ..., xn]. Fix a monomial order ≺. For a nonzero polynomial f, LT(f) denotes its leading term with respect to ≺. An ideal I ⊆ k[x1, ..., xn] has a Groebner basis G with respect to ≺ if the ideal of leading terms LT(I) is generated by {LT(g) : g ∈ G}. In that case, many questions about I can be reduced to combinatorial questions about LT(I).

A foundational construction is Buchberger’s algorithm. Starting from a generating set of I, the algorithm computes S-polynomials of pairs of current basis elements and reduces them modulo the current basis. If every S-polynomial reduces to zero, the set is a Groebner basis for I. For any ideal I in a polynomial ring over a field, a finite Groebner basis exists for any fixed monomial order. The reduced Groebner basis is unique for a given I and ≺; it is monic and no leading term is redundant.

Monomial orders and elimination are important aspects. Different orders yield different Groebner bases. A common order, lexicographic, can be chosen to facilitate elimination of variables, enabling computation of elimination ideals and solving polynomial systems via back-substitution from the basis.

Applications are broad: solving systems of polynomial equations, testing ideal membership, computing dimensions and Hilbert functions, performing intersection or saturation operations, and more within algebraic geometry and computational algebra.

Computational aspects include extensive implementations in computer algebra systems such as Magma, Macaulay2, Singular, Maple, and Mathematica. The algorithms are powerful but can have high computational complexity in general.