Gröbner

Gröbner is a German surname that is widely associated in mathematics with Gröbner bases. The term refers to a concept in computational algebraic geometry and commutative algebra named after Austrian mathematician Wolfgang Gröbner (1903–1989). The computational development of the idea was carried forward by Bruno Buchberger, who in the 1960s introduced an algorithm to compute such bases, now known as Buchberger's algorithm.

A Gröbner basis for an ideal I in a polynomial ring k[x1, ..., xn] with respect to a chosen monomial order is a finite subset G of I such that the leading term of every polynomial in I is divisible by the leading term of some polynomial in G. This property allows algorithmic manipulation of polynomials, including testing membership in I and performing reductions to a normal form. Different monomial orders yield different Gröbner bases, and a reduced Gröbner basis is a canonical form with minimal leading coefficients and no redundant elements.

Gröbner bases have broad applications in solving systems of polynomial equations, performing eliminations to project solutions onto subspaces, and studying properties of ideals in algebraic geometry. They are implemented in many computer algebra systems and underpin many modern algorithms in symbolic computation.

Historically, the concept reflects Gröbner’s early work on polynomial ideals, while the practical computation and widespread use stem from Buchberger’s algorithm. Variants and terminology, including Groebner or Grӧbner bases, arise from different spellings of the name in various languages and texts.