computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...

, the Gröbner fan of an

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

in the

ring of polynomials In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...

is a concept in the theory of Gröbner bases. It is defined to be a fan consisting of

cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

s that correspond to different

monomial order In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all ( monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., * If u \leq v and ...

s on that ideal. The concept was introduced by

Mora Mora may refer to: People * Mora (surname) Places Sweden * Mora, Säter, Sweden * Mora, Sweden, the seat of Mora Municipality * Mora Municipality, Sweden United States * Mora, Louisiana, an unincorporated community * Mora, Minnesota, a city * M ...

and Robbiano in 1988. The result is a weaker version of the result presented in the same issue of the journal by Bayer and Morrison. Gröbner fan is a base for the nowadays active field of

tropical geometry In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: : x \oplus y = \min\, : x \otimes y = x + y. So f ...

. One implementation of the Gröbner fan is called Gfan, based on an article of Fukuda, et al. which is included in some computer algebra systems such as

Singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...

Macaulay2 Macaulay2 is a free computer algebra system created by Daniel Grayson (from the University of Illinois at Urbana–Champaign) and Michael Stillman (from Cornell University) for computation in commutative algebra and algebraic geometry. Overview ...

, and

CoCoA Cocoa may refer to: Chocolate * Chocolate * ''Theobroma cacao'', the cocoa tree * Cocoa bean, seed of ''Theobroma cacao'' * Chocolate liquor, or cocoa liquor, pure, liquid chocolate extracted from the cocoa bean, including both cocoa butter and ...

References

Computer algebra Algebraic geometry Commutative algebra {{commutative-algebra-stub

See also

References