In
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 expression (mathematics), ...
, the Gröbner fan of an
ideal in the
ring of polynomials
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often ...
is a concept in the theory of
Gröbner bases. It is defined to be a
fan consisting of
cone
In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''.
A cone is formed by a set of line segments, half-lines ...
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 an ...
s on that ideal. The concept was introduced by
Mora 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 for exampl ...
.
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 or sounder, a group of boar, see List of animal names
* Singular (band), a Thai jazz pop duo
*'' Singula ...
,
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.
See also
*
Gröbner basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K _1,\ldots,x_n/math> ove ...
*
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 for exampl ...
References
Computer algebra
Algebraic geometry
Commutative algebra
{{commutative-algebra-stub