documentation:latest:ideal:ideal

Available versions of this document: latest release, release 4.0, release 3.6, release 3.5, nightly master

Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2

# BigObject Ideal

from application ideal

An ideal in a polynomial ring.

Properties of an ideal computed via commutative algebra.

DEPTH

The depth of the ideal.

Type:
Int
depends on extension:

DIM

The dimension of the ideal, i.e. the Krull dimension of Polynomial ring/Ideal.

Type:
Int
depends on extension:

GROEBNER

Subobject containing properties that depend on the monomial ordering of the ring.

Type:
Groebner
depends on extension:

HILBERT_POLYNOMIAL

The Hilbert polynomial of the ideal. For toric ideals this is linked with the Ehrhart polynomial.

Type:
Polynomial<Rational,Int>

HOMOGENEOUS

True if the ideal can be generated by homogeneous polynomials.

Type:
Bool

MONOMIAL

True if the ideal can be generated by monomials.

Type:
Bool

N_VARIABLES

The number of variables of the polynomial ring containing the ideal.

Type:
Int

PRIMARY

True if the ideal is a primary ideal. I.e. its RADICAL is PRIME and in the quotient ring by the ideal every zero divisor is nilpotent.

Type:
Bool

PRIMARY_DECOMPOSITION

An array containing the primary decomposition of the given ideal, i.e. the contained ideals are PRIMARY and their intersection is the given ideal.

Type:
Array<Ideal>
depends on extension:

PRIME

True if the is ideal a prime ideal.

Type:
Bool

RADICAL

Type:
Ideal
depends on extension:

ZERO

True if the ideal is the zero ideal.

Type:
Bool

Properties defining an ideal.

GENERATORS

A set of generators usually given by the user and not unique.

Type:
Array<Polynomial<Rational,Int>>

SATURATION

UNDOCUMENTED

from extension:

SOLVE

UNDOCUMENTED

from extension:

contains_monomial(String s)

Check via saturation whether the ideal contains a monomial. Returns a monomial from the ideal or the trivial monomial if there is none.

Parameters:

String s: Optional term order (see ORDER_NAME) for intermediate Groebner bases, default: “dp”

Returns:
Polynomial
from extension:

• documentation/latest/ideal/ideal.txt