Available versions of this document: latest release, 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.

Commutative algebra

Properties of an ideal computed via commutative algebra.

DEPTH

The depth of the ideal.

Type:
depends on extension:

DIM

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

Type:
depends on extension:

GROEBNER

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

Type:
depends on extension:

HILBERT_POLYNOMIAL

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

Type:

HOMOGENEOUS

True if the ideal can be generated by homogeneous polynomials.

Type:

MONOMIAL

True if the ideal can be generated by monomials.

Type:

N_VARIABLES

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

Type:

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:

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:
depends on extension:

PRIME

True if the is ideal a prime ideal.

Type:

The radical of the ideal.

Type:
depends on extension:

ZERO

True if the ideal is the zero ideal.

Type:

Input properties

Properties defining an ideal.

GENERATORS

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

Type:

no category

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:
from extension:

• documentation/latest/ideal/ideal.txt