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:

singular

DIM

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

Type:

Int

depends on extension:

singular

GROEBNER

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

Type:

Groebner

depends on extension:

singular

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:

singular

PRIME

True if the is ideal a prime ideal.

Type:

Bool

RADICAL

The radical of the ideal.

Type:

Ideal

depends on extension:

singular

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:

bundled:singular

SOLVE

UNDOCUMENTED

from extension:

bundled:singular

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:

bundled:singular