BigObject Ideal

from application ideal

An ideal in a polynomial ring.

Specializations:

Ideal::Binomial: A binomial ideal represents an ideal which is generated by polynomials of the form p(X) - q(X), where p(X) and q(X) are both multivariate monomials. For example, x1*x2^2 - x1x3x4^10 would be a polynomial of this form, but x1^2 + x1 and 2×1 - x3 are not polynomials of this form. Toric ideals of lattice polytopes are one example of an ideal which may be represented by such a generating set. Since these generator sets have a special form, they may be represented compactly with a matrix.

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

Type:

Groebner

depends on extension:

singular

Properties of GROEBNER:

BINOMIAL_BASIS

An integer matrix representation of a binomial groebner basis. Rows correspond to polynomials, and columns to variables. For example, the row (1, -3, -1, 0, 2) corresponds to the polynomial x0*x4^2 - x^2*x3.

Type:

Matrix<Int,NonSymmetric>

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>>

BINOMIAL

UNDOCUMENTED

Type:

Bool

BINOMIAL_GENERATORS

An integer matrix representation of a generating set of the binomial ideal. Rows correspond to polynomials, and columns to variables. The absolute value of an entry determines the degree of the coefficient of the corresponding column variable in the row polynomial. The parity determines whether it is in the positive or negative monomial. For example, the row (1, -3, -1, 0, 2) corresponds to the polynomial x0*x4^2 - x^2*x3.

Type:

Matrix<Int,NonSymmetric>

Example:

The following declares a binomial ideal via its matrix encoding, and reencodes it into polynomials.

 > $mat = new Matrix<Int>([1,2,0,-4],[3,1,0,1],[-4,-3,0,0]);
 > $ideal = new Ideal(BINOMIAL_GENERATORS=>$mat);

print $ideal→GENERATORS; x_0*x_1^2 - x_3^4 x_0^3*x_1*x_3 - 1 - x_0^4*x_1^3 + 1

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