from application ideal
An ideal in a polynomial ring.
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.
DIM
The dimension of the ideal, i.e. the Krull dimension of Polynomial ring/Ideal.
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.
HILBERT_POLYNOMIAL
The Hilbert polynomial of the ideal. For toric ideals this is linked with the Ehrhart polynomial.
HOMOGENEOUS
True if the ideal can be generated by homogeneous polynomials.
MONOMIAL
True if the ideal can be generated by monomials.
N_VARIABLES
The number of variables of the polynomial ring containing the ideal.
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.
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.
PRIME
True if the is ideal a prime ideal.
RADICAL
The radical of the ideal.
ZERO
True if the ideal is the zero ideal.
Properties defining an ideal.
GENERATORS
A set of generators usually given by the user and not unique.
BINOMIAL
UNDOCUMENTED
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.
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
SOLVE
UNDOCUMENTED
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.
String
s
: Optional term order (see ORDER_NAME
) for intermediate Groebner bases, default: “dp”