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# Copyright (c) 1997-2021
# Ewgenij Gawrilow, Michael Joswig, and the polymake team
# Technische Universität Berlin, Germany
# https://polymake.org
#
# This program is free software; you can redistribute it and/or modify it
# under the terms of the GNU General Public License as published by the
# Free Software Foundation; either version 2, or (at your option) any
# later version: http://www.gnu.org/licenses/gpl.txt.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#-------------------------------------------------------------------------------
# Note:
# So far we assume all ideals living in a polynomial ring with a global
# monomial ordering. If you need something else please contact the authors.
object Ideal {
# @category Commutative algebra
# The depth of the ideal.
# @depends singular
property DEPTH : Int;
# @category Commutative algebra
# The dimension of the ideal, i.e. the Krull dimension of Polynomial ring/Ideal.
# @depends singular
property DIM : Int;
# @category Commutative algebra
# The number of variables of the polynomial ring containing the ideal.
property N_VARIABLES : Int;
# @category Input properties
# A set of generators usually given by the user and not unique.
property GENERATORS : Array<Polynomial> {
sub equal {
defined(find_permutation(@_))
}
}
# @category Commutative algebra
# Subobject containing properties that depend on the monomial ordering of the ring.
# @depends singular
property GROEBNER : Groebner : multiple;
# @category Commutative algebra
# The Hilbert polynomial of the ideal. For toric ideals this is linked with
# the Ehrhart polynomial.
property HILBERT_POLYNOMIAL : Polynomial;
# @category Commutative algebra
# True if the ideal can be generated by homogeneous polynomials.
property HOMOGENEOUS : Bool;
# @category Commutative algebra
# True if the ideal can be generated by monomials.
property MONOMIAL : Bool;
# @category Commutative algebra
# True if the is ideal a prime ideal.
property PRIME : Bool;
# @category Commutative algebra
# 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.
property PRIMARY : Bool;
# @category Commutative algebra
# An array containing the primary decomposition of the given ideal, i.e. the
# contained ideals are [[PRIMARY]] and their intersection is the given ideal.
# @depends singular
property PRIMARY_DECOMPOSITION : Array<Ideal>;
# @category Commutative algebra
# The radical of the ideal.
# @depends singular
property RADICAL : Ideal;
# @category Commutative algebra
# True if the ideal is the zero ideal.
property ZERO : Bool;
property BINOMIAL : Bool;
}
# This object contains all properties of an ideal that depend on the monomial ordering.
object Groebner {
# @category Commutative algebra
# The elements of the Groebner basis corresponding to the given order. This
# may vary for different algorithms, even if the order stays the same.
# @depends singular
property BASIS : Array<Polynomial>;
# @category Commutative algebra
# The initial order corresponding to the given order. This is always a
# [[ideal::Ideal::MONOMIAL|MONOMIAL]] ideal, even if only a weight vector is provided. Internally this
# weight vector will be concatenated with a total order.
# @depends singular
property INITIAL_IDEAL : Ideal;
# @category Commutative algebra
# The initial forms of all polynomials in the [[BASIS]], with respect to either the [[ORDER_VECTOR]] or
# the first row of the [[ORDER_MATRIX]].
# @depends singular
property INITIAL_FORMS : Array<Polynomial>;
# @category Input properties
# The matrix defining the monomial ordering.
# For performance reasons this is realized via several weight vectors preceding a lexicographic order.
# (Singular: a(row), a(row),...,lp)
#
# Note that only one of [[ORDER_MATRIX]], [[ORDER_VECTOR]], [[ORDER_NAME]]
# should be given.
property ORDER_MATRIX : Matrix<Int>;
# @category Input properties
# A weight vector for the monomial ordering, a reverse lexicographic order will be used
# as tie-breaker. (Singular: ''wp(vector)'')
# This vector is expected to consist of positive integers only.
#
# Note that only one of [[ORDER_MATRIX]], [[ORDER_VECTOR]], [[ORDER_NAME]]
# should be given.
property ORDER_VECTOR : Vector<Int>;
# @category Input properties
# A string containing the name of the monomial ordering. Currently we follow
# the singular conventions, i.e. dp, lp, rp, ds, etc.
#
# Note that only one of [[ORDER_MATRIX]], [[ORDER_VECTOR]], [[ORDER_NAME]]
# should be given.
property ORDER_NAME : String;
}
# 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 2x1 - 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.
declare object_specialization Binomial = Ideal {
precondition : BINOMIAL;
# 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.
#
# @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
property BINOMIAL_GENERATORS : Matrix<Int>;
property GROEBNER {
# 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.
property BINOMIAL_BASIS : Matrix<Int>;
}
}
# Local Variables:
# mode: perl
# cperl-indent-level:3
# indent-tabs-mode:nil
# End:
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