application ideal

This application allows to define ideals and enables other applications to use these. For example we can compute the tropical variety of an ideal via gfan in the application 'tropical'. Using this application with the bundled extension Singular adds a lot more commutative algebra power.

imports from:

• application common

• Groebner:
  The Groebner object collects information about an ideal depending on the monomial ordering of the ambient ring.

• Ideal:
  An ideal in a polynomial ring.

• SlackIdeal:
  The type SlackIdeal is a thin wrapper around Ideal that allows avoiding saturation. The current algorithm for computing the SLACK_IDEAL of a Polytope first computes an ideal whose saturation is the actual slack ideal. This last step is very expensive and probably necessary for many applications. Hence the SlackIdeal contains a second ideal NON_SATURATED as a property and the actual saturation step is only done once one asks for the GENERATORS of the SlackIdeal.

With these clients you can create ideals belonging to various parameterized families which occur frequently in comumutative algebra.

pluecker_ideal(Int d, Int n)

Generates the ideal of all Grassmann-Plücker relations of dxd minors of an dxn matrix. For the algorithm see Sturmfels: Algorithms in invariant theory, Springer, 2nd ed., 2008

Parameters:

Int d

Int n

Returns:

Ideal

pluecker_ideal(Matroid m)

Generates the ideal of all Grassmann-Plücker relations of the given matroid. For the algorithm see Sturmfels: Algorithms in invariant theory, Springer, 2nd ed., 2008

Parameters:

Matroid m

Returns:

Ideal

Functions, methods and objects and attached from/to Singular.

load_singular_library(String s)

Loads a SINGULAR library

Parameters:

String s

from extension:

bundled:singular

singular_eval(String s)

Executes given string with Singular

Parameters:

String s

from extension:

bundled:singular

singular_get_var(String s)

Retrieves a variable from 'Singular'

Parameters:

String s: variable name

Returns:

List

from extension:

bundled:singular

slack_ideal_non_saturated

Computes the non-saturated slack ideal of a polytope, as described in > João Gouveia, Antonio Macchia, Rekha R. Thomas, Amy Wiebe: > The Slack Realization Space of a Polytope > (https://arxiv.org/abs/1708.04739)

from extension:

bundled:singular

Functions, methods and objects and attached from/to Singular.

SingularIdeal

An intermediate object wrapping the ideal on the Singular side and providing its methods.

from extension:

bundled:singular