gCorners -- generators of the initial ideal of a polynomial ideal

Synopsis

Usage:

G = gCorners(p, I)
Inputs:
- p, a point,
- I, an ideal, or a one-row matrix of generators
Optional inputs:
- StandardBasis => ..., default value false
- Tolerance => ..., default value null, optional argument for numerical tolerance
Outputs:
- G, a matrix, generators of the initial ideal in a one-row matrix

Description

This method computes the generators of the initial ideal of an ideal, with respect to a local order. These are precisely the monomials in the corners of the staircase diagram of the initial ideal. The ring of the ideal should be given a (global) monomial order and the local order will be taken to be the reverse order. The point p is moved to the origin, so the monomial generators represent terms of the Taylor expansion at p.

i1 : R = CC[x,y];

i2 : I = ideal{x^2-y^2}

            2    2
o2 = ideal(x  - y )

o2 : Ideal of R

i3 : p = point matrix{{1,1}};

i4 : gCorners(p, I)
-- warning: experimental computation over inexact field begun
--          results not reliable (one warning given per session)

o4 = | y |

             1      1
o4 : Matrix R  <-- R

If the optional argument StandardBasis is set to true, the output is instead a matrix of elements of the ideal with the point p translated to the origin such that the lead terms generate the initial ideal, i.e., a standard basis. Note that the coordinates of the standard basis elements are translated to be centered at the point p.

i5 : S = gCorners(p, I, StandardBasis=>true)

o5 = | -.5x2+.5y2-1x+y |

             1      1
o5 : Matrix R  <-- R

i6 : R = CC[x,y,z];

i7 : J = ideal{z*(x*y-4), x-y}

o7 = ideal (x*y*z - 4z, x - y)

o7 : Ideal of R

i8 : q = point matrix{{1.4142136, 1.4142136, 0}};

i9 : gCorners(q, J, Tolerance=>1e-5)

o9 = | y z |

             1      2
o9 : Matrix R  <-- R

i10 : gCorners(q, J, StandardBasis=>true)

o10 = | -1x+y -.5x2z-1.41421xz+1z |

              1      2
o10 : Matrix R  <-- R

Ways to use gCorners :

gCorners(AbstractPoint,Ideal)
gCorners(AbstractPoint,Matrix)

For the programmer

The object gCorners is a method function with options.