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NoetherianOperators :: gCorners

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

Synopsis

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+z |

              1       2
o10 : Matrix R  <--- R

Ways to use gCorners :

For the programmer

The object gCorners is a method function with options.