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

## Synopsis

• Usage:
G = gCorners(p, I)
• Inputs:
• Optional inputs:
• StandardBasis => ..., default value false
• Tolerance => ..., default value null, optional argument for numerical tolernace
• Outputs:
• G, , 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(Point,Ideal)"
• "gCorners(Point,Matrix)"

## For the programmer

The object gCorners is .