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
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i3 : p = point matrix{{1,1}};
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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
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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
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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
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i8 : q = point matrix{{1.4142136, 1.4142136, 0}};
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i9 : gCorners(q, J, Tolerance=>1e-5)
o9 = | y z |
1 2
o9 : Matrix R <--- R
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i10 : gCorners(q, J, StandardBasis=>true)
o10 = | -1x+y -.5x2z-1.41421xz+1z |
1 2
o10 : Matrix R <--- R
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The object gCorners is a method function with options.