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I am looking for an algebraic software package that provides a routine that computes the vanishing ideal of a finite set of points. So far i am working with Macaulay2 but i have not been able to find any such routine. Does anyone know otherwise? Is anyone familiar with some other software that provides this feature?

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Can't write down the vanishing ideal directly? –  Martin Brandenburg Oct 4 '13 at 21:13
    
@MartinBrandenburg: I can write it abstractly; it is the intersection of the vanishing ideals of all points. But computing it, e.g. finding a Groebner basis, is far from trivial in terms of computational complexity. For example, a simple for loop in Macaulay2 that computes this intersection (which is the same as the product) incrementally, gets stuck after 50 points in $\mathbb{Q}^3$. What i want, is to be able to compute the vanishing ideal of as many as 500 points in an ambient space as big as $\mathbb{Q}^7$. –  Manos Oct 4 '13 at 21:31
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Maple has the VanishingIdeal command in the PolynomialIdeals package. E.g. for the points {[5,4,4], [4,0,2], [6,4,1]} in [x,y,z] you would do:

with(PolynomialIdeals):
J := VanishingIdeal({[5,4,4], [4,0,2], [6,4,1]}, [x,y,z]);
Simplify(PrimeDecomposition(J));  # check
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Thanks. Do you have any idea how fast this routine is? What if we have 100 points in $\mathbb{R}^3$? How fast does it do the computation? –  Manos Oct 7 '13 at 14:48
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