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In this subsection we consider the following setting. Let X ⊂ Rn be a finite set of points whose coordinates are known only with limited precision, and let I (X) = { f ∈ R[x1 , . . , xn ] | f (p) = 0 for all p ∈ X} be its vanishing ideal. Our goal is to compare the different residue class rings P/I (X) where P = R[x1 , . . e. a set made up of points differing by less than the data uncertainty from the corresponding points of X . 32 Kreuzer, Poulisse, Robbiano Given two distinct admissible perturbations X1 and X2 of X , it can happen that their affine coordinate rings P/I (X1 ) and P/I (X2 ) as well as their vanishing ideals I (X1 ) and I (X2 ) have very different bases – this is a well known phenomenon in Gr¨obner basis theory.

Iμ of MxEi E for i = 1, . . , n and then to check for all points (λ1 j1 , . . , λn jn ) such that j1 , . . , jn ∈ {1, . . , μ } whether they are zeros of I . Clearly, this approach has several disadvantages: 1. Usually, the K -eigenvalues of the multiplication matrices MxEi E can only be determined approximately. 2. The set of candidate points is a grid which is typically much larger than the set Z (I). A better approach uses the next theorem. For a K -linear map ϕ : A −→ A , we let ϕ¯ = ϕ ⊗K K : A −→ A .

3) Starting with the last row and working upwards, use the first non-zero entry of each row of R to clean out the non-zero entries above it. 1 From Oil Fields to Hilbert Schemes 25 (4) For i = 1, . . , m , compute the norm ρi of the i -th row of R . If ρi < τ , set this row to zero. Otherwise, divide this row by ρi . Then return the matrix R . This is an algorithm which computes a matrix R in reduced row echelon form. The row space of R is contained in the row space of the matrix A which is obtained from A by setting the columns whose norm is less than τ to zero.