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PolynomialIdeals[VanishingIdeal] - compute the vanishing ideal for finite a set of points
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Calling Sequence
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VanishingIdeal(S, X)
VanishingIdeal(S, X, T, p)
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Parameters
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S
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list or set of points
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X
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list of variable names
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T
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(optional) monomial order
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p
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(optional) characteristic, a non-negative integer
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Description
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The VanishingIdeal command constructs the vanishing ideal for a set of points in affine space. The output of this command is the ideal of polynomials that vanish (that is, are identically zero) on S.
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The first argument must be a list or set of points in affine space. Each point is given as a list with coordinates corresponding to the variables in X.
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The third argument is optional, and specifies a monomial order for which a Groebner basis is computed. If omitted, VanishingIdeal chooses lexicographic order, which is generally the fastest order.
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The field characteristic can be specified with an optional last argument. The default is characteristic zero.
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Multiple occurrences of the same point in S are ignored, so that VanishingIdeal always returns a radical ideal.
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Examples
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![L := [[5, 4, 4], [4, 0, 2], [6, 4, 1], [3, 0, 5], [3, 1, 3]]](/support/helpjp/helpview.aspx?si=8516/file02599/math118.png)
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![[[5, 4, 4], [4, 0, 2], [6, 4, 1], [3, 0, 5], [3, 1, 3]]](/support/helpjp/helpview.aspx?si=8516/file02599/math121.png)
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![J := VanishingIdeal(L, [x, y, z])](/support/helpjp/helpview.aspx?si=8516/file02599/math125.png)
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![POLYNOMIALIDEAL(24*x+48+351*z^2-454*z-98*z^3+9*z^4, 4*y+18*z-40+13*z^2-8*z^3+z^4, z^5-15*z^4+85*z^3-225*z^2+274*z-120, variables = {x, y, z}, characteristic = 0, known_groebner_bases = (table( [( plex(x, y, z) ) = [[1, z^5, z^5-15*z^4+85*z^3-225*z^2+274*z-120], [4, y, 4*y+18*z-40+13*z^2-8*z^3+z^4], [24, x, 24*x+48+351*z^2-454*z-98*z^3+9*z^4]] ] )))](/support/helpjp/helpview.aspx?si=8516/file02599/math128.png)
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![VanishingIdeal(L, [x, y, z], tdeg(x, y, z))](/support/helpjp/helpview.aspx?si=8516/file02599/math139.png)
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![M := [[1, alpha], [alpha^2/(alpha-1), 0], [1, 1/alpha]]](/support/helpjp/helpview.aspx?si=8516/file02599/math153.png)
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![[[1, alpha], [alpha^2/(alpha-1), 0], [1, 1/alpha]]](/support/helpjp/helpview.aspx?si=8516/file02599/math156.png)
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![K := VanishingIdeal(M, [x, y], 2)](/support/helpjp/helpview.aspx?si=8516/file02599/math160.png)
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![POLYNOMIALIDEAL(x+1+alpha^2+y+y^2*alpha^2, y+y^3+y^2*alpha^2+y^2*alpha+y^2, variables = {x, y}, characteristic = 2, known_groebner_bases = (table( [( plex(x, y) ) = [[1, y^3, y+y^3+y^2*RootOf(_Z^3+_Z+1)^2+y^2*RootOf(_Z^3+_Z+1)+y^2], [1, x, x+1+RootOf(_Z^3+_Z+1)^2+y+y^2*RootOf(_Z^3+_Z+1)^2]] ] )))](/support/helpjp/helpview.aspx?si=8516/file02599/math163.png)
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](/support/helpjp/helpview.aspx?si=8516/file02599/math167.png)
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References
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Farr, Jeff. Computing Grobner bases, with applications to Pade approximation and algebraic coding theory. Ph.D. Thesis, Clemson University, 2003.
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