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.

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.

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.

The field characteristic can be specified with an optional last argument.  The default is characteristic zero.

Multiple occurrences of the same point in S are ignored, so that VanishingIdeal always returns a radical ideal.

$\mathrm{with}\left(\mathrm{PolynomialIdeals}\right)\:$

$L≔\left[\left[5\,4\,4\right]\,\left[4\,0\,2\right]\,\left[6\,4\,1\right]\,\left[3\,0\,5\right]\,\left[3\,1\,3\right]\right]$

$L≔\left[\left[5\,4\,4\right]\,\left[4\,0\,2\right]\,\left[6\,4\,1\right]\,\left[3\,0\,5\right]\,\left[3\,1\,3\right]\right]$

$J≔\mathrm{VanishingIdeal}\left(L\,\left[x\,y\,z\right]\right)$

$J≔⟨z^5-15z^4+85z^3-225z^2+274z-120\,9z^4-98z^3+351z^2+24x-454z+48\,z^4-8z^3+13z^2+4y+18z-40⟩$

$\mathrm{Simplify}\left(\mathrm{PrimeDecomposition}\left(J\right)\right)$

$⟨y\,x-3\,z-5⟩,⟨-5+x\,-4+y\,z-4⟩,⟨-1+y\,x-3\,z-3⟩,⟨y\,-4+x\,z-2⟩,⟨-6+x\,-4+y\,z-1⟩$

$\mathrm{VanishingIdeal}\left(L\,\left[x\,y\,z\right]\,\mathrm{tdeg}\left(x\,y\,z\right)\right)$

$⟨13y^2-36x-37y-12z+168\,13yz+26z^2-48x-19y-198z+484\,78xz+91z^2-354x+50y-937z+2302\,39yx-26z^2-36x-193y+170z-92\,39x^2-26z^2-255x-40y+188z+124\,13z^3-117z^2-12x+18y+308z-204⟩$

$\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left(z^3+z+1\right)\right)$

$\mathrm{\alpha }$

$M≔\left[\left[1\,\mathrm{\alpha }\right]\,\left[\frac{{\mathrm{\alpha }}^2}{\mathrm{\alpha }-1}\,0\right]\,\left[1\,\frac{1}{\mathrm{\alpha }}\right]\right]$

$M≔\left[\left[1\,\mathrm{\alpha }\right]\,\left[\frac{{\mathrm{\alpha }}^2}{\mathrm{\alpha }-1}\,0\right]\,\left[1\,\frac{1}{\mathrm{\alpha }}\right]\right]$

$K≔\mathrm{VanishingIdeal}\left(M\,\left[x\,y\right]\,2\right)$

$K≔⟨{\mathrm{\alpha }}^2{y}^2+{\mathrm{\alpha }}^2+x+y+1\,{\mathrm{\alpha }}^2{y}^2+y^2\mathrm{\alpha }+y^3+y^2+y⟩$

$\mathrm{IdealInfo}\left[\mathrm{Characteristic}\right]\left(K\right)$

$2$

$\mathrm{Simplify}\left(\mathrm{PrimeDecomposition}\left(K\right)\right)$

$⟨y\,{\mathrm{\alpha }}^2+x+1⟩,⟨x+1\,y+\mathrm{\alpha }⟩,⟨x+1\,{\mathrm{\alpha }}^2+y+1⟩$

Farr, Jeff. Computing Grobner bases, with applications to Pade approximation and algebraic coding theory. Ph.D. Thesis, Clemson University, 2003.