VanishingIdeal - Maple Help

PolynomialIdeals

 VanishingIdeal
 compute the vanishing ideal for finite a set of points

 Calling Sequence VanishingIdeal(S, X) VanishingIdeal(S, X, T, p)

Parameters

 S - list or set of points X - list of variable names T - (optional) monomial order p - (optional) characteristic, a non-negative integer

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $L≔\left[\left[5,4,4\right],\left[4,0,2\right],\left[6,4,1\right],\left[3,0,5\right],\left[3,1,3\right]\right]$
 ${L}{≔}\left[\left[{5}{,}{4}{,}{4}\right]{,}\left[{4}{,}{0}{,}{2}\right]{,}\left[{6}{,}{4}{,}{1}\right]{,}\left[{3}{,}{0}{,}{5}\right]{,}\left[{3}{,}{1}{,}{3}\right]\right]$ (1)
 > $J≔\mathrm{VanishingIdeal}\left(L,\left[x,y,z\right]\right)$
 ${J}{≔}⟨{{z}}^{{5}}{-}{15}{}{{z}}^{{4}}{+}{85}{}{{z}}^{{3}}{-}{225}{}{{z}}^{{2}}{+}{274}{}{z}{-}{120}{,}{9}{}{{z}}^{{4}}{-}{98}{}{{z}}^{{3}}{+}{351}{}{{z}}^{{2}}{+}{24}{}{x}{-}{454}{}{z}{+}{48}{,}{{z}}^{{4}}{-}{8}{}{{z}}^{{3}}{+}{13}{}{{z}}^{{2}}{+}{4}{}{y}{+}{18}{}{z}{-}{40}⟩$ (2)
 > $\mathrm{Simplify}\left(\mathrm{PrimeDecomposition}\left(J\right)\right)$
 $⟨{y}{,}{x}{-}{3}{,}{z}{-}{5}⟩{,}⟨{-}{5}{+}{x}{,}{-}{4}{+}{y}{,}{z}{-}{4}⟩{,}⟨{-}{1}{+}{y}{,}{x}{-}{3}{,}{z}{-}{3}⟩{,}⟨{y}{,}{-}{4}{+}{x}{,}{z}{-}{2}⟩{,}⟨{-}{6}{+}{x}{,}{-}{4}{+}{y}{,}{z}{-}{1}⟩$ (3)
 > $\mathrm{VanishingIdeal}\left(L,\left[x,y,z\right],\mathrm{tdeg}\left(x,y,z\right)\right)$
 $⟨{13}{}{{y}}^{{2}}{-}{36}{}{x}{-}{37}{}{y}{-}{12}{}{z}{+}{168}{,}{13}{}{y}{}{z}{+}{26}{}{{z}}^{{2}}{-}{48}{}{x}{-}{19}{}{y}{-}{198}{}{z}{+}{484}{,}{78}{}{x}{}{z}{+}{91}{}{{z}}^{{2}}{-}{354}{}{x}{+}{50}{}{y}{-}{937}{}{z}{+}{2302}{,}{39}{}{y}{}{x}{-}{26}{}{{z}}^{{2}}{-}{36}{}{x}{-}{193}{}{y}{+}{170}{}{z}{-}{92}{,}{39}{}{{x}}^{{2}}{-}{26}{}{{z}}^{{2}}{-}{255}{}{x}{-}{40}{}{y}{+}{188}{}{z}{+}{124}{,}{13}{}{{z}}^{{3}}{-}{117}{}{{z}}^{{2}}{-}{12}{}{x}{+}{18}{}{y}{+}{308}{}{z}{-}{204}⟩$ (4)
 > $\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({z}^{3}+z+1\right)\right)$
 ${\mathrm{\alpha }}$ (5)
 > $M≔\left[\left[1,\mathrm{\alpha }\right],\left[\frac{{\mathrm{\alpha }}^{2}}{\mathrm{\alpha }-1},0\right],\left[1,\frac{1}{\mathrm{\alpha }}\right]\right]$
 ${M}{≔}\left[\left[{1}{,}{\mathrm{\alpha }}\right]{,}\left[\frac{{{\mathrm{\alpha }}}^{{2}}}{{\mathrm{\alpha }}{-}{1}}{,}{0}\right]{,}\left[{1}{,}\frac{{1}}{{\mathrm{\alpha }}}\right]\right]$ (6)
 > $K≔\mathrm{VanishingIdeal}\left(M,\left[x,y\right],2\right)$
 ${K}{≔}⟨{{\mathrm{\alpha }}}^{{2}}{}{{y}}^{{2}}{+}{{\mathrm{\alpha }}}^{{2}}{+}{x}{+}{y}{+}{1}{,}{{\mathrm{\alpha }}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{2}}{}{\mathrm{\alpha }}{+}{{y}}^{{3}}{+}{{y}}^{{2}}{+}{y}⟩$ (7)
 > $\mathrm{IdealInfo}\left[\mathrm{Characteristic}\right]\left(K\right)$
 ${2}$ (8)
 > $\mathrm{Simplify}\left(\mathrm{PrimeDecomposition}\left(K\right)\right)$
 $⟨{y}{,}{{\mathrm{\alpha }}}^{{2}}{+}{x}{+}{1}⟩{,}⟨{x}{+}{1}{,}{y}{+}{\mathrm{\alpha }}⟩{,}⟨{x}{+}{1}{,}{{\mathrm{\alpha }}}^{{2}}{+}{y}{+}{1}⟩$ (9)

References

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