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PolynomialIdeals

 Saturate
 saturate an ideal

 Calling Sequence Saturate(J, f, s)

Parameters

 J - polynomial ideal f - polynomial, or list or set of polynomials s - (optional) name

Description

 • The Saturate command computes the saturation of an ideal J with respect to a polynomial f, denoted J:${f}^{\mathrm{\infty }}$.  Saturation removes all the solutions of f from J, and is equivalent to a repeated application of Quotient. This functionality is also available through the Simplify command.
 • If the second argument is a list or set of polynomials, then the Saturate command removes the solutions of each polynomial, or equivalently their product.
 • If the optional third argument s is given, it is assigned a positive integer exponent with the property that J:f^infinity = J:f^s. This value is not guaranteed to be minimal.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨{x}^{2},{\left(y-1\right)}^{2}\left(y+1\right)⟩$
 ${J}{≔}⟨{{x}}^{{2}}{,}{\left({y}{-}{1}\right)}^{{2}}{}\left({y}{+}{1}\right)⟩$ (1)
 > $\mathrm{Saturate}\left(J,y-1\right)$
 $⟨{{x}}^{{2}}{,}{y}{+}{1}⟩$ (2)
 > $K≔⟨{x}^{3}{y}^{2}+x{y}^{2},{x}^{3}y+{x}^{3}{y}^{3}⟩$
 ${K}{≔}⟨{{x}}^{{3}}{}{{y}}^{{2}}{+}{x}{}{{y}}^{{2}}{,}{{x}}^{{3}}{}{{y}}^{{3}}{+}{{x}}^{{3}}{}{y}⟩$ (3)
 > $\mathrm{Saturate}\left(K,x\right)$
 $⟨{{y}}^{{3}}{+}{y}{,}{{x}}^{{2}}{}{y}{+}{y}⟩$ (4)
 > $\mathrm{q1}≔\mathrm{Quotient}\left(K,x\right)$
 ${\mathrm{q1}}{≔}⟨{-}{{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{,}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{2}}{,}{{x}}^{{4}}{}{y}{+}{{x}}^{{2}}{}{y}⟩$ (5)
 > $\mathrm{q2}≔\mathrm{Quotient}\left(\mathrm{q1},x\right)$
 ${\mathrm{q2}}{≔}⟨{-}{{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{,}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{2}}{,}{{x}}^{{3}}{}{y}{+}{x}{}{y}⟩$ (6)
 > $\mathrm{q3}≔\mathrm{Quotient}\left(\mathrm{q2},x\right)$
 ${\mathrm{q3}}{≔}⟨{{y}}^{{3}}{+}{y}{,}{{x}}^{{2}}{}{y}{+}{y}⟩$ (7)
 > $\mathrm{Saturate}\left(\mathrm{q3},y\right)$
 $⟨{{x}}^{{2}}{+}{1}{,}{{y}}^{{2}}{+}{1}⟩$ (8)
 > $\mathrm{Saturate}\left(K,\left\{x,y\right\},'s'\right)$
 $⟨{{x}}^{{2}}{+}{1}{,}{{y}}^{{2}}{+}{1}⟩$ (9)
 > $s$
 ${3}$ (10)

References

 Becker, T., and Weispfenning, V. Groebner Bases. New York: Springer-Verlag, 1993.