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.

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

$J≔⟨x^2\,{\left(y-1\right)}^2\left(y+1\right)⟩$

$J≔⟨x^2\,{\left(y-1\right)}^2\left(y+1\right)⟩$

$\mathrm{Saturate}\left(J\,y-1\right)$

$⟨x^2\,y+1⟩$

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

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

$\mathrm{Saturate}\left(K\,x\right)$

$⟨y^3+y\,x^{2}y+y⟩$

$\mathrm{q1}≔\mathrm{Quotient}\left(K\,x\right)$

$\mathrm{q1}≔⟨-x^{2}y+y^3\,x^2{y}^2+y^2\,x^{4}y+x^{2}y⟩$

$\mathrm{q2}≔\mathrm{Quotient}\left(\mathrm{q1}\,x\right)$

$\mathrm{q2}≔⟨-x^{2}y+y^3\,x^2{y}^2+y^2\,x^{3}y+xy⟩$

$\mathrm{q3}≔\mathrm{Quotient}\left(\mathrm{q2}\,x\right)$

$\mathrm{q3}≔⟨y^3+y\,x^{2}y+y⟩$

$\mathrm{Saturate}\left(\mathrm{q3}\,y\right)$

$⟨x^2+1\,y^2+1⟩$

$\mathrm{Saturate}\left(K\,\left\{x\,y\right\}\,'s'\right)$

$⟨x^2+1\,y^2+1⟩$

$s$

$3$

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