PolynomialIdeals

 test whether an ideal is radical
 compute the radical of an ideal
 test for membership in the radical

Parameters

 J - polynomial ideal f - polynomial

Description

 • The IsRadical command tests whether a given ideal is radical. An ideal J is radical if ${f}^{m}$ in J implies f in J for all f in the polynomial ring. Similarly, the radical of J is the ideal of polynomials f such that ${f}^{m}$ is in J for some integer m. This can be computed using the Radical command.
 • The RadicalMembership command tests for membership in the radical without explicitly computing the radical.  This command can be useful in cases where computation of the radical cannot be performed.
 • The algorithms employed by Radical and IsRadical are based on the algorithm for prime decomposition, and require only a single lexicographic Groebner basis in the zero-dimensional case.  In practice, this means that computing the radical is no harder than computing a decomposition, and that both can be computed using the same information.
 • The Radical and IsRadical commands require polynomials over a perfect field.  Infinite fields of positive characteristic are not supported, and over finite fields only zero-dimensional ideals can be handled because the dimension reducing process generates infinite fields.  These restrictions do not apply to the RadicalMembership command.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨{\left({x}^{3}-y\right)}^{2},{y}^{3}-1⟩$
 ${J}{≔}⟨{\left({{x}}^{{3}}{-}{y}\right)}^{{2}}{,}{{y}}^{{3}}{-}{1}⟩$ (1)
 > $\mathrm{IsRadical}\left(J\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IdealMembership}\left({x}^{3}-y,J\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{RadicalMembership}\left({x}^{3}-y,J\right)$
 ${\mathrm{true}}$ (4)
 > $R≔\mathrm{Simplify}\left(\mathrm{Radical}\left(J\right)\right)$
 ${R}{≔}⟨{{y}}^{{3}}{-}{1}{,}{{x}}^{{3}}{-}{y}⟩$ (5)
 > $\mathrm{Simplify}\left(\mathrm{PrimeDecomposition}\left(J\right)\right)$
 $⟨{-}{1}{+}{y}{,}{x}{-}{1}⟩{,}⟨{-}{1}{+}{y}{,}{{x}}^{{2}}{+}{x}{+}{1}⟩{,}⟨{-}{{x}}^{{3}}{+}{y}{,}{{x}}^{{6}}{+}{{x}}^{{3}}{+}{1}⟩$ (6)
 > $\mathrm{Intersect}\left(\right)$
 $⟨{{y}}^{{3}}{-}{1}{,}{{x}}^{{3}}{-}{y}⟩$ (7)
 > $\mathrm{IdealContainment}\left(,R,\right)$
 ${\mathrm{true}}$ (8)
 > $K≔⟨{x}^{3}-{y}^{2},{y}^{3}-z{x}^{2},{z}^{2}-2xyz+{x}^{2}{y}^{2}⟩$
 ${K}{≔}⟨{{x}}^{{3}}{-}{{y}}^{{2}}{,}{-}{z}{}{{x}}^{{2}}{+}{{y}}^{{3}}{,}{{x}}^{{2}}{}{{y}}^{{2}}{-}{2}{}{x}{}{y}{}{z}{+}{{z}}^{{2}}⟩$ (9)
 > $\mathrm{IsRadical}\left(K\right)$
 ${\mathrm{false}}$ (10)
 > $\mathrm{IsPrimary}\left(K\right)$
 ${\mathrm{false}}$ (11)
 > $R≔\mathrm{Radical}\left(K\right)$
 ${R}{≔}⟨{-}{{x}}^{{3}}{+}{{y}}^{{2}}{,}{-}{x}{}{y}{+}{z}⟩$ (12)
 > $\mathrm{IsPrime}\left(R\right)$
 ${\mathrm{true}}$ (13)

References

 Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms. 2nd ed. New York: Springer-Verlag, 1997.
 Gianni, P.; Trager, B.; and Zacharias, G. "Grobner bases and primary decompositions of polynomial ideals." J. Symbolic Comput. Vol. 6, (1988): 149-167.

Compatibility