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.

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

$J≔⟨{\left(x^3-y\right)}^2\,y^3-1⟩$

$J≔⟨{\left(x^3-y\right)}^2\,y^3-1⟩$

$\mathrm{IsRadical}\left(J\right)$

$\mathrm{false}$

$\mathrm{IdealMembership}\left(x^3-y\,J\right)$

$\mathrm{false}$

$\mathrm{RadicalMembership}\left(x^3-y\,J\right)$

$\mathrm{true}$

$R≔\mathrm{Simplify}\left(\mathrm{Radical}\left(J\right)\right)$

$R≔⟨y^3-1\,x^3-y⟩$

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

$⟨-1+y\,x-1⟩,⟨-1+y\,x^2+x+1⟩,⟨-x^3+y\,x^6+x^3+1⟩$

$\mathrm{Intersect}\left(\right)$

$⟨y^3-1\,x^3-y⟩$

$\mathrm{IdealContainment}\left(\,R\,\right)$

$\mathrm{true}$

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

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

$\mathrm{IsRadical}\left(K\right)$

$\mathrm{false}$

$\mathrm{IsPrimary}\left(K\right)$

$\mathrm{false}$

$R≔\mathrm{Radical}\left(K\right)$

$R≔⟨-x^3+y^2\,-xy+z⟩$

$\mathrm{IsPrime}\left(R\right)$

$\mathrm{true}$

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.

The PolynomialIdeals[IsRadical], PolynomialIdeals[Radical] and PolynomialIdeals[RadicalMembership] commands were updated in Maple 16.