test whether an ideal is radical - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Algebra : Polynomials : PolynomialIdeals : PolynomialIdeals/IsRadical

PolynomialIdeals[IsRadical] - test whether an ideal is radical

PolynomialIdeals[Radical] - compute the radical of an ideal

PolynomialIdeals[RadicalMembership] - test for membership in the radical

Calling Sequence

IsRadical(J)

Radical(J)

RadicalMembership(f, J)

Parameters

J

-

polynomial ideal

f

-

polynomial

Description

• 

The IsRadical command tests whether a given ideal is radical. An ideal J is radical if fm 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 fm 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

withPolynomialIdeals:

J:=x3y2,y31

J:=x3y2,y31

(1)

IsRadicalJ

false

(2)

IdealMembershipx3y,J

false

(3)

RadicalMembershipx3y,J

true

(4)

R:=SimplifyRadicalJ

R:=x91,x3+y

(5)

SimplifyPrimeDecompositionJ

1+y,x1,1+y,x2+x+1,x3+y,x6+x3+1

(6)

Intersect

y31,x3y

(7)

IdealContainment,R,

true

(8)

K:=x3y2,y3zx2,z22xyz+x2y2

K:=x3y2,x2z+y3,x2y22xyz+z2

(9)

IsRadicalK

false

(10)

IsPrimaryK

false

(11)

R:=RadicalK

R:=x3+y2,xy+z

(12)

IsPrimeR

true

(13)

See Also

Groebner[Basis], map, PolynomialIdeals, PolynomialIdeals[HilbertDimension], PolynomialIdeals[IdealContainment], PolynomialIdeals[IdealMembership], PolynomialIdeals[Intersect], PolynomialIdeals[PrimeDecomposition], PolynomialIdeals[ZeroDimensionalDecomposition]

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.


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam