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isHomogeneous(NCIdeal) -- Determines whether the input defines a homogeneous object

Synopsis

Description

Many methods in the NCAlgebra package require inputs to be homogeneous. The meaning of "homogeneous" depens on the type of object.

If x is an NCRingElement, the method returns true if all terms of x have the same degree.

If x is an NCIdeal, NCLeftIdeal, or NCRightIdeal, the method returns true if all generators of the ideal are homogeneous (not necessarily of the same degree).

If x is an NCPolynomialRing, the method returns true. If x is any other NCRing, the method returns true if and only if the defining ideal of x is homogeneous.

If x is an NCMatrix, the method returns true if integer weights were assigned to the source and target of the associated map of free right modules such that the map is graded (degree 0). See assignDegrees.

i1 : A=QQ{x,y,z}

o1 = A

o1 : NCPolynomialRing
 
i2 : w=x^3-y^2

       2  3
o2 = -y +x

o2 : A
 
i3 : isHomogeneous w

o3 = false
 
i4 : setWeights(A, {2,3,1})

o4 = A

o4 : NCPolynomialRing
 
i5 : isHomogeneous w

o5 = true
 
i6 : I = ncIdeal{w,x+z^2}

                        2  3   2
o6 = Two-sided ideal {-y +x , z +x}

o6 : NCIdeal
 
i7 : isHomogeneous I

o7 = true
 
i8 : B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
--Calling Bergman for NCGB calculation.
Complete!

o8 = B

o8 : NCQuotientRing
 
i9 : M = ncMatrix {{x,y,z,0}, {-y*z-2*x^2,-y*x,z*x-x*z,x},{x*y-2*y*x,x*z,-x^2,y}, {-y^2-z*x,x^2,-x*y,z}}

o9 = | x          y    z         0 |
     | -y*z-2*x^2 -y*x y^2-2*x*z x |
     | -2*y*x+x*y x*z  -x^2      y |
     | -2*y^2+x*z x^2  -x*y      z |

o9 : NCMatrix
 
i10 : isHomogeneous M

o10 = false
 
i11 : assignDegrees(M,{1,0,0,0},{2,2,2,1})

o11 = | x          y    z         0 |
      | -y*z-2*x^2 -y*x y^2-2*x*z x |
      | -2*y*x+x*y x*z  -x^2      y |
      | -2*y^2+x*z x^2  -x*y      z |

o11 : NCMatrix
 
i12 : isHomogeneous M

o12 = true
 
i13 : N = ncMatrix {gens B}

o13 = | x y z |

o13 : NCMatrix
 
i14 : isHomogeneous N

o14 = true

See also

Ways to use this method: