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PencilsOfQuadrics :: matrixFactorizationK

matrixFactorizationK -- Knoerrer matrix factorization from a bilinear form X*transpose Y

Synopsis

Description

Produces a matrix factorization (M1,M2) of the bilinear form X*transpose Y. It does this by specializing the formula given by Knoerrer for $\sum X_i*Y_i$.

i1 : kk=ZZ/101

o1 = kk

o1 : QuotientRing
i2 : n=2

o2 = 2
i3 : R=kk[a_0..a_(binomial(n+2,2))]

o3 = R

o3 : PolynomialRing
i4 : S=kk[x_0..x_(n-1),a_0..a_(binomial(n+2,2))]

o4 = S

o4 : PolynomialRing
i5 : M=genericSymmetricMatrix(S,a_0,n)

o5 = | a_0 a_1 |
     | a_1 a_2 |

             2       2
o5 : Matrix S  <--- S
i6 : X=(vars S)_{0..n-1}

o6 = | x_0 x_1 |

             1       2
o6 : Matrix S  <--- S
i7 : Y=X*M

o7 = | x_0a_0+x_1a_1 x_0a_1+x_1a_2 |

             1       2
o7 : Matrix S  <--- S
i8 : (M1,M2)=matrixFactorizationK(X,Y)

o8 = ({1} | -x_0          -x_1           |, {2} | -x_0a_0-x_1a_1 x_1  |)
      {0} | x_0a_1+x_1a_2 -x_0a_0-x_1a_1 |  {2} | -x_0a_1-x_1a_2 -x_0 |

o8 : Sequence
i9 : M12=M1*M2

o9 = {1} | x_0^2a_0+2x_0x_1a_1+x_1^2a_2 0                            |
     {0} | 0                            x_0^2a_0+2x_0x_1a_1+x_1^2a_2 |

             2       2
o9 : Matrix S  <--- S

Ways to use matrixFactorizationK :

For the programmer

The object matrixFactorizationK is a method function.