Macaulay2 » Documentation
Packages » PencilsOfQuadrics :: quadraticForm
next | previous | forward | backward | up | index | toc

# quadraticForm -- part of a RandomNicePencil

- Usage:
    `qq = rNP.quadraticForm`
- Outputs:
  - qq, a ring element, an element of S, quadratic in x_0..z_2 and linear in s, t.

## Description

the polynomial that represents a pencil of quadrics qq=s*q1+t*q2 with a fixed isotropic subspace and a fixed corank one quadric in normal form q1.

    i1 : setRandomSeed 0
     -- setting random seed to 0

    o1 = 0

    i2 : kk=ZZ/101;

    i3 : g=1;

    i4 : rNP=randNicePencil(kk,g);

    i5 : S=rNP.qqRing;

    i6 : qq=rNP.quadraticForm

                  2      2                           2
    o6 = x y s - z s - 5x t + 12x z t - 18y z t - 30z t - 12x z t + 10y z t +
          0 0     1      0       0 1       0 1       1       0 2       0 2
         ------------------------------------------------------------------------
                     2
         6z z t - 48z t
           1 2       2

    o6 : S

    i7 : q1=diff(S_(2*g+2), qq)

                 2
    o7 = x y  - z
          0 0    1

    o7 : S

    i8 : q2=diff(S_(2*g+3), qq)

             2                        2                                2
    o8 = - 5x  + 12x z  - 18y z  - 30z  - 12x z  + 10y z  + 6z z  - 48z
             0      0 1      0 1      1      0 2      0 2     1 2      2

    o8 : S

    i9 : qq==S_(2*g+2)*q1+S_(2*g+3)*q2

    o9 = true

## See also

- qqRing -- part of a RandomNicePencil
- baseRing of a RandomNicePencil -- part of a RandomNicePencil
- matFact1 -- part of a RandomNicePencil
- matFact2 -- part of a RandomNicePencil
- matFactu1 -- part of a RandomNicePencil
- matFactu2 -- part of a RandomNicePencil
- isotropicSpace -- part of a RandomNicePencil

## For the programmer

The object quadraticForm is a symbol.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PencilsOfQuadrics.m2:2448:0.