Determinantal representations of quadrics

Usage:

detRep f

detRep(f, Tolerance => 1e-6)
Inputs:
- f, a ring element, a quadric with real coefficients
Outputs:
- a matrix, giving a determinantal representation of $f$

This page demonstrates how the method detRep computes a monic symmetric determinantal representation of a real quadric $f$ (in any number of variables), or returns false if no such representation exists.

If a quadratic determinantal representation of size $2$ exists, then it is returned. Otherwise, the method will find a determinantal representation of size $n+1$, where $n$ is the number of variables (if it exists). If no monic symmetric determinantal representation exists, then null is returned.

When working over an InexactFieldFamily like RR or CC, the option Tolerance can be used to specify the internal threshold for checking equality (any floating point number below the tolerance is treated as numerically zero).

i1 : R = RR[x1, x2, x3, x4]

o1 = R

o1 : PolynomialRing

i2 : f = 260*x1^2+180*x1*x2-25*x2^2-140*x1*x3-170*x2*x3-121*x3^2+248*x1*x4+94*x2*x4-142*x3*x4+35*x4^2+36*x1+18*x2+2*x3+20*x4+1

          2                  2                              2             
o2 = 260x1  + 180x1*x2 - 25x2  - 140x1*x3 - 170x2*x3 - 121x3  + 248x1*x4 +
     ------------------------------------------------------------------------
                              2
     94x2*x4 - 142x3*x4 + 35x4  + 36x1 + 18x2 + 2x3 + 20x4 + 1

o2 : R

i3 : A = first detRep f

o3 = | 17.8243x1+13.8011x2-.241342x3+5.84723x4+1
     | 7.99807x1+9.10764x2+10.9754x3+6.91046x4  
     ------------------------------------------------------------------------
     7.99807x1+9.10764x2+10.9754x3+6.91046x4   |
     18.1757x1+4.19887x2+2.24134x3+14.1528x4+1 |

             2      2
o3 : Matrix R  <-- R

i4 : clean(1e-10, f - det A)

o4 = 0

o4 : R

i5 : g = -61*x1^2-96*x1*x2-177*x2^2-126*x1*x3-202*x2*x3-86*x3^2-94*x1*x4-190*x2*x4-140*x3*x4-67*x4^2+8*x1+3*x2+5*x3+3*x4+1

           2                  2                             2            
o5 = - 61x1  - 96x1*x2 - 177x2  - 126x1*x3 - 202x2*x3 - 86x3  - 94x1*x4 -
     ------------------------------------------------------------------------
                               2
     190x2*x4 - 140x3*x4 - 67x4  + 8x1 + 3x2 + 5x3 + 3x4 + 1

o5 : R

i6 : B = first detRep g

o6 = | 1                                       0                            
     | 0                                       1                            
     | 0                                       0                            
     | 0                                       0                            
     | 7.81025x1+6.14577x2+8.06632x3+6.01773x4 11.7996x2+4.35832x3+4.91683x4
     ------------------------------------------------------------------------
     0                    0         7.81025x1+6.14577x2+8.06632x3+6.01773x4 |
     0                    0         11.7996x2+4.35832x3+4.91683x4           |
     1                    0         1.39265x3+.0214748x4                    |
     0                    1         2.57123x4                               |
     1.39265x3+.0214748x4 2.57123x4 8x1+3x2+5x3+3x4+1                       |

             5      5
o6 : Matrix R  <-- R

i7 : clean(1e-10, g - det B)

o7 = 0

o7 : R

Determinantal representations of quadrics

See also