coefficient field -- the role of the coefficient field

The SumsOfSquares package works with two coefficient rings: the rational numbers $\QQ$ and the real numbers $\RR$. Almost all operations in this package rely on a numerical semidefinite programming Solver. When calling such a solver, even if the input was a polynomial with rational coefficients, the result is numerical. The package makes some effort to round and return a rational result, but this can fail, independent of whether a rational sum-of-squares decomposition exists or not. In case of failure, a real result is returned. The following example of Scheiderer is a sum of squares, but does not admit any rational decomposition. Consequently the package must return a real solution.

i1 : f = library("Scheiderer", QQ[x,y,z])

      4      3    4     2          2      2 2      3      3    4
o1 = x  + x*y  + y  - 3x y*z - 4x*y z + 2x z  + x*z  + y*z  + z

o1 : QQ[x..z]

i2 : sol = solveSOS (f);

i3 : sosPoly sol

                        2                         2                                     2 2                       2                          2                                       2 2                      2                        2                                    2 2                     2                       2                                     2 2                    2                       2                                     2 2                     2                        2                                     2 2
o3 = 3.30453e-9*(.34757x  - .0338185x*y + .768917y  + .336255x*z - .203874y*z + .363573z )  + 5.71257e-9*(.318117x  + .679504x*y - .00674091y  + .148907x*z + .644086y*z - .00319975z )  + .612611*(- .489504x  + .515357x*y - .174266y  + .126103x*z - .33001y*z + .582766z )  + .88919*(- .389146x  + .18471x*y + .191405y  + .590213x*z - .140295y*z - .640139z )  + 1.85689*(.622038x  + .195424x*y - .44107y  + .199983x*z - .564971y*z - .145429z )  + 3.9924*(.00407928x  - .446352x*y - .383631y  + .678611x*z + .309527y*z + .311867z )

o3 : SOSPoly

Given a sum-of-squares decomposition with real coefficients, it is often useful to ignore the squares with very small coefficients. The function clean(RR,SOSPoly) removes the squares whose coefficients are smaller than a given tolerance.

i4 : clean (0.001, sosPoly sol)

                        2                        2                                    2 2                     2                       2                                     2 2                    2                       2                                     2 2                     2                        2                                     2 2
o4 = .612611*(- .489504x  + .515357x*y - .174266y  + .126103x*z - .33001y*z + .582766z )  + .88919*(- .389146x  + .18471x*y + .191405y  + .590213x*z - .140295y*z - .640139z )  + 1.85689*(.622038x  + .195424x*y - .44107y  + .199983x*z - .564971y*z - .145429z )  + 3.9924*(.00407928x  - .446352x*y - .383631y  + .678611x*z + .309527y*z + .311867z )

o4 : SOSPoly

coefficient field -- the role of the coefficient field

See also