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randomSd -- a random homogeneous system of polynomial equations

Synopsis

Description

Generates a system of homogeneous polynomials T_i such that deg T_i = d_i. The system is normalized, so that it is on a unit sphere in the Bombieri-Weyl norm.
i1 : T = randomSd {2,3}

                                 2                                    
o1 = {(- .0781727 + .152968*ii)x1  + (- .00590623 + .612107*ii)x1*x2 +
     ------------------------------------------------------------------------
                             2                                             
     (.180157 + .300935*ii)x2  + (- .318601 + .243438*ii)x1*x3 + (.347119 +
     ------------------------------------------------------------------------
                                                    2              
     .0104095*ii)x2*x3 + (- .111537 - .0736084*ii)x3 , (- .079529 +
     ------------------------------------------------------------------------
                   3                              2               
     .0448545*ii)x1  + (- .312576 + .0972835*ii)x1 x2 + (.249309 +
     ------------------------------------------------------------------------
                     2                            3               
     .437306*ii)x1*x2  + (.115614 - .0889281*ii)x2  + (- .339802 +
     ------------------------------------------------------------------------
                  2                                                     
     .383275*ii)x1 x3 + (- .749754 - .514692*ii)x1*x2*x3 + (- .0642848 -
     ------------------------------------------------------------------------
                  2                               2             
     .380404*ii)x2 x3 + (.27409 + .243245*ii)x1*x3  + (.260476 +
     ------------------------------------------------------------------------
                      2                            3
     .0583774*ii)x2*x3  + (.0152886 - .090098*ii)x3 }

o1 : List
i2 : (S,solsS) = goodInitialPair T;
i3 : M = track(S,T,solsS,gamma=>0.6+0.8*ii,Software=>M2)

o3 = {{{-.158193-.0683287*ii, .120619+.26915*ii, .939065+.038372*ii}, #steps
     ------------------------------------------------------------------------
     => 11, status  => REGULAR, last t => 1, cond#^{-1} => .530168}}

o3 : List

Ways to use randomSd :