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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 = {(- .0476166 - .337745*ii)x1  + (.245755 - .22824*ii)x1*x2 + (- .0941414
     ------------------------------------------------------------------------
                    2                                             
     - .220444*ii)x2  + (.0583399 - .0790842*ii)x1*x3 + (.336081 -
     ------------------------------------------------------------------------
                                                    2              
     .0799001*ii)x2*x3 + (- .153003 - .0263427*ii)x3 , (- .377358 +
     ------------------------------------------------------------------------
                  3                             2               
     .106388*ii)x1  + (- .146354 - .473076*ii)x1 x2 + (.582629 +
     ------------------------------------------------------------------------
                      2                              3             
     .0685094*ii)x1*x2  + (- .325748 + .0789237*ii)x2  + (.305306 +
     ------------------------------------------------------------------------
                  2                                                     
     .171776*ii)x1 x3 + (- .397696 - .252939*ii)x1*x2*x3 + (- .0304875 -
     ------------------------------------------------------------------------
                  2                                  2             
     .211591*ii)x2 x3 + (- .242122 - .338933*ii)x1*x3  + (.256886 -
     ------------------------------------------------------------------------
                     2                             3
     .323256*ii)x2*x3  + (.0722553 - .0761494*ii)x3 }

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

o3 = {{{.0093985-.404387*ii, .250671+.275339*ii, .829713+.096498*ii}, #steps
     ------------------------------------------------------------------------
     => 11, status  => REGULAR, last t => 1, cond#^{-1} => .355823}}

o3 : List

Ways to use randomSd :