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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 = {(.0917533 + .0653588*ii)x1  + (- .533904 + .175756*ii)x1*x2 + (-
     ------------------------------------------------------------------------
                            2                                                
     .103357 + .299188*ii)x2  + (- .432783 - .458833*ii)x1*x3 + (- .0627913 -
     ------------------------------------------------------------------------
                                                 2           
     .116727*ii)x2*x3 + (.0377321 - .137539*ii)x3 , (.24738 -
     ------------------------------------------------------------------------
                    3                           2               
     .00984484*ii)x1  + (.187665 + .568984*ii)x1 x2 + (.488559 +
     ------------------------------------------------------------------------
                     2                           3               
     .472357*ii)x1*x2  + (.106432 - .175298*ii)x2  + (- .217384 -
     ------------------------------------------------------------------------
                2                                                     
     .2757*ii)x1 x3 + (- .00179197 + .364914*ii)x1*x2*x3 + (.0335197 +
     ------------------------------------------------------------------------
                  2                                   2               
     .136318*ii)x2 x3 + (- .259974 + .0252209*ii)x1*x3  + (- .125784 -
     ------------------------------------------------------------------------
                     2                            3
     .170266*ii)x2*x3  + (.101186 - .0799529*ii)x3 }

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

o3 = {{{.15531-.238471*ii, .396975-.15623*ii, .632972+.579965*ii}, #steps =>
     ------------------------------------------------------------------------
     15, status  => REGULAR, last t => 1, cond#^{-1} => .410874}}

o3 : List

Ways to use randomSd :