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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 = {(.0774848 - .0908319*ii)x1  + (- .257753 - .00695657*ii)x1*x2 + (-
     ------------------------------------------------------------------------
                             2                                              
     .0851006 + .610413*ii)x2  + (- .194058 + .0308036*ii)x1*x3 + (.239737 -
     ------------------------------------------------------------------------
                                                   2             
     .125677*ii)x2*x3 + (- .118617 + .0728049*ii)x3 , (.0670061 -
     ------------------------------------------------------------------------
                   3                             2                 
     .0745453*ii)x1  + (- .506067 - .186508*ii)x1 x2 + (.00599343 +
     ------------------------------------------------------------------------
                     2                               3                
     .191315*ii)x1*x2  + (- .0274498 - .0377487*ii)x2  + (- .0464112 -
     ------------------------------------------------------------------------
                  2                                                  
     .391628*ii)x1 x3 + (.809253 + .336106*ii)x1*x2*x3 + (- .101624 +
     ------------------------------------------------------------------------
                  2                                2                 
     .455778*ii)x2 x3 + (.466092 - .206831*ii)x1*x3  + (- .00940661 +
     ------------------------------------------------------------------------
                     2                               3
     .212668*ii)x2*x3  + (- .147093 - .00517371*ii)x3 }

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

o3 = {{{.0470534-.0621565*ii, .253704-.266387*ii, .926595+.004048*ii}, #steps
     ------------------------------------------------------------------------
     => 11, status  => REGULAR, last t => 1, cond#^{-1} => .569985}}

o3 : List

Ways to use randomSd :