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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 = {(- .152766 - .428164*ii)x1  + (- .0919176 - .0618796*ii)x1*x2 +
     ------------------------------------------------------------------------
                              2                                             
     (.231015 - .0604934*ii)x2  + (.485426 + .0214613*ii)x1*x3 + (- .11144 -
     ------------------------------------------------------------------------
                                                 2             
     .138746*ii)x2*x3 + (.044617 + .0829878*ii)x3 , (.0773703 +
     ------------------------------------------------------------------------
                  3                           2               
     .108708*ii)x1  + (.254193 - .556547*ii)x1 x2 + (.703913 +
     ------------------------------------------------------------------------
                      2                               3             
     .0573067*ii)x1*x2  + (- .0097376 + .0515268*ii)x2  + (.116168 -
     ------------------------------------------------------------------------
                  2                                                  
     .276927*ii)x1 x3 + (- .481445 + .209223*ii)x1*x2*x3 + (.471566 -
     ------------------------------------------------------------------------
                   2                                2            
     .0392319*ii)x2 x3 + (.245722 - .412346*ii)x1*x3  + (.20363 -
     ------------------------------------------------------------------------
                     2                            3
     .160328*ii)x2*x3  + (.0650523 - .147514*ii)x3 }

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

o3 = {{{-.14417-.0761343*ii, -.0300544+.3001*ii, .939035-.025865*ii}, #steps
     ------------------------------------------------------------------------
     => 11, status  => REGULAR, last t => 1, cond#^{-1} => .167887}}

o3 : List

Ways to use randomSd :