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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 = {(.0103959 - .14808*ii)x1  + (.502296 + .325519*ii)x1*x2 + (- .0969204 -
     ------------------------------------------------------------------------
                  2                                                 
     .337334*ii)x2  + (- .207863 + .523893*ii)x1*x3 + (- .00688413 +
     ------------------------------------------------------------------------
                                                  2            
     .318099*ii)x2*x3 + (- .199814 + .113443*ii)x3 , (.274292 +
     ------------------------------------------------------------------------
                   3                            2               
     .0267973*ii)x1  + (- .13408 - .107976*ii)x1 x2 + (.414352 -
     ------------------------------------------------------------------------
                     2                              3              
     .598218*ii)x1*x2  + (- .134067 - .0656269*ii)x2  + (.0977538 -
     ------------------------------------------------------------------------
                   2                                               
     .0450636*ii)x1 x3 + (.0201219 + .50454*ii)x1*x2*x3 + (- .184 +
     ------------------------------------------------------------------------
                   2                              2              
     .0923449*ii)x2 x3 + (.12683 + .12588*ii)x1*x3  + (.0214234 -
     ------------------------------------------------------------------------
                     2                             3
     .357959*ii)x2*x3  + (.0898723 + .0818304*ii)x3 }

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

o3 = {{{-.29169-.156918*ii, -.0196194-.157094*ii, .930174-.002532*ii}, #steps
     ------------------------------------------------------------------------
     => 11, status  => REGULAR, last t => 1, cond#^{-1} => .416743}}

o3 : List

Ways to use randomSd :