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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 = {(.364674 + .125156*ii)x1  + (- .346355 + .0771661*ii)x1*x2 + (.35447 +
     ------------------------------------------------------------------------
                   2                                               
     .0437713*ii)x2  + (- .262701 + .122194*ii)x1*x3 + (- .189524 +
     ------------------------------------------------------------------------
                                                  2              
     .162166*ii)x2*x3 + (.0372904 - .0562262*ii)x3 , (.00250291 +
     ------------------------------------------------------------------------
                  3                            2                 
     .176519*ii)x1  + (- .047729 + .09665*ii)x1 x2 + (- .476481 +
     ------------------------------------------------------------------------
                     2                            3             
     .445832*ii)x1*x2  + (.0880866 - .210318*ii)x2  + (.360004 -
     ------------------------------------------------------------------------
                  2                                                   
     .315163*ii)x1 x3 + (.474151 - .0861519*ii)x1*x2*x3 + (- .339531 +
     ------------------------------------------------------------------------
                   2                                  2                
     .0890004*ii)x2 x3 + (- .137195 - .602912*ii)x1*x3  + (- .0241373 -
     ------------------------------------------------------------------------
                     2                             3
     .337598*ii)x2*x3  + (.180174 + .00742785*ii)x3 }

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

o3 = {{{.334674+.506379*ii, -.46558+.106018*ii, .620638-.135565*ii}, #steps
     ------------------------------------------------------------------------
     => 13, status  => REGULAR, last t => 1, cond#^{-1} => .389096}}

o3 : List

Ways to use randomSd :