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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 = {(.0835454 - .204119*ii)x1  + (- .195614 + .188968*ii)x1*x2 + (- .185578
     ------------------------------------------------------------------------
                    2                                              
     - .260108*ii)x2  + (.504835 + .0130924*ii)x1*x3 + (- .128936 -
     ------------------------------------------------------------------------
                                                     2             
     .0421711*ii)x2*x3 + (- .433618 + .00215475*ii)x3 , (- .21592 +
     ------------------------------------------------------------------------
                   3                              2               
     .0630143*ii)x1  + (- .0253269 - .177422*ii)x1 x2 + (.107961 +
     ------------------------------------------------------------------------
                     2                               3               
     .281348*ii)x1*x2  + (- .0105597 + .0354112*ii)x2  + (- .440333 +
     ------------------------------------------------------------------------
                  2                                                   
     .191115*ii)x1 x3 + (- .557394 - .323044*ii)x1*x2*x3 + (.0837769 +
     ------------------------------------------------------------------------
                 2                                    2              
     .41187*ii)x2 x3 + (- .0783481 - .0228042*ii)x1*x3  + (.0504062 -
     ------------------------------------------------------------------------
                     2                              3
     .371698*ii)x2*x3  + (- .371724 - .0492756*ii)x3 }

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

o3 = {{{.323712-.470199*ii, .195788-.586429*ii, .518647+.151318*ii}, #steps
     ------------------------------------------------------------------------
     => 11, status  => REGULAR, last t => 1, cond#^{-1} => .626352}}

o3 : List

Ways to use randomSd :