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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 = {(- .448591 - .00110758*ii)x1  + (.159643 + .160813*ii)x1*x2 + (.12092 +
     ------------------------------------------------------------------------
                   2                                             
     .0115438*ii)x2  + (.0598688 - .0460059*ii)x1*x3 + (.057621 -
     ------------------------------------------------------------------------
                                                 2                          3
     .257745*ii)x2*x3 + (.128567 + .0502202*ii)x3 , (.286205 + .238824*ii)x1 
     ------------------------------------------------------------------------
                               2                                2            
     + (.351064 - .306467*ii)x1 x2 + (.195727 + .253507*ii)x1*x2  + (- .08274
     ------------------------------------------------------------------------
                    3                            2                 
     + .205498*ii)x2  + (.0988738 + .236227*ii)x1 x3 + (- .157206 +
     ------------------------------------------------------------------------
                                                     2               
     .0265652*ii)x1*x2*x3 + (.323553 + .0962029*ii)x2 x3 + (.774655 +
     ------------------------------------------------------------------------
                     2                              2             
     .148922*ii)x1*x3  + (.302588 - .308404*ii)x2*x3  + (.228647 -
     ------------------------------------------------------------------------
                  3
     .144736*ii)x3 }

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

o3 = {{{-.531511+.039068*ii, .191672+.355886*ii, .741928-.046040*ii}, #steps
     ------------------------------------------------------------------------
     => 11, status  => REGULAR, last t => 1, cond#^{-1} => .382529}}

o3 : List

Ways to use randomSd :