randomSd(List) -- a random homogeneous system of polynomial equations

Synopsis

Function: randomSd
Usage:

T = randomSd d
Inputs:
- d, a list, contains the degrees
Outputs:
- T, contains polynomials

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 the unit sphere in the Bombieri-Weyl norm.

i1 : T = randomSd {2,3}

                              2                                              
o1 = {(.125583 - .231255*ii)x1  + (- .0106283 + .495511*ii)x1*x2 + (.140689 -
     ------------------------------------------------------------------------
                   2                                             
     .0470904*ii)x2  + (- .395888 - .177428*ii)x1*x3 + (- .3032 -
     ------------------------------------------------------------------------
                                                  2              
     .259731*ii)x2*x3 + (.00470858 - .244829*ii)x3 , (- .130206 -
     ------------------------------------------------------------------------
                  3                            2                 
     .186531*ii)x1  + (- .17129 + .368432*ii)x1 x2 + (- .246583 +
     ------------------------------------------------------------------------
                     2                             3               
     .719498*ii)x1*x2  + (- .143827 + .153796*ii)x2  + (- .157571 -
     ------------------------------------------------------------------------
                2                                                    
     .1962*ii)x1 x3 + (- .0910157 + .825927*ii)x1*x2*x3 + (.0624194 -
     ------------------------------------------------------------------------
                  2                                   2                
     .242607*ii)x2 x3 + (- .234055 - .0910887*ii)x1*x3  + (- .0833408 +
     ------------------------------------------------------------------------
                     2                            3
     .155138*ii)x2*x3  + (.134672 + .0401753*ii)x3 }

o1 : List

i2 : (S,solsS) = goodInitialPair T;

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

o3 = {{.751732+.463283*ii, .284351-.0717164*ii, .348144-.114302*ii}}

o3 : List

Ways to use this method: