# 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