Resultant of polynomial into real and imaginary

https://math.stackexchange.com/q/4292657/8530


It is possible, but not very practical. Taking as an example the polynomial p(z)=z3−1, let z=x+iy then equating the real and imaginary parts to 0 and eliminating y between the two equations using polynomial resultants gives resultant(x^3 - 3xy^2 - 1, 3x^2y - y^3, y) q(x)=(x3−1)(8x3+1)2. The real roots of q(x) are the real parts of the roots of p(x), in this case {1,−1/2}. –
dxiv yesterday


(*start*)
Clear[x, a, b];
Solve[x^3 - 1 == 0, x]
"x^3-1"
Resultant[(Expand[(a + I*b)^3 - 1] + Expand[(a - I*b)^3 - 1])/
  2, (Expand[(a + I*b)^3 - 1] - Expand[(a - I*b)^3 - 1])/2/I, a]
Solve[% == 0, b]
Clear[x];
x = b /. %%
x^3 - 1
"x^3-1"
Resultant[(Expand[(a + I*b)^3 - 1] + Expand[(a - I*b)^3 - 1])/
  2, (Expand[(a + I*b)^3 - 1] - Expand[(a - I*b)^3 - 1])/2/I, b]
Solve[% == 0, a]
Clear[x];
x = a /. %%
x^3 - 1
"x^3-1"
(*end*)