Complex polynomial variable transformation

Question

I have two real polynomials that depend in variables (xi1, xi2) and parameters (eps, e2). I apply the following complexification:

w(xi1, xi2, eps, e2) = gamma1(xi1, xi2, eps, e2) + I gamma2(xi1, xi2, eps, e2)

Now I wish to define the variable z = xi1 + I xi2, and find a neat expression for w in terms of z and it's conjugate, zc, as follows:

w(z, zc) = c30 z^3 + c21 z^2 zc+ c12 z zc^2 + c03 zc^3 + o(3)

In order to find the coefficients.

How may I ask this to Mathematica?

---------------------------- actual code -----------------------

gamma1[xi1,xi2,eps,e2]=(0. - 1.72945 I) e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 - (0. + 0.5 I) (eps (0.5929 xi1 - 0.3806 xi2 - 1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);
gamma2[xi1,xi2,eps,e2]=(-1.72945 + 0. I) e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 + (0.5 + 0. I) (eps (0.5929 xi1 - 0.3806 xi2 - 1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);

w[xi1, xi2, eps, e2] = gamma1[xi1, xi2, eps, e2] + I gamma2 [xi1, xi2, eps, e2];

Answer 1

Start with the given definitions.

gamma1[xi1, xi2, eps, 
   e2] = (0. - 
      1.72945 I) e2 (-0.3567 xi1 + 0.229 xi2 + 
       0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 - (0. + 
      0.5 I) (eps (0.5929 xi1 - 0.3806 xi2 - 
         1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 
      38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 
          0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);
gamma2[xi1, xi2, eps, 
   e2] = (-1.72945 + 
      0. I) e2 (-0.3567 xi1 + 0.229 xi2 + 
       0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 + (0.5 + 
      0. I) (eps (0.5929 xi1 - 0.3806 xi2 - 
         1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 
      38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 
          0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);
w[xi1, xi2, eps, e2] = 
  gamma1[xi1, xi2, eps, e2] + I gamma2[xi1, xi2, eps, e2];

Now form a Groebner basis from the reducing relations, ordered in such a way that the x variables are "larger" than the z variables. Use that to reduce the expression of interest.

gb = GroebnerBasis[{xi1+I*xi2-z, xi1-I*xi2-zc}, {xi1,xi2,zc,z}];
Expand[PolynomialReduce[w[xi1,xi2,eps,e2], gb, {xi1,xi2,zc,z}][[2]]]

(* Out[546]= (-0.0326007293462 - 
    0.00464029976304 I) e2 z^3 - (0.0369688588939 + 
    0.0632291173231 I) e2 eps z^3 + (0.0197005770425 - 
    0.0506048635599 I) e2 eps^2 z^3 + (0.012824498915 - 
    0.00395607930627 I) e2 eps^3 z^3 - (0.0533696753722 - 
    0.0831308436911 I) e2 z^2 zc - (0.122052644558 - 
    0.0838872079541 I) e2 eps z^2 zc - (0.0964865190488 - 
    0.0524172122716 I) e2 eps^2 z^2 zc - (0.0400623552714 - 
    0.00400907884547 I) e2 eps^3 z^2 zc + (0.0533696753722 + 
    0.0831308436911 I) e2 z zc^2 + (0.122052644558 + 
    0.0838872079541 I) e2 eps z zc^2 + (0.0964865190488 + 
    0.0524172122716 I) e2 eps^2 z zc^2 + (0.0400623552714 + 
    0.00400907884547 I) e2 eps^3 z zc^2 + (0.0326007293462 - 
    0.00464029976304 I) e2 zc^3 + (0.0369688588939 - 
    0.0632291173231 I) e2 eps zc^3 - (0.0197005770425 + 
    0.0506048635599 I) e2 eps^2 zc^3 - (0.012824498915 + 
    0.00395607930627 I) e2 eps^3 zc^3 *)

Answer 2

You have,

gamma1[xi1_, xi2_] = (0. - 1.72945 I) e2 (-0.3567 xi1 + 0.229 xi2 + 
       0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 - (0. + 
      0.5 I) (eps (0.5929 xi1 - 0.3806 xi2 - 
         1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 
      38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 
          0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);
gamma2[xi1_, xi2_] = (-1.72945 + 0. I) e2 (-0.3567 xi1 + 0.229 xi2 + 
       0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 + (0.5 + 
      0. I) (eps (0.5929 xi1 - 0.3806 xi2 - 
         1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 
      38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 
          0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);

Just rewrite your variables in terms of z and zc,

Solve[{z == xi1 + I xi2, zc == xi1 - I xi2}, {xi1, xi2}]
(* {{xi1 -> (z + zc)/2, xi2 -> -(1/2) I (z - zc)}} *)

Then substitute them into the original expression:

gamma1[(z + zc)/2, -(1/2) I (z - zc)] // FullSimplify
gamma2[(z + zc)/2, -(1/2) I (z - zc)] // FullSimplify
(* (0.00641225 - 0.00197804 I) e2 ((0.838054 - 1.0568 I) z + 
    1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
       0.198158 I) eps zc)^3 - (0. + 
    0.5 I) (eps (((0.29645 + 
            0.1903 I) + (0.026006 + 0.259876 I) eps) z + ((0.29645 - 
            0.1903 I) + (0.026006 - 
             0.259876 I) eps) zc) + (0.043984 + 
       0.142584 I) e2 ((0.838054 - 1.0568 I) z + 
       1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
          0.198158 I) eps zc)^3) *)
(* (-0.00197804 - 0.00641225 I) e2 ((0.838054 - 1.0568 I) z + 
    1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
       0.198158 I) eps zc)^3 + (0.5 + 
    0. I) (eps (((0.29645 + 
            0.1903 I) + (0.026006 + 0.259876 I) eps) z + ((0.29645 - 
            0.1903 I) + (0.026006 - 
             0.259876 I) eps) zc) + (0.043984 + 
       0.142584 I) e2 ((0.838054 - 1.0568 I) z + 
       1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
          0.198158 I) eps zc)^3) *)

That's the basic strategy, but you want a Taylor series in z and zc for w, so let's try this:

Normal[
 Series[gamma1[(z + zc)/2, -(1/2) I (z - zc)] + 
   I gamma2[(z + zc)/2, -(1/2) I (z - zc)], {z, 0, 3}, {zc, 0, 3}]]
(* ((-0.0326007 - 0.0046403 I) e2 - (0.0369689 + 
       0.0632291 I) e2 eps + (0.0197006 - 
       0.0506049 I) e2 eps^2 + (0.0128245 - 
       0.00395608 I) e2 eps^3) z^3 + ((-0.0533697 + 
       0.0831308 I) e2 - (0.122053 - 
       0.0838872 I) e2 eps - (0.0964865 - 
       0.0524172 I) e2 eps^2 - (0.0400624 - 
       0.00400908 I) e2 eps^3) z^2 zc + ((0.0533697 + 
       0.0831308 I) e2 + (0.122053 + 
       0.0838872 I) e2 eps + (0.0964865 + 
       0.0524172 I) e2 eps^2 + (0.0400624 + 
       0.00400908 I) e2 eps^3) z zc^2 + ((0.0326007 - 
       0.0046403 I) e2 + (0.0369689 - 
       0.0632291 I) e2 eps - (0.0197006 + 
       0.0506049 I) e2 eps^2 - (0.0128245 + 
       0.00395608 I) e2 eps^3) zc^3 *)

But this isn't exactly what you want, it has terms like z^2 zc^2, so if you want it to match your expression exactly, you should use the SeriesCoefficient function.

wz = gamma1[(z + zc)/2, -(1/2) I (z - zc)] + 
   I gamma2[(z + zc)/2, -(1/2) I (z - zc)];
Total[SeriesCoefficient[
     wz, {z, 0, #1}, {zc, 0, #2}] z^#1 zc^#2 & @@@ {{3, 0}, {2, 
    1}, {1, 2}, {0, 3}}]
(* (0.0128245 - 
    0.00395608 I) ((-2.21927 - 1.04643 I) e2 - (1.24345 + 
       5.31391 I) e2 eps + (2.51416 - 3.17039 I) e2 eps^2 + (1. + 
       0. I) e2 eps^3) z^3 + ((-0.0533697 + 
       0.0831308 I) e2 - (0.122053 - 
       0.0838872 I) e2 eps - (0.0964865 - 
       0.0524172 I) e2 eps^2 - (0.0400624 - 
       0.00400908 I) e2 eps^3) z^2 zc + ((0.0533697 + 
       0.0831308 I) e2 + (0.122053 + 
       0.0838872 I) e2 eps + (0.0964865 + 
       0.0524172 I) e2 eps^2 + (0.0400624 + 
       0.00400908 I) e2 eps^3) z zc^2 + ((0.0326007 - 
       0.0046403 I) e2 + (0.0369689 - 
       0.0632291 I) e2 eps - (0.0197006 + 
       0.0506049 I) e2 eps^2 - (0.0128245 + 
       0.00395608 I) e2 eps^3) zc^3 *)