During some calculations, I have arrived at complex variable polynomial functions of the following type:

$f_{41}(x_1, x_2, x_3, x_4) = x_1 + x_2 + x_3 + x_4 = \displaystyle\sum_{1\le i\le 4} x_i$

$f_{42}(x_1, x_2, x_3, x_4) = x_1\bar x_2 + x_1\bar x_3 + x_1\bar x_4 + x_2\bar x_3 + x_2\bar x_4 + x_3\bar x_4 = \displaystyle\sum_{1\le i < j \le 4} x_i\bar x_j$

$f_{43}(x_1, x_2, x_3, x_4) = x_1\bar x_2 x_3 + x_1\bar x_2 x_4 + x_1\bar x_3 x_4 + x_2\bar x_3 x_4 = \displaystyle\sum_{1\le i < j < k \le 4} x_i\bar x_j x_k$

$f_{44}(x_1, x_2, x_3, x_4) = x_1\bar x_2 x_3 \bar x_4 = \displaystyle\sum_{1\le i < j < k < l \le 4} x_i\bar x_j x_k\bar x_l$

i.e., elementary symmetric polynomials, but with the variables at even position appearing conjugated. Do you know whether these functions have already been studied? I need to simplify them as much as possible. Thank you very much in advanced.

  • $\begingroup$ simplify them as much as possible An example would help of the simplifications you have in mind, perhaps for the $n=2$ case $f_{21}=x_1+x_2$ and $f_{22}=x_1 \overline{x_2}$. $\endgroup$ – dxiv Sep 30 '16 at 1:17
  • $\begingroup$ The problem is that $n$ can be arbitrarily large. Using these polynomials, I have to obtain some related quantities. When $n$ is small, I can manipulate them by hand, but that is not enough for my purposes, so I was wondering whether they have already been considered by somebody else. $\endgroup$ – User Sep 30 '16 at 9:59

I don't know that these polynomials have been studied, and that may be because they don't have the "nice" properties which make the symmetric polynomials appealing and useful:

  • these are not symmetrical;

  • there is no direct relation between $x_i$ and the roots of an associated function.

In the $n=2$ case for example $f_{21}=x_1+x_2$ and $f_{22}=x_1 \overline{x_2}$ give after simple manipulations:

$$|x_1|^2 - \overline{f_{21}} \; x_1 + f_{22} = 0$$ $$|x_2|^2 - \overline{f_{21}} \; x_2 + \overline{f_{22}} = 0$$

where the equations are not polynomial, and are not the same for the two variables.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.