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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.

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  • $\begingroup$ 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
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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.

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