Write function $F$, based on parameters?

Question

Let $$U=x+y+z\, ,$$ $$V=xy+yz+zx\, ,$$ $$W=xyz\, .$$ and we have $F(U,V,W)=x^4+y^4+z^4$. My question it is, How to write function $F$, based on parameters $U$ , $V$ and $W$?

Answer 1

Because $x^4+y^4+z^4$ is a bit complicated, we try to go back to lower powers. You can see for example that $$x^4+y^4+z^4=(x+y+z)(x^3+y^3+z^3)-(xy+yz+xz)(x^2+y^2+z^2)+xyz(x+y+z).$$ Hence $$x^4+y^4+z^4=U(x^3+y^3+z^3)-V(x^2+y^2+z^2)+UW.$$

But $$x^2+y^2+z^2=(x+y+z)^2-2xy-2yz-2xz=U^2-2V$$ and $$x^3+y^3+z^3=(x+y+z)^3-3(xy+yz+xz)(x+y+z)+3xyz=U^3-3UV+3W.$$ All that together gives $$x^4+y^4+z^4=U^4-3U^2V+3UW-U^2V+2V^2+UW=U^4-4U^2V+2V^2+4UW.$$

Answer 2

Based on answer of @C. Dubussy , I want to generalize his answer, But I dont know, how to prove it. If we want to expand this equation: $$f:={(x_1^n+x_2^n+\cdots+x_k^n)}\, .$$ We can use this formula: $$f=(\sum_i^1{x_i})(x_1^{n-1}+x_2^{n-1}+\cdots+x_k^{n-1})- (\sum_{i_1\neq i_2}^2{x_{i_1}x_{i_2}})(x_1^{n-2}+x_2^{n-2}+\cdots+x_k^{n-2})$$ $$+(\sum_{i_1\neq i_2\neq i_3}^3{x_{i_1}x_{i_2}x_{i_3}})(x_1^{n-3}+x_2^{n-3}+\cdots+x_k^{n-3}) -\cdots$$

$$+{(-1)}^{k-1}(x_1x_2\cdots x_k)(x_1^{n-k}+x_2^{n-k}+\cdots+x_k^{n-k})\, .$$

My mean for example of $(\sum_{i_1\neq i_2\neq i_3}^3{x_{i_1}x_{i_2}x_{i_3}})$ is all chooses cases of order $3$ from the set $(x_1,x_2,\cdots,x_k)$.

For example, based on the above formula, we can see: $$ {x}^{4}+{y}^{4}+{z}^{4}+{w}^{4}= \left( x+y+z+w \right) \left( {x}^{3 }+{y}^{3}+{z}^{3}+{w}^{3} \right) $$ $$- \left( xy+yz+zx+xw+yw+zw \right) \left( {x}^{2}+{y}^{2}+{z}^{2}+{w}^{2} \right) $$ $$+ \left( xyz+xyw+yzw+z xw \right) \left( x+y+z+w \right) -4\,xyzw\, . $$ Another example make above formula clear: $$ {x}^{5}+{y}^{5}+{z}^{5}= \left( x+y+z \right) \left( {x}^{4}+{y}^{4}+{z}^{4} \right) - \left( xy+xz+yz \right) \left( {x}^{3}+{y}^{3}+{z}^{3} \right) +xyz \left( {x}^{2}+{y}^{2}+{z}^{2} \right) \, . $$