I have a question about generating a certain set of symmetric polynomials. I believe that what I'm looking for is known result, but I'm not 100% sure. 


Suppose that I have two sets of indeterminates $X\_1,\ldots,X\_n$ and $Y\_1, \ldots, Y\_n$, of the same (finite) cardinality. Now let $e\_i(A,B,C,\ldots)$ denote the $i$th elementary symmetric polynomial on $A,B,C,\ldots$ and define,
$$ A\_i = e\_i(X\_1,\ldots,X\_n) + e\_i(Y\_1,\ldots,Y\_n)$$


Now define the following:


1. $$ B\_{i} = X\_i + Y\_i $$
2. $$ C\_{i} = \left(\prod\_{j\neq i} X\_j\right) + \left(\prod\_{j\neq i} Y\_j \right) $$


Note that $\sum\_{i=1}^n B\_i = A\_1, \sum\_{i=1}^n C\_i = A\_{n-1}$.


I was wondering the following:


If I am given $B\_i, C\_i, \forall i$ and $A\_n$, can I construct $A\_i$ for all $i \in\{2,\ldots,n-2\}$?


I'm not sure if it is possible, but I think we should be able to construct a basis for the symmetric polynomials with the $B\_i, C\_i$ and $A\_n$, and as such we should be able to get $A\_i, \forall i$. I must admit, I don't have much experience with these type of problems, so I might be barking up the wrong tree.


Thanks!



Constructing the sum of two pairs of symmetric polynomials
What you want is not possible. Suppose $n>3$, and consider making $A\_2=\sum\_{i < j}(X\_iX\_j+Y\_iY\_j)$. All polynomials available are homogeneous in the total degree, and the degree of the $C\_i$ is too high to be of any use, so you'll have to make do with the $B\_i$. But the only way to get a monomial $X\_iX\_j$ is to multiply $B\_iB\_j$; however that gives you mixed terms $X\_iY\_j$ and $X\_jY\_i$ as well, which you don't want. There is no way to get only the terms in $A\_2$.


