Subrings of rational functions invariant under change of sign Let $R$ be a ring generated by $k$ rational functions in the
variables $x_1,...,x_n$  over the real numbers. 
Is there an algorithm that computes a set of rational functions
$f_1,...,f_l \in R$ which generate the subring $Q$ of $R$ consisting of all
those rational functions in $R$ which are invariant under the transformation
$(x_1,...,x_n) \rightarrow (-x_1,...,-x_n)$? 
Background of the question:
I have a birational map $F$ on n dimensional real affine space and I would like to find 
conserved quantities of the map, that is a function H such that $H \circ f = H$. Now, the map f is the composition of two involutions $I_1$, $I_2$, where $I_1$ is simply defined by $x \rightarrow -x$. The involution $I_2$ is a quite complicated map, but I am nevertheless able to find a set of rational functions generating the ring of all conserved quantities of $I_2$. Hence, I would like to know whether $I_1$ and $I_2$ have common conserved quantities, or in other words whether $I_2$ has conserved quantities which are invariant under $x \rightarrow -x$.
 A: Let $S$ be the rational functions in variables $x_i^2$. They are all in your invariants. The point is is that $R>S$ is a galois extensions with a group $C_2^n$. In particular, $R$ is a $2^n$-dimensional vector space over $S$ with basis $x_{a_1} \ldots x_{a_k}$ with $a_1 < \ldots < a_k$.
The field of invariants you are interested in is a $2^{n-1}$-dimensional subspace spanned by all monomials with even $k$. Proofs are easy exercises in Galois theory as Charles has pointed out.
A: So at the level of function fields, you have a group of order 2 acting on L, the field of fractions of R, and there is a subfield K fixed by it. The extension L/K will be of degree 2, so quadratic. In other words L is generated by adjoining a square root of a function. All that is happening at the field level is that this square root is being replaced by its negative. (All this is standard Galois theory.)
So you want to chase this down to the level of R. By analogy with rings of algebraic integers, how hard will it be? In effect you are asking how to compute the field trace in a quadratic extension? To clarify the problem, taking any one of the variables shows how to adjoin a square root (the square of any one of the x's is in K). 
