# 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 $x \rightarrow -x$.

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Are you doing this for arbitrary $n$ or for a fixed $n$? Also, does the other involution act linearly? –  Charles Siegel Jun 15 '10 at 9:47
Arbitrary $n$ would be great, but most of my examples are for $n=3,4,5,6$. The other involution does not act linearly. It is birational. –  Andi Jun 15 '10 at 9:53

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.

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As far as I can see, the group is not finite (except in some very trivial examples). Does this mean that I should also consider alternatives to Galois theory? Also, I am very much dependent on algorithmic solutions to this problem, since the map $F$ is quite complicated, that is numerators and denominators are polynomials of total degree $n-1$ or $n$ in the $n$ variables. Hence, it stands out of the question to deal with these maps without using a CAS like Maple or Mathematica. –  Andi Jun 15 '10 at 10:35