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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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  • $\begingroup$ Are you doing this for arbitrary $n$ or for a fixed $n$? Also, does the other involution act linearly? $\endgroup$
    – Charles Siegel
    Commented Jun 15, 2010 at 9:47
  • $\begingroup$ 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. $\endgroup$
    – Andi
    Commented Jun 15, 2010 at 9:53

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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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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).

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  • $\begingroup$ Thank you for your answer. I do however have no idea of Galois Theory (my traing has been mostly in integrable systems, mathematical physics, etc.) , so I do not even understand if - by your last two sentences - you mean whether this is a hard or easy problem...I guess I should start reading books on Galois theory :-) $\endgroup$
    – Andi
    Commented Jun 15, 2010 at 10:09
  • $\begingroup$ I don't know how hard. The Galois theory isn't profound at all. I wanted to say that this is a recognisable type of problem. $\endgroup$
    – Charles Matthews
    Commented Jun 15, 2010 at 10:12
  • $\begingroup$ alright, thanks for the clarification. $\endgroup$
    – Andi
    Commented Jun 15, 2010 at 10:13
  • $\begingroup$ But it is possible that you have a group action (whatever your two involutions generate). If that group is finite, Galois theory might be more revealing. The formulation does suggest a slightly more algebraic approach to what is happening. It depends on the nature of F. $\endgroup$
    – Charles Matthews
    Commented Jun 15, 2010 at 10:22
  • $\begingroup$ 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. $\endgroup$
    – Andi
    Commented Jun 15, 2010 at 10:35

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