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More a reference request / more information.

I was reading some websites about hyperbolic geometry and got to thinking about how would polynomials $(x^2-2)$ behave in such a geometry.

So, I need to know of a good book or reference to further study this. I have only found basic websites, but none with problems and theorems.

So help me, please.

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I don't understand what you mean by "polynomial" in this context. – Qiaochu Yuan Sep 25 '12 at 2:48
@WillJagy Know of a good introduction book? – yiyi Sep 25 '12 at 3:51
up vote 4 down vote accepted

First, a definition. A polynomial on $\mathbb{R}^n$ is a function $p:\mathbb{R}^n\to\mathbb{R}$ which takes the form $$p(x_1,\ldots,x_n) = \sum_{\alpha} a_\alpha x^\alpha,$$ where $\alpha = (\alpha_1,\alpha_2,\ldots,\alpha_n)$ is a multi-index (ordered $n$-tuple of integers), $a_\alpha$ is a constant, and $x^\alpha = x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n}$, a product of powers of coordinates.

Note that polynomials depend very essentially on the coordinate functions you have chosen. If you transform coordinates by a diffeomorphism $f:\mathbb{R}^n\to\mathbb{R}^n$, there is absolutely no guarantee that polynomials transform to polynomials (i.e., for any polynomial $p$, we would have $p\circ f$ also a polynomial). For instance, on $\mathbb{R}^2$, the polynomial $p(x) = x_1^2$ transforms to $p(r,\theta) = r^2\cos^2{\theta}$. This is not a polynomial in $r$ and $\theta$!

Now to the meat of the question: "polynomials" on $\mathbb{H}^n$. Hyperbolic space is the unique (up to isometry) complete simply connected $n$-dimensional Riemannian manifold of constant curvature $-1$. As a Riemannian manifold, there is no canonical choice of coordinates. In fact, there are several common and very useful coordinate systems on hyperbolic space which I'm sure you've seen in your reading: the ball model, the hyperboloid model, the upper half space model.

In each of these models, there are useful coordinate functions. Once you've chosen coordinates, you can define polynomials, and then you get polynomials. For example, on the upper half space model of $\mathbb{H}^n$, polynomials are simply the restrictions of polynomials on $\mathbb{R}^n$. But until you've chosen coordinates, you cannot define polynomials, since they depend on your chosen coordinates - and there is no one "right" choice of coordinates for hyperbolic space!

I think this should settle your question. To read more about hyperbolic space, I highly recommend the first couple of chapters of Volume 1 of Geometry and Topology of $3$-Manifolds by William Thurston (edited by Silvio Levy).

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