A question regarding an application of the Seidenberg-Tarski Theorem

Question

I'm reading J. N. Bernstein's Paper; The Analytic Continuation of Generalized Functions With Respect to a Parameter. You can find it here: http://www.math1.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-a-cont-FAN.pdf

In Lemma 4.1 he states the following:

From the Seidenberg-Tarski theorem it follows that $P(y) > C\left| \left| y \right| \right|^{\alpha}$ for all $y \in Y$ and some $\alpha, C >0.$

Here $Y$ is a finite dimensional spacer over $R,$ that is $Y = R^n$ for some $n$ and $P$ is a strictly positive polynomial in $Y.$

I looked for the Seidenberg-Tarski theorem and tried to apply in order to prove the claim but was unable to prove it. I don't see how to use it here, nor how to prove the claim with it. Can somebody please help me?

Answer 1

Here is a sketch of an almost complete proof.

$\text{}$1. Let$$f(r) := \max_{\|x\| \leq r} {1\over{p(x)}}.$$$\text{}$2. Clearly $f$ is continuous.

$\text{}$3. By the Tarski–Seidenberg theorem $f$ is semialgebraic, i.e. its graph is a semialgebraic subset of $\mathbb{R}^n$.

$\text{}$4. This should imply rather easily that $f$ is bounded by a power of $x$.