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Let $P$ be a unitary polynomial with rational coefficients in one variable $x$, such that $P(x) \geq 0$ for all $x \in \mathbb R$. Then $P$ is of even degree, say $2d$. Is it true that there always exist $d$ rational numbers $q_1,q_2, \ldots ,q_d$ such that

$$ P(x) \geq \bigg( \prod_{k=1}^{d} (x-q_k)^2\bigg), $$

for all $x\in \mathbb R$ ?

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No. It's not even true for $d=1$: $(x-\sqrt{2})^2$ is a counterexample.

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  • $\begingroup$ Oops. I had mistakenly put "real coefficients" instead of "rational coefficients" in the OP ; it's corrected now. Note that this new question is always true for d=1 (this is what the canonical form is all about). $\endgroup$ Jan 30 '12 at 15:36

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