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I am working through a Calculus book and I found an exercise, which I am not able to solve:

Let $K$ be a field of rational functions over $\mathbb{R}$ and let $a,b$ be arbitrary in $\mathbb{Q}$ but not all $b$ are $0$. And let $m,n$ be arbitrary in $\mathbb{N}$. We define $$P:=\left.\left\{\frac{\sum_{i=0}^n a_ix^i}{\sum_{i=0}^m b_ix^i}\;\right|\;a_nb_m\lt 0\right\}.$$

Is $P$ a prepositive cone?

Does anybody have a hint. My idea was to express $-1$ as a sum of squares but that didn't really work for me.

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    I am really curious as to what "calculus book" this is taken from.2011-05-10
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    It's a german calculus book "Analysis Band 1, Erhardt Behrends", first chapter :S2011-05-10

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