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When $a<b<c$ are three positive integers, let

$$ P_{a,b,c}(x)=x^c-(x^a+x^b)+1 $$

and denote by $N(a,b,c)$ the number of positive real roots of $P_{a,b,c}$ (note that $1$ is always a root).

What is the maximal value for $N(a,b,c)$ (pheraps it is $+\infty$, if it can get arbitrarily large ?) Judging from a few random examples, it would seem that this maximum value is $2$.

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It is in fact 2, by Descartes' rule of signs.

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