Call a real number $\lambda>1$ **special** if it is a root of a polynomial $f(x)$ such that


1. $f(x)$ is monic with integer coefficients,
2. all roots of $f(x)$ are distinct,
3. for all $z\in\mathbb{C}$, if $f(z)=0$ and $z\neq \lambda$ then $|z|<1$.


For example, the Golden Ratio $\frac{1+\sqrt{5}}{2}$ is special.


**Question**: Are there special numbers arbitrarily close to 1?


*Some Remarks*:


1. After messing around on Wolfram Alpha, I was able to beat the Golden Ratio using the unique real root of $x^3-x-1$, which is approximately 1.3247.
2. I have found lots of work on polynomials such that *all* roots are on the unit disc, and some things where all but *two* roots are on the unit disc. But nothing close to what I want: exactly one real root not on the unit disc, and the rest strictly inside the unit disc.



Polynomials with all but one root inside the unit disc
This is a well-known topic. Such numbers are called Pisot numbers. Salem (1945) proved that they form a closed subset $S$ of $(1,+\infty)$, and Siegel (1944) proved that its lowest element is the positive root of $X^3-X-1$. Therefore the answer to your question is **No**.


The Golden ratio is however the smallest among the cluster points of $S$. The derived set $S'$ was described by Dufresnoy & Pisot (1953). A Pisot number $\theta$ (root of $f$ as above) belongs to $S'$ iff there exist another polynomial $g\in{\mathbb Z}[X]$ such that $|g|\le|f|$ over the unit circle, and $\pm g$ is neither $f$, nor its reciprocal polynomial $f^\*(X)=X^nf(1/X)$.


