# Positive real root separation

Let $$\beta\in(1,2)$$ and $$\gamma\in(1,2)$$ be Galois conjugates of height 1. That is, there exists a polynomial $$p$$ with coefficients $$-1,0,1$$ such that $$p(\beta)=p(\gamma)=0$$ (not necessarily minimal).

Numerically, there appears to be an absolute constant $$C>0$$ such that $$|\gamma-\beta|\ge C$$. Is this true/known? If it is, what is the best known value for $$C$$?

I've looked into some literature on root separation but couldn't find this claim.

• Chapter 9 deals with a similar problem, but the bound depends on $\beta$ (naturally, there are many results like that). The hight could be a red herring, as any polynomial with Mahler measure less than 2, always divides a polynomial of height 1. However, asking for an absolute bound might complicate the matter, and could potentially relate to Lehmer's conjecture. – pavl0 Nov 6 '18 at 0:18

$$1 + x^n + x^{2n} - x^{3n} - x^{5n} - x^{6n} + x^{7n}$$
is irreducible and has two Galois conjugate roots $$\beta_n$$ and $$\gamma_n$$ in $$(1,2)$$ with
$$| \beta_n - \gamma_n| \sim \frac{\log(\beta_1/\gamma_1)}{n} \rightarrow 0.$$
• Simple and beautiful! Maybe the irreducibilty (at least for infinitely many $n$, like the primes) deserves an explanation. – Peter Mueller Nov 8 '18 at 10:46
• Since the Galois group of the polynomial $p(x)$ for $n = 1$ is $S_7$, the irreducibility of $p(x^n)$ is immediate as long as a (any) root of $p(x)$ is not a perfect $n$th power for some $n > 1$. But any root is a fundamental unit in the corresponding degree $7$ field. – user131093 Nov 8 '18 at 18:05