Salem numbers and Lehmer's decic Given,
$$x^{12}-x^7-x^6-x^5+1 = 0\tag1$$
This has Lehmer’s decic polynomial as a factor,
$$x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1=0\tag2$$
hence one of its roots is the smallest known Salem number.  All ten roots obey the beautiful cyclotomic relation,
$$x^{630}-1=\frac{(x^{315}-1)(x^{210}-1)(x^{126}-1)^2(x^{90}-1)(x^{3}-1)^3(x^{2}-1)^5(x-1)^3} {(x^{35}-1)(x^{15}-1)^2(x^{14}-1)^2(x^{5}-1)^6\, x^{68}}$$
found by D. Broadhurst. But this was back in 1999 (paper here). Has anything similar for other Salem numbers been found since then?

Revisiting this old question, now armed with Mathematica's "Integer Relations", I find that it is quite easy to look for similar polynomial relations. For example, let $x$ be a root of Lehmer's decic, then it is also the case that,
$$x^{630}-1 = \frac{(x^{315}-1)(x^{210}-1)(x^{126}-1)^2(x^{90}-1)(x^{10}-1)(x^{9}-1)}{(x^{70}-1)(x^{63}-1)(x^{45}-1)(x^{42}-1)(x^{30}-1)(x^{6}-1)}$$
Given the fifth smallest known Salem number $y$ and a root of the decic,
$$y^{10}-y^6-y^5-y^4+1=0$$
then,
$$y^{210}-1 =\frac{(y^{105}-1)(y^{70}-1)(y^{42}-1)(y^{30}-1)(y^{14}-1)(y^{6}-1)(y^{3}-1)^2}{ (y^{35}-1)(y^{10}-1)(y^{7}-1)^2(y^{2}-1)\,y}$$
