Not a proof of my generalization, but a demonstration: Clear[a, b, c, d, c0, p, q, qq] (* Pisot polynomial to n=10*) a = {1, x - 1, x^2 - x - 1, x^3 - x - 1, x^4 - x^3 - 1, -1 + x^2 - x^3 - x^4 + x^5, -1 + x^2 - x^4 - x^5 + x^6, -1 + x^2 - x^5 - x^6 + x^7, -1 + x^2 - x^6 - x^7 + x^8, -1 + x^2 - x^7 - x^8 + x^9, -1 + x^2 - x^8 - x^9 + x^10}; (* beta roots of the polynomials: largest positive root*) b = {0, 1, GoldenRatio, 1.324717957244746`, 1.3802775690976334`, 1.443268791270373`, 1.5015948035390874`, 1.5452156497327552`, 1.573678968393517`, 1.5911843056671024`, 1.6017558616969834`}; (* Pisot cyclotomics in n>=5 generalization: dividing out the beta root*) p[n_] = (x^n - x^(n - 1) - x^(n - 2) + x^2 - 1)/(x - b[[n]]); (* regular cyclotomics*) c0[n_] := Sum[x^m, {m, 0, n - 1}] (* solution of the Pisot cyclotomics roots in Mathematica order*) Table[ Table[x /. NSolve[p[n] == 0, x][[m]], {m, 1, n - 1}], {n, 5, 10}] (* solution of the Pisot cyclotomics roots in Mathematica order*) Table[ Table[x /. NSolve[c0[n] == 0, x][[m]], {m, 1, n - 1}], {n, 5, 10}] (* subtraction of absolute values of the root arrays*) Table[ Apply[Plus, -Abs[Table[x /. NSolve[p[n] == 0, x][[m]], {m, 1, n - 1}]] + Abs[Table[x /. NSolve[c0[n] == 0, x][[m]], {m, 1, n - 1}]]], {n, 5, 10}] {-0.2021067948778119`, -0.2710747447758055`, -0.22801234335138398`, -0.2772521279528657`, -0.23811499354308818`, -0.27206623095251126`} ListLinePlot[%] The differences alternate by even and odd, but are always going to be negative, so that the Pisot roots (excluding the beta root) are always inside the unit disk of cyclotomic roots. The limiting beta does appear to be the golden mean for these Polynomials: Show[Plot[N[GoldenRatio], {x, 1, 18}, PlotRange -> {{0, 18}, {1.2, 1.8}}], ListLinePlot[{1.324717957244746`, 1.3802775690976334`, 1.443268791270373`, 1.5015948035390874`, 1.5452156497327552`, 1.573678968393517`, 1.5911843056671024`, 1.6017558616969834`, 1.6081283851873869`, 1.611983421246492`, 1.6143264149391272`, 1.6157565175408435`, 1.616632435387905`, 1.6171703361720169`, 1.6175012998129095`, 1.6177052198884552`, 1.6178309858778124`, 1.6179086035278054`}, PlotRange -> {{0, 18}, {1.2, 1.8}}]]