Short & Sweet Math Challenge #21: Powers that be

[J-F Garnier]

Following Paul's approach, I modified my Emu71 program to explore these solutions.
I kept the possibility for the least parameter a0 to be +/-1.

Results:
Order 2
1.61803398875 ok x^2-x-1
Order 3
1.83928675521 ok x^3-x^2-x-1
1.46557123188 ok x^3-x^2-1
1.32471795724 ok x^3-x-1
Order 4
1.92756197548 ok x^4-x^3-x^2-x-1
1.38027756910 ok x^4-x^3-1
Order 5
1.96594823665 ok x^5-x^4-x^3-x^2-x-1
1.88851884548 ok x^5-x^4-x^3-x^2-1
1.77847961614 ok x^5-x^4-x^3-x^2+1
1.81240361927 ok x^5-x^4-x^3-x-1
1.70490277604 ok x^5-x^4-x^3-1
1.44326879127 ok x^5-x^4-x^3+x^2-1
1.57014731220 ok x^5-x^4-x^2-1
1.53415774491 ok x^5-x^3-x^2-x-1
Order 6
1.98358284342 ok x^6-x^5-x^4-x^3-x^2-x-1
1.91118343669 ok x^6-x^5-x^4-x^3-x-1
1.80750202302 ok x^6-x^5-x^4-x^3+1
1.71428532914 ok x^6-x^5-x^4-x^2+1
1.74370016590 ok x^6-x^5-x^4-x-1
1.50159480354 ok x^6-x^5-x^4+x^2-1
1.66040772247 ok x^6-x^5-x^3-x^2-1
Order 7
1.99196419661 ok x^7-x^6-x^5-x^4-x^3-x^2-x-1
1.97504243425 ok x^7-x^6-x^5-x^4-x^3-x^2-1
1.92212800436 ok x^7-x^6-x^5-x^4-x^2-x-1
1.88004410997 ok x^7-x^6-x^5-x^4-x-1
1.85454747658 ok x^7-x^6-x^5-x^4-1
1.74745742449 ok x^7-x^6-x^5-x^4+x^3-1
1.77452005864 ok x^7-x^6-x^5-x^3-1
1.67752174784 ok x^7-x^6-x^5-x^2+1
1.65363458677 ok x^7-x^6-x^5-1
1.54521564973 ok x^7-x^6-x^5+x^2-1
1.60134733379 ok x^7-x^6-x^4-x^2-1
1.59000537390 ok x^7-x^5-x^4-x^3-x^2-x-1
Order 8
1.99603117974 ok x^8-x^7-x^6-x^5-x^4-x^3-x^2-x-1
1.97061782036 ok x^8-x^7-x^6-x^5-x^4-x^3-1
1.96113576083 ok x^8-x^7-x^6-x^5-x^4-x^3+1
1.92172206590 ok x^8-x^7-x^6-x^5-x^4+1
1.90988759678 ok x^8-x^7-x^6-x^5-x^4+x+1
1.88166942009 ok x^8-x^7-x^6-x^5-x^3+1
1.88670294847 ok x^8-x^7-x^6-x^5-x^2-x-1
1.86221396917 ok x^8-x^7-x^6-x^5-x-1
1.84771602509 ok x^8-x^7-x^6-x^5-1
1.83032136835 ok x^8-x^7-x^6-x^5+1
1.83524591514 ok x^8-x^7-x^6-x^4-x^3-x-1
1.74243322086 ok x^8-x^7-x^6-x^4+1
1.65524582449 ok x^8-x^7-x^6-x^2+1
1.57367896839 ok x^8-x^7-x^6+x^2-1
1.72778030821 ok x^8-x^7-x^5-x^4-x^3-x^2-1
1971 candidates (1<root<2)
48 candidates (dominant root)

An interesting point is that solutions with the last parameter a0=+1 are found.
If solutions are restricted to a0=-1, there are only 37 candidates.
Another open point is that some solutions found by my pure numerical method are not there, for instance:
1.75487766625 ok x^4-x^3-x^2-1

My HP71 program (actually close to Gerson's program, but I explored all roots):
10 ! --- SSMC21 ---
20 OPTION BASE 0 @ DIM A(10)
30 C=0 @ C2=0
40 FOR D=2 TO 8
50 DISP "Order";D
60 DIM A(D) @ COMPLEX R(D-1)
70 A(0)=1
80 ! A(D)=-1 ! a0=-1 assumed...
90 K=3^(D-1) ! numbers of coefficient combinaisons
100 K=K*2 ! for a0=+/-1
110 FOR J=0 TO K-1
120 L=J
130 ! build the coefficients
140 IF MOD(L,2) THEN A(D)=1 ELSE A(D)=-1 ! for a0=+/-1
150 L=L DIV 2 ! for a0=+/-1
160 FOR I=D-1 TO 1 STEP -1
170 A(I)=MOD(L,3)-1 @ L=L DIV 3
180 NEXT I
190 ! find roots of polynomial A
200 MAT R=PROOT(A)
210 FOR K=0 TO D-1
220 X=REPT(R(K))
230 IF IMPT(R(K))=0 AND X>1.000001 AND X<2 THEN GOSUB 300
240 NEXT K
250 NEXT J ! next polynomial of order D
260 NEXT D ! next order polynomials
270 DISP C;"candidates (1<root<2)"
280 DISP C2;"candidates (dominant root)"
290 END
300 'T': ! evaluate candidate
310 C=C+1
320 F=1
330 FOR I=0 TO D-1
340 IF I<>K AND ABS(R(I))>=1 THEN F=0
350 NEXT I
360 IF F=0 THEN 480
370 C2=C2+1
380 FIX 11 @ DISP X;"ok"; @ STD
390 DISP " x^";STR$(D);
400 FOR I=1 TO D-1
410 IF A(I)=1 THEN DISP "+";
420 IF A(I)=-1 THEN DISP "-";
430 IF A(I)<>0 THEN DISP "x";
440 IF A(I)<>0 AND D-I<>1 THEN DISP "^";STR$(D-I);
450 NEXT I
460 IF A(D)>0 THEN DISP "+";
470 DISP STR$(A(D))
480 RETURN

J-F