Web Supplement for Complex Chebyshev Polynomials (2012-14):

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Description [to be completed].

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Results 1: α=π/2 (semicircle/semidisk)

deg 1:
P_1a=m-1, root 1 (τ^(C,1)(0)=τ^(S,1)(0)=Root[P_1a,1]=1)

deg 2:
P_2a=m²+4m-4, root 2 (τ^(C,2)(0)=τ^(S,2)(0)=Root[P_2a,2]~.828427)

deg 3:
P_3a=m⁴+44m²-16, root=2 (τ^(C,3)(0)=τ^(S,3)(0)=Root[P_3a,2]~.600566)

deg 4:
P_4a=m⁶-160m⁵+896m⁴-2240m³-768m²+3072m-1024, root=2 (τ^(C,4)(0)=Root[P_4a,2]~.426417)
P_4b=m⁷-41m⁶-13m⁵-7m⁴-64m³+56m²-48m+16, root=2 (τ^(S,4)(0)=Root[P_4b,2]~.428592)

deg 5:
P_5a=m⁴+168m³+640m²+640m-256, root=2 (τ^(C,5)(0)=Root[P_5a,2]~.301733)
P_5b=2121843m²²-80430176m²⁰-447447275m¹⁸-1030213279m¹⁶-1363646416m¹⁴-1172262368m¹²-660934400m¹⁰-218989824m⁸-31686656m⁶-647168m⁴+65536, root=3 (τ^(S,5)(0)=Root[P_5b,3]~.314564)

P_6a=m¹²+11568m¹¹+32064m¹⁰+30375168m⁹-279310336m⁸+213975040m⁷-1156890624m⁶+7164657664m⁵-11530141696m⁴+4777312256m³+2348810240m²-1879048192m+268435456,
root=3 (τ^(C,6)(0)=Root[P_6m1,3]~.213383)
P_6b=--(~.234697)

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Results 2: α=π/3 (arc/sector)

deg 1:
P_1a=4m²-3, root 2 (τ^(C,1)(1/2)=τ^(S,1)(1/2)=Root[P_1a,2]~.866025)

deg 2:
P_2a=m²+6m-3, root=2 (τ^(C,2)(1/2)=Root[P_2a,2]~.464102)
P_2b=2m-1, root 1 (τ^(S,2)(1/2)=Root[P_2b,1]=.5)

deg 3:
P_3a=16m⁸+648m⁶+14553m⁴-14202m²+729, root=3 (τ^(C,3)(1/2)=Root[P_3a,3]~.233167)
P_3b=8m²-12m+3, root 1 (τ^(S,3)(1/2)=Root[P_3b,1]~.316987)

deg 4:
P_4a=m⁶-252m⁵+714m⁴+306m³-3015m²+2430m-243, root=2 (τ^(C,4)(1/2)=Root[P_4a,2]~.116624)
P_4b=1744m⁸+19872m⁷-104652m⁶+185004m⁵-147447m⁴+44388m³+6210m²-5832m+729, root=3 (τ^(S,4)(1/2)=Root[P_4b,3]~.189080)

deg 5:
P_5a=m¹⁶-6234m¹⁴+251683209m¹²-1085250906m¹⁰+36947937780m⁸-74276829738m⁶+53201685897m⁴-12839260266m²+43046721, root=3 (τ^(C,5)(1/2)=Root[P_5a,3]~.0583132)
P_5b=--(~.107174)

deg 6:
P_6a=m¹²+22530m¹¹-847629m¹⁰+20704122m⁹-228947391m⁸+385650666m⁷-2052104652m⁶+1552297608m⁵+2442565530m⁴-3315640716m³+1358107317m²-200884698m+4782969, root=3 (τ^(C,6)(1/2)=Root[P_6a,3]~.029157)
P_6b=--(~.063304)

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Results 3: α=2π/3 (arc/sector)

deg 1:
P_1a=m-1, root 1 (τ^(C,1)(-1/2 )=τ^(S,1)(-1/2)=Root[P_1a,1]=1)

deg 2:
P_2a=m-1, root 1 (τ^(C,2)(-1/2)=τ^(S,2)(-1/2)=Root[P_2a,1]=1)

deg 3:
P_3a=16m⁸+648m⁶+14553m⁴-14202m²+729, root=4 (τ^(C,3)(-1/2)=τ^(S,3)(-1/2)=Root[P_3a,4]~.940035)

deg 4:
P_4a=m³-72m²+27m+27, root=2 (τ^(C,4)(-1/2)=τ^(S,4)(-1/2)=Root[P_4a,2]~.834198)

deg 5:
P_5a=m¹⁶-6234m¹⁴+251683209m¹²-1085250906m¹⁰+36947937780m⁸-74276829738m⁶+53201685897m⁴-12839260266m²+43046721, root=4 (τ^(C,5)(-1/2)=τ^(S,5)(-1/2)=Root[P_5a,4]~.727999)

deg 6:
P_6a=m⁶+95m⁵+4659m⁴+8524m³+2052m²-7047m+729, root=2 (τ^(C,6)(-1/2)=τ^(S,6)(-1/2)=Root[P_6a,2]~.632036)

deg 7:
P_7a=256m³²+...+984770902183611232881, root=5 (τ^(A,7)(-1/2)=Root[P_7a,2]~.547808)
984770902183611232881 - 4709846277829618283702538 m^2 + 360758311333745564221764561 m^4 - 9784118639755436807948540904 m^6 + 105636987810917193568445380872 m^8 - 344439152590304644226459232498 m^10 + 314088761681036804012958869364 m^12 - 70474534757063012413064367810 m^14 + 15498109430414144893479481992 m^16 - 3462091044030437387120431128 m^18 + 722278003427167466060358705 m^20 - 12326389974716365066405338 m^22 + 365781838652489741544369 m^24 + 672710142830280914448 m^26 + 299942074868994144 m^28 - 5407659264 m^30 + 256 m^32

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Full Description of the Solutions


Results 1: α=π/2 (semicircle, semicircular arc)

deg 1:
P_1m1=m-1, root 1 (τ^(C,1)(0)=τ^(S,1)(0)=Root[P_1m1,1]=1)
P_1m2=m-1, root 1 ((τ^(C,1)(0))²=(τ^(S,1)(0))²=Root[P_1m2,1]=1)

P_1a0=a, root 1 (a0^(C,1)(0)=a0^(S,1)(0)=0)

Remark. No inner extrema

deg 2:
P_2m1=m²+4m-4, root 2 (τ^(C,2)(0)=τ^(S,2)(0)=Root[P_2m1,2]~.828427)
P_2m2=m²-24m+16, root 1 ((τ^(C,2)(0))²=(τ^(S,2)(0))²=Root[P_2m2,1]~.686292)

P_2a0=a²+2a-1, root 2 (a0^(C,2)(0)=a0^(S,2)(0)=Root[P_2a0,2]~.414214)
P_2a1=a²+4a+2, root 2 (a1^(C,2)(0)=a1^(S,2)(0)=Root[P_2a1,2]~-.585786)

P_2r1=r⁴-4r³-1, root 3 (r1^(C,2)(0)=r1^(S,2)(0)=Root[P_2r1,3]~.292893-.573086i)
P_2r2=r⁴-4r³-1, root 4 (r2^(C,2)(0)=r2^(S,2)(0)=Root[P_2r2,4]~.292893+.573086i)

Remark. No inner extrema





deg 3:
P_3m1=m⁴+44m²-16, root=2 (τ^(C,3)(0)=τ^(S,3)(0)=Root[P_3m1,2]~.600566)
P_3m2=m²+44m-16, root=2 ((τ^(C,3)(0))²=(τ^(S,3)(0))²=Root[P_3m2,2]~.360680)

P_3a0=a⁴+4a²-1, root 1 (a0^(C,3)(0)=a0^(S,3)(0)=Root[P_3a0,1]~-.485868)
P_3a1=a-1, root 1 (a1^(C,3)(0)=a1^(S,3)(0)=Root[P_3a1,1]=1)
P_3a2=a⁴+20a²-25, root 1 (a2^(C,3)(0)=a2^(S,3)(0)=Root[P_3a2,1]~-1.08643)

P_3r1=r¹²+24r¹⁰+r⁸-12r⁶-r⁴+4r²-1, root 2 (r1^(C,3)(0)=r1^(S,3)(0)=Root[P_2r1,2]~.673139)
P_3r2=r¹²+24r¹⁰+r⁸-12r⁶-r⁴+4r²-1, root 9 (r2^(C,3)(0)=r2^(S,3)(0)=Root[P_2r2,9]~.206648-.824070i)
P_3r3=r¹²+24r¹⁰+r⁸-12r⁶-r⁴+4r²-1, root 10 (r3^(C,3)(0)=r3^(S,3)(0)=Root[P_2r2,10]~.206648+.824070i)

P_3x1=x⁸+8x⁶-2x⁴+8x²+1, root 8 (x1^(C,3)(0)=x1^(S,3)(0)=Root[P_3x1,8]~.786151+.618034i)
P_3xr1=x⁴+x²-1, root=2 (xr1^(C,3)(0)=xr1^(S,3)(0)=Root[P_3xr1,2]~.786151)





deg 4:
P_4m1=m⁶-160m⁵+896m⁴-2240m³-768m²+3072m-1024, root=2 (τ^(C,4)(0)=Root[P_4m1,2]~.426417)
P_4m2=m⁶-23808m⁵+84480m⁴-5412864m³12517376m²-7864320m+1048576, root=1 ((τ^(C,4)(0))²=Root[P_4m2,1]~.181831)

P_4a0=a⁶+6a⁵-7a⁴+4a³+7a²-2a-1, root=4 (a0^(C,4)(0)=Root[P_4a0,4]~.497610)
P_4a1=a⁶-80a⁵-262a⁴-320a³-196a²-64a-8, root=1 (a1^(C,4)(0)=Root[P_4a1,1]~-1.28440)
P_4a2=a⁶-104a⁵+98a⁴+40a³+164a²+80a-8, root=3 (a2^(C,4)(0)=Root[P_4a2,3]~1.79913)
P_4a3=a⁶+24a⁵-46a⁴+96a³+300a²-96a-8, root=2 (a3^(C,4)(0)=Root[P_4a3,2]~-1.58592)

P_4r1=r²⁴-24r²³+58r²²+156r²¹+964r²⁰-1472r¹⁹+486r¹⁸-228r¹⁷-1003r¹⁶+1008r¹⁵-924r¹⁴+56r¹³+
        +160r¹²-608r¹¹+428r¹⁰-40r⁹-149r⁸+104r⁷-62r⁶+44r⁵+28r⁴-32r³+14r²+12r-1, root=17 (r1^(C,4)(0)=Root[P_4r1,17]~.137779-.916121i)
P_4r2=r²⁴-24r²³+58r²²+156r²¹+964r²⁰-1472r¹⁹+486r¹⁸-228r¹⁷-1003r¹⁶+1008r¹⁵-924r¹⁴+56r¹³+
        +160r¹²-608r¹¹+428r¹⁰-40r⁹-149r⁸+104r⁷-62r⁶+44r⁵+28r⁴-32r³+14r²+12r-1, root=18 (r2^(C,4)(0)=Root[P_4r3,18]~.137779+.916121i)
P_4r3=r²⁴-24r²³+58r²²+156r²¹+964r²⁰-1472r¹⁹+486r¹⁸-228r¹⁷-1003r¹⁶+1008r¹⁵-924r¹⁴+56r¹³+
        +160r¹²-608r¹¹+428r¹⁰-40r⁹-149r⁸+104r⁷-62r⁶+44r⁵+28r⁴-32r³+14r²+12r-1, root=23 (r3^(C,4)(0)=Root[P_4r3,23]~.655182-.387974i)
P_4r4=r²⁴-24r²³+58r²²+156r²¹+964r²⁰-1472r¹⁹+486r¹⁸-228r¹⁷-1003r¹⁶+1008r¹⁵-924r¹⁴+56r¹³+
        +160r¹²-608r¹¹+428r¹⁰-40r⁹-149r⁸+104r⁷-62r⁶+44r⁵+28r⁴-32r³+14r²+12r-1, root=24 (r4^(C,4)(0)=Root[P_4r4,24]~.655182+.387974i)

P_4x1=x¹²-12x¹¹-86x¹⁰-276x⁹-465x⁸-512x⁷-484x⁶-512x⁵-465x⁴-276x³-86x²-12x+1, root=12 (x1^(C,4)(0)=Root[P_4x1,12]~.541766+.840529i)
P_4xr1=4x⁶-24x⁵-92x⁴-108x³-28x²+32x+17, root=3 (xr1^(C,4)(0)=Root[P_4xr1,3]~.541766)




deg 5:
P_5m1=m⁴+168m³+640m²+640m-256, root=2 (τ^(C,5)(0)=Root[P_5m1,2]~.301733)
P_5m2=m⁴-26944m³+194048m²-737280m+65536, root=1 ((τ^(C,5)(0))²=Root[P_5m2,1]~.0910429)


P_5a0=a⁸+24a⁶-22a⁴+16a²-3, root=1 (a0^(C,5)(0)=Root[P_5a0,1]~-.499591)
P_5a1=a⁴-64a³+134a²-64a+9, root=1 (a1^(C,5)(0)=Root[P_5a1,1]~1.54124)
P_5a2=a⁸+296a⁶-1872a⁴-896a²-768, root=1 (a2^(C,5)(0)=Root[P_5a2,1]~-2.58540)
P_5a3=a⁴+100a³-132a²-304a-432, root=2 (a3^(C,5)(0)=Root[P_5a3,2]~2.84297)
P_5a4=a⁸+496a⁶-2302a⁴+3640a²-13467, root=1 (a4^(C,5)(0)=Root[P_5a4,1]~-2.08581)

P_5x1=x³²+216x³⁰+1080x²⁸-7160x²⁶+38172x²⁴-59688x²²+87432x²⁰-64440x¹⁸+
        +74310x¹⁶-64440x¹⁴+87432x¹²-59688x¹⁰+38172x⁸-7160x⁶+1080x⁴+216x²+1, root=22 (x1^(C,5)(0)=Root[P_5x1,22]~.378549+.925581i)
P_5x2=x³²+216x³⁰+1080x²⁸-7160x²⁶+38172x²⁴-59688x²²+87432x²⁰-64440x¹⁸+
        +74310x¹⁶-64440x¹⁴+87432x¹²-59688x¹⁰+38172x⁸-7160x⁶+1080x⁴+216x²+1, root=30 (x2^(C,5)(0)=Root[P_5x2,30]~.914152+.405371i)
P_5rx1=x¹⁶+50x¹⁴-115x¹²-60x¹⁰+482x⁸-658x⁶+400x⁴-108x²+9, root=3 (rx1^(C,5)(0)=Root[P_5rx1,3]~.378549)
P_5rx2=x¹⁶+50x¹⁴-115x¹²-60x¹⁰+482x⁸-658x⁶+400x⁴-108x²+9, root=4 (rx2^(C,5)(0)=Root[P_5rx2,4]~.914152)

P_5r1=r⁴⁰+296r³⁸+10562r³⁶-29680r³⁴+256457r³²+170208r³⁰-332552r²⁸-258816r²⁶+124354r²⁴+177264r²²+
        +18860r²⁰-48608r¹⁸-32998r¹⁶-13344r¹⁴+23480r¹²+6784r¹⁰-5283r⁸-3672r⁶+2658r⁴-432r²-3, root=2 (r1^(C,5)(0)=Root[P_5r1,2]~.787688)
P_5r2=r⁴⁰+296r³⁸+10562r³⁶-29680r³⁴+256457r³²+170208r³⁰-332552r²⁸-258816r²⁶+124354r²⁴+177264r²²+
        +18860r²⁰-48608r¹⁸-32998r¹⁶-13344r¹⁴+23480r¹²+6784r¹⁰-5283r⁸-3672r⁶+2658r⁴-432r²-3, root=25 (r2^(C,5)(0)=Root[P_5r2,25]~.0951271-.955008i)
P_5r3=r⁴⁰+296r³⁸+10562r³⁶-29680r³⁴+256457r³²+170208r³⁰-332552r²⁸-258816r²⁶+124354r²⁴+177264r²²+
        +18860r²⁰-48608r¹⁸-32998r¹⁶-13344r¹⁴+23480r¹²+6784r¹⁰-5283r⁸-3672r⁶+2658r⁴-432r²-3, root=26 (r3^(C,5)(0)=Root[P_5r3,26]~.0951271+.955008i)
P_5r4=r⁴⁰+296r³⁸+10562r³⁶-29680r³⁴+256457r³²+170208r³⁰-332552r²⁸-258816r²⁶+124354r²⁴+177264r²²+
        +18860r²⁰-48608r¹⁸-32998r¹⁶-13344r¹⁴+23480r¹²+6784r¹⁰-5283r⁸-3672r⁶+2658r⁴-432r²-3, root=31 (r4^(C,5)(0)=Root[P_5r4,31]~.553935-.617854i)
P_5r5=r⁴⁰+296r³⁸+10562r³⁶-29680r³⁴+256457r³²+170208r³⁰-332552r²⁸-258816r²⁶+124354r²⁴+177264r²²+
        +18860r²⁰-48608r¹⁸-32998r¹⁶-13344r¹⁴+23480r¹²+6784r¹⁰-5283r⁸-3672r⁶+2658r⁴-432r²-3, root=32 (r5^(C,5)(0)=Root[P_5r5,32]~.553935+.617854i)



deg 6:
P_6m1=m¹²+11568m¹¹+32064m¹⁰+30375168m⁹-279310336m⁸+213975040m⁷-1156890624m⁶+7164657664m⁵-11530141696m⁴+4777312256m³+2348810240m²-1879048192m+268435456,
root=3 (τ^(C,6)(0)=Root[P_6m1,3]~.213383)
P_6m2=m¹²-133754496m¹¹-702290407424m¹⁰-945515284561920m⁹+64775234558885888m⁸+164372411527987200m⁷+4423226371862429696m⁶-27896468892882567168m⁵+59708105934143225856m⁴-50682569024474185728m³+17280311770220593152m²-2269814212194729984m+72057594037927936, root=1 ((τ^(C,6)(0))²=Root[P_6m2,1]~.0455323)


P_6a0=a¹²+12a¹¹-26a¹⁰+44a⁹+127a⁸-40a⁷-60a⁶-40a⁵+31a⁴+28a³-10a²-4a+1, root=6 (a0^(C,6)(0)=Root[P_6a0,6]~.499930)
P_6a1=a¹²-728a¹¹-10164a¹⁰-302608a⁹-851060a⁸+1061440a⁷+8033792a⁶+15182080a⁵+15025840a⁴+8706688a³+2928960a²+514304a+33856, root=1 (a1^(C,6)(0)=Root[P_6a1,1]~-1.79257)
P_6a2=a¹²+6452a¹¹+45042a¹⁰-142012a⁹-1825265a⁸-403352a⁷+10754492a⁶+22844200a⁵+23526895a⁴+14362980a³+5129970a²+1021076a+74593, root=5 (a2^(C,6)(0)=Root[P_6a2,5]~3.48506)
P_6a3=a¹²+7504a¹¹-95608a¹⁰+662592a⁹+1400704a⁸-15837440a⁷+16467584a⁶+1610752a⁵-10668032a⁴-3162112a³+335872a²+196608a+16384, root=2 (a3^(C,6)(0)=Root[P_6a3,2]~-4.52925)
P_6a4=a¹²-1732a¹¹-27010a¹⁰-1783332a⁹+9671975a⁸-3969864a⁷-11451628a⁶-10065352a⁵+17542607a⁴-882420a³+3445518a²-3770644a-22871, root=5 (a4^(C,6)(0)=Root[P_6a4,5]~4.13601)
P_6a5=a¹²+72a¹¹-652a¹⁰+6864a⁹+72908a⁸+78016a⁷+25344a⁶-382976a⁵-945104a⁴+2217600a³-1448256a²+229120a+188992, root=3 (a5^(C,6)(0)=Root[P_6a5,3]~-2.58579)

P_6x1=x⁴⁸-48x⁴⁷-72x⁴⁶+6256x⁴⁵+108660x⁴⁴+635088x⁴³+1589256x⁴²-876432x⁴¹-21828798x⁴⁰-99279184x³⁹-327097176x³⁸-919213680x³⁷-
        -2222790236x³⁶-4592413008x³⁵-8284374504x³⁴-13568500592x³³-20781036049x³²-29982156640x³¹-40654829136x³⁰-51998358560x²⁹-63314097688x²⁸-73787616608x²⁷-82230454320x²⁶-87559689248x²⁵-
        -89345239652x²⁴-87559689248x²³-82230454320x²²-73787616608x²¹-63314097688x²⁰-51998358560x¹⁹-40654829136x¹⁸-29982156640x¹⁷-20781036049x¹⁶-13568500592x¹⁵-8284374504x¹⁴-4592413008x¹³-
        -2222790236x¹²-919213680x¹¹-327097176x¹⁰-99279184x⁹-21828798x⁸-876432x⁷+1589256x⁶+635088x⁵+108660x⁴+6256x³-72x²-48x+1, root=38 (x1^(C,6)(0)=Root[P_6x1,38]~.274098+.961702i)
P_6x2=x⁴⁸-48x⁴⁷-72x⁴⁶+6256x⁴⁵+108660x⁴⁴+635088x⁴³+1589256x⁴²-876432x⁴¹-21828798x⁴⁰-99279184x³⁹-327097176x³⁸-919213680x³⁷-
        -2222790236x³⁶-4592413008x³⁵-8284374504x³⁴-13568500592x³³-20781036049x³²-29982156640x³¹-40654829136x³⁰-51998358560x²⁹-63314097688x²⁸-73787616608x²⁷-82230454320x²⁶-87559689248x²⁵-
        -89345239652x²⁴-87559689248x²³-82230454320x²²-73787616608x²¹-63314097688x²⁰-51998358560x¹⁹-40654829136x¹⁸-29982156640x¹⁷-20781036049x¹⁶-13568500592x¹⁵-8284374504x¹⁴-4592413008x¹³-
        -2222790236x¹²-919213680x¹¹-327097176x¹⁰-99279184x⁹-21828798x⁸-876432x⁷+1589256x⁶+635088x⁵+108660x⁴+6256x³-72x²-48x+1, root=46 (x1^(C,6)(0)=Root[P_6x2,46]~.768762+.639534i)
P_6r1=r⁷²-72z⁷¹+1080z⁷⁰+3280z⁶⁹+42382z⁶⁸-784440z⁶⁷+247448z⁶⁶-470576z⁶⁵+46064089z⁶⁴-88003008z⁶³
        +161498624z⁶²+409090432z⁶¹-836412272z⁶⁰-929463872z⁵⁹+1147256704z⁵⁸-2313654400z⁵⁷+2104584628z⁵⁶+1184895776z⁵⁵-2740721824z⁵⁴+3229709504z⁵³
        -2993526712z^52+469961952z^51+1678564192z^50-1949975360z^49+2556660164z^48-1612368448z^47+452764544z^46+774391424z^45-920919696z^44+1302943296z^43
        -556749056*z^42+119666816*z^41+7666350*z^40-147165872*z^39+236109456*z^38-306918688*z^37+286808596*z^36-159991376*z^35-20113840*z^34+145905888*z^33
        -284458834*z^32+53718976*z^31-71134976*z^30-95597952*z^29+107783408*z^28-63076288*z^27-30044800*z^26+5964928*z^25-55581756*z^24-13179616*z^23
        +13414368*z^22-8148800*z^21-20859192*z^20+4255456*z^19-150816*z^18-10354496*z^17+4748212*z^16-2838464*z^15-3716736*z^14+2067840*z^13
        -2445808z¹²+580544z¹¹+1349632z¹⁰-1594496z⁹-174119z⁸+513720z⁷-138760z⁶-85296z⁵+20558z⁴+1736z³+216z²-48z+1
root=45 (r1^(C,6)(0)=Root[P_6r1,45]~.00685555-.973557i)
root=46 (r2^(C,6)(0)=Root[P_6r2,46]~.00685555+.973557i)
root=57 (r3^(C,6)(0)=Root[P_6r3,57]~.451706-.752965i)
root=58 (r4^(C,6)(0)=Root[P_6r4,58]~.451706+.752965i)
root=67 (r5^(C,6)(0)=Root[P_6r5,67]~.772634-.289456i)
root=68 (r6^(C,6)(0)=Root[P_6r6,68]~.772634+.289456i)

---

Complex Chebyshev Polynomials with Different Methods

Description.
Code: QE1001
Name: Sec2a (Chebyshev on sectors (1st boundary arc), d=2, k\in [1/sqrt2, 1])
Vars: 4
Formula: (Ea1)(Ax)[(0<=x<=1) ==> [T^2(x,m,a1)(/.y->Sqrt[1/k^2-1] x)<=m^2]]
Output:
SW/HW: Mma9/Qepcad/Corei7-920
Best Time (confirmed): 1.5 sec [M9quine]
Remarks. qepcad 203 sec proj-lev-factors 1 17(12)

Description.
Code: QE1101
Name: Sec2a (Chebyshev on sectors (direct), d=2, k=0, half disk)
Vars: 5
Formula: (Ea0)(Ea1)(Ax)(Ay)[(x^2+y^2<=1 /\ x>=0) ==> [T^2(x,y,a0,a1)<=m^2]]
Output: 2(Sqrt[2]-1)<=m<9/10
SW/HW: Mma9/Qepcad/Corei7-920
Best Time (confirmed): 1486 sec [M9quine]
Remarks. 5GB!, !m\in[82/100,9/10] qepcad ?? sec proj-lev-factors 1 ??(??)



Description.
Code: QE1102
Name: Sec2a (Chebyshev on sectors (direct), coeffs, d=2, k=0, half disk)
Vars: 4
Formula: (Ax)(Ay)[(x^2+y^2<=1 /\ x>=0) ==> [T^2(x,y,a0,a1)<=(2(Sqrt[2]-1))^2]]
Output: a0=Sqrt[2]-1, a1=Sqrt[2]-2
SW/HW: Mma9/Qepcad/Corei7-920
Best Time (confirmed): 27 sec [M9quine]
Remarks. qepcad ?? sec proj-lev-factors 1 ??(??)

---

Real Chebyshev Polynomials with Different Methods
I=[0,1] (similar result should be obtaiend for [-1,1],[-1,0],[0,s])

deg=2
QE: MMA9
m=1/8 (.17s one stroke, ass m \in[0,1])
coeffs: a0=1/8,a1=-1. (0.s)
details-


deg=3
QE MMA9 m=1/32 (6.4s in two stroke(m|a0a10a2), ass m \in[0,1])
coeffs: a0=-1/32, a1=9/16, a2=-3/2. (95s)
details-

GB1 (EquiOsc):
2048m^7-6720m^6+2640m^5+4564m^4+1263m^3+84m^2-4m=(..)(32m-1) (0.01s)
Coeffs:=a0=-1/32, a1=9/16, a2=-3/2. (0s)


deg=4
QE-

GB1 (EquiOsc+RabTrick):
8192m^4-2112m^3+144m^2-m=(m)*(8m-1)^2*(128m-1) (.1s)
Coeffs:=a0=1/128,a1=-1/4,a2=5/4,a3=-2. (also with QE in two stroke and with CAD 40sec)


deg=5
QE-

GB1 (EquiOsc+RabTrick):
(m)*(m+2)^3*(512m-1)(m^2-246m+4)^2(256m^2-176m-1)^3 (365s)
Coeffs:=a0=-1/512,a1=25/256,a2=-25/32,a3=35/16,a4=-5/2.

Remark. Lex variable ordering: v,w,x,y,z,z2,a2,a3,a4,m
{e5c,e5c/.x->y,(e5c/.x->z)-2m,(e5c/.x->z2)-2m, D[e5c,x],D[e5c/.x->y,y],D[e5c/.x->z,z],D[e5c/.x->z2,z2],1+v(x-y),1+w(z-z2)}
where e5c=(2m-1-a2-a3-a4)x+a2x^2+a3x^3+a4x^4+a5x^5


Robert Vajda