Web Repository for Zolotarev's First Problem (ZFP) (2018-2023):

Introduction and Description. The goal of this project is to systematically investigate and describe the terms which enable one to compute algebraically the solution of Zolotarev's First Problem (ZFP) for 4≤n≤13. ZFP reads as follows: For a prescribed real number s>0 determine the monic polynomial Zn,s of degree n≥2, of form xn+(-ns)xn-1+lower degree terms, which deviates least from zero (in the uniform norm ||.||) on the interval [-1,1]. If 0≤s≤(tan(π/(2n)))2, then Zn,s is called an improper Zolotarev polynomial. We consider here only proper Zolotarev polynomials.

In the current form this project is a web repository which contains a list of Results and Benchmark Problems for symbolic computation, with the aid of the CAS Mathematica and Maple. This page is also intended as a web supplement of research articles by H.-J Rack (Hagen, Germany) and by R. Vajda (Szeged, Hungary) on Zolotarev polynomials.

[1] V. A. Malyshev, Algebraic solution of Zolotarev's problem, Algebra i Analiz, 14:4 (2002), 238-240; St. Petersburg Math. J., 14:4 (2003), 711-712
[2] H.-J. Rack, R. Vajda, Explicit algebraic solution of Zolotarev's First Problem for low-degree polynomials J. Numer. Anal. Approx. Theory, 48(2) (2019), 175-201.
[3] H.-J. Rack, R. Vajda, An explicit univariate and radical parametrization of the sextic proper Zolotarev polynomials in power form, Dolomites Research Notes on Approximation Padova University Press, 12(1), pp. 43-50, 2019.
[4] H.-J. Rack, R. Vajda, An explicit univariate and radical parametrization of the septic proper Zolotarev polynomials in power form (2020) (preprint), arXiv:2002.00503
[5] H.-J. Rack, R. Vajda, An explicit radical parametrization of Zolotarev polynomials of degree 7 in terms of nested square roots, Advanced Studies: Euro-Tbilisi Mathematical Journal, 14(4) (2021), pp. 37-60. DOI: 10.3251/asetmj/1932200813
[6] H.-J. Rack, R. Vajda. Explicit algebraic solution of Zolotarev's First Problem for low-degree polynomials, Part II. Dolomites Research Notes on Approximation Padova University Press, 16(3), pp. 75-103, 2023.

For details on ZFP and information on improper and proper Zolotarev polynomials see the References above (in particular [6]), and the References given therein.
Below information is provided to solve ZFP (for 4≤n≤13 and the proper case only) algebraically; for n = 2, 3 see [6, Section 1].

Details about the (radical) parametrizations of Zolotarev polynomials of degree 7 can be found here.
The page is under construction. Extensions and updates are in progress.
2 Three tentative algebraic forms of Zolotarev polynomials depending on additional parameters α and β, 1<α<β (see References [2] and [6])

2.1 The polynomials Rn,α,β used in the 1st algorithm of [2] and [6]

$R_{4,\alpha,\beta}(x)$:
 [Mma input R4alphabeta]

$R_{5,\alpha,\beta}(x)$:
 [Mma input R5alphabeta]

$R_{6,\alpha,\beta}(x)$:
 [Mma input R6alphabeta]

$R_{7,\alpha,\beta}(x)$:
 [Mma input R7alphabeta]

$R_{8,\alpha,\beta}(x)$:
 [Mma input R8alphabeta]

$R_{9,\alpha,\beta}(x)$:
 [Mma input R9alphabeta]

$R_{10,\alpha,\beta}(x)$:
 [Mma input R10alphabeta]

$R_{11,\alpha,\beta}(x)$:
 [Mma input R11alphabeta]

$R_{12,\alpha,\beta}(x)$:
 [Mma input R12alphabeta]

$R_{13,\alpha,\beta}(x)$:
 [Mma input R13alphabeta]
2.2 The polynomials Sn,α,β used in the 2nd algorithm of [2] and [6]

$S_{4,\alpha,\beta}(x)$:
 [Mma input S4alphabeta]

$S_{5,\alpha,\beta}(x)$:
 [Mma input S5alphabeta]

$S_{6,\alpha,\beta}(x)$:
 [Mma input S6alphabeta]

$S_{7,\alpha,\beta}(x)$:
 [Mma input S7alphabeta]

$S_{8,\alpha,\beta}(x)$:
 [Mma input S8alphabeta]

$S_{9,\alpha,\beta}(x)$:
 [Mma input S9alphabeta]

$S_{10,\alpha,\beta}(x)$:
 [Mma input S10alphabeta]

$S_{11,\alpha,\beta}(x)$:
 [Mma input S11alphabeta]

$S_{12,\alpha,\beta}(x)$:
 [Mma input S12alphabeta]

$S_{13,\alpha,\beta}(x)$:
 [Mma input S13alphabeta]

Calculation Rule to get Formula (12) from Formula (11) in [6] (Equioscillation points)
 [From Formula (12) to Formula (11)]
2.3 The polynomials Pn,α,β used in the 3rd algorithm of [6]

These polynomials are defined, for all n≥4, by a Formula involving two determinants, see [6, Formula 14]. The determinants in turn contain the moments σk=σk(α,β) .

$ \begin{split} &\sigma_0= \\ &1 \end{split} $

$ \begin{split} &\sigma_1= \\ &\frac{1}{2}(-2+\alpha+\beta) \end{split} $

$ \begin{split} &\sigma_2= \\ &\frac{1}{8}(4 - 4 \alpha + 3 \alpha^2 - 4 \beta + 2 \alpha \beta + 3 \beta^2) \end{split} $

$ \begin{split} &\sigma_3= \\ &\frac{1}{16}(5 \alpha ^3+3 \alpha^2 \beta -6 \alpha^2+3 \alpha \beta^2-4 \alpha \beta +4 \alpha +5 \beta^3-6 \beta^2+4 \beta -8) \end{split} $

$\begin{split} &\sigma_4=\\ &\frac{1}{128}(35 \alpha^4+20 \alpha^3 \beta -40 \alpha ^3+18 \alpha ^2 \beta ^2-24 \alpha ^2 \beta +24 \alpha ^2+20 \alpha \beta ^3-24 \alpha \beta ^2+ 16 \alpha \beta -32 \alpha +35 \beta ^4-40 \beta ^3+24 \beta ^2-32 \beta +48) \end{split} $

$\begin{split} &\sigma_5=\\ &\frac{1}{256}(63 \alpha ^5+35 \alpha ^4 \beta -70 \alpha ^4+30 \alpha ^3 \beta ^2-40 \alpha ^3 \beta +40 \alpha ^3+30 \alpha ^2 \beta ^3-36 \alpha ^2 \beta ^2+24 \alpha ^2 \beta -48 \alpha ^2+ &35 \alpha \beta ^4-40 \alpha \beta ^3+24 \alpha \beta ^2-32 \alpha \beta +48 \alpha +63 \beta ^5-70 \beta ^4+40 \beta ^3-48 \beta ^2+48 \beta -96) \end{split} $

$ \begin{split} &\sigma_6= \\ &\frac{1}{1024}(231 \alpha ^6+126 \alpha ^5 \beta -252 \alpha ^5+105 \alpha ^4 \beta ^2-140 \alpha ^4 \beta +140 \alpha ^4+100 \alpha ^3 \beta ^3-120 \alpha ^3 \beta ^2+80 \alpha ^3 \beta -160 \alpha ^3 +105 \alpha ^2 \beta ^4-120 \alpha ^2 \beta ^3+72 \alpha ^2 \beta ^2-96 \alpha ^2 \beta +144 \alpha ^2+126 \alpha \beta ^5-140 \alpha \beta ^4+80 \alpha \beta ^3-96 \alpha \beta ^2+96 \alpha \beta -192 \alpha +231 \beta ^6-252 \beta ^5+140 \beta ^4-160 \beta ^3+144 \beta ^2-192 \beta +320) \end{split} $

$ \begin{split} &\sigma_7=\\ &\frac{1}{2048}(429 \alpha ^7+231 \alpha ^6 \beta -462 \alpha ^6+189 \alpha ^5 \beta ^2-252 \alpha ^5 \beta +252 \alpha ^5+175 \alpha ^4 \beta ^3-210 \alpha ^4 \beta ^2+140 \alpha ^4 \beta -280 \alpha ^4+175 \alpha ^3 \beta ^4-200 \alpha ^3 \beta ^3+120 \alpha ^3 \beta ^2-160 \alpha ^3 \beta + \\ &240 \alpha ^3+189 \alpha ^2 \beta ^5-210 \alpha ^2 \beta ^4+120 \alpha ^2 \beta ^3-144 \alpha ^2 \beta ^2+144 \alpha ^2 \beta -288 \alpha ^2+231 \alpha \beta ^6-252 \alpha \beta ^5+140 \alpha \beta ^4-160 \alpha \beta ^3+144 \alpha \beta ^2-192 \alpha \beta +320 \alpha +429 \beta ^7-462 \beta ^6+252 \beta ^5-280 \beta ^4+240 \beta ^3-288 \beta ^2+320 \beta -640) \end{split} $

$ \begin{split} &\sigma_8= \\ &\frac{1}{32768}(6435 \alpha ^8+3432 \alpha ^7 \beta -6864 \alpha ^7+2772 \alpha ^6 \beta ^2-3696 \alpha ^6 \beta +3696 \alpha ^6+2520 \alpha ^5 \beta ^3-3024 \alpha ^5 \beta ^2 +2016 \alpha ^5 \beta -4032 \alpha ^5+2450 \alpha ^4 \beta ^4-2800 \alpha ^4 \beta ^3+1680 \alpha ^4 \beta ^2-2240 \alpha ^4 \beta +3360 \alpha ^4+2520 \alpha ^3 \beta ^5-2800 \alpha ^3 \beta ^4 +1600 \alpha ^3 \beta ^3-1920 \alpha ^3 \beta ^2+1920 \alpha ^3 \beta -3840 \alpha ^3+ \\ &2772 \alpha ^2 \beta ^6-3024 \alpha ^2 \beta ^5+1680 \alpha ^2 \beta ^4-1920 \alpha ^2 \beta ^3+1728 \alpha ^2 \beta ^2-2304 \alpha ^2 \beta +3840 \alpha ^2+3432 \alpha \beta ^7-3696 \alpha \beta ^6+2016 \alpha \beta ^5-2240 \alpha \beta ^4+1920 \alpha \beta ^3-2304 \alpha \beta ^2+2560 \alpha \beta -5120 \alpha +6435 \beta ^8-6864 \beta ^7+3696 \beta ^6-4032 \beta ^5+3360 \beta ^4-3840 \beta ^3+3840 \beta ^2-5120 \beta +8960) \end{split} $

$ \begin{split} &\sigma_9= \\ &\frac{1}{65536}(12155 \alpha ^9+6435 \alpha ^8 \beta -12870 \alpha ^8+5148 \alpha ^7 \beta ^2-6864 \alpha ^7 \beta +6864 \alpha ^7+4620 \alpha ^6 \beta ^3-5544 \alpha ^6 \beta ^2+3696 \alpha ^6 \beta -7392 \alpha ^6+4410 \alpha ^5 \beta ^4-5040 \alpha ^5 \beta ^3+3024 \alpha ^5 \beta ^2-4032 \alpha ^5 \beta +6048 \alpha ^5+4410 \alpha ^4 \beta ^5-4900 \alpha ^4 \beta ^4 +2800 \alpha ^4 \beta ^3-3360 \alpha ^4 \beta ^2+3360 \alpha ^4 \beta -6720 \alpha ^4+4620 \alpha ^3 \beta ^6- \\ &5040 \alpha ^3 \beta ^5+2800 \alpha ^3 \beta ^4-3200 \alpha ^3 \beta ^3+ 2880 \alpha ^3 \beta ^2-3840 \alpha ^3 \beta +6400 \alpha ^3+5148 \alpha ^2 \beta ^7-5544 \alpha ^2 \beta ^6+3024 \alpha ^2 \beta ^5-3360 \alpha ^2 \beta ^4+2880 \alpha ^2 \beta ^3-3456 \alpha ^2 \beta ^2+3840 \alpha ^2 \beta -7680 \alpha ^2+6435 \alpha \beta ^8-6864 \alpha \beta ^7+3696 \alpha \beta ^6-4032 \alpha \beta ^5+3360 \alpha \beta ^4-3840 \alpha \beta ^3+3840 \alpha \beta ^2-5120 \alpha \beta + \\ &8960 \alpha +12155 \beta ^9-12870 \beta ^8+6864 \beta ^7-7392 \beta ^6+6048 \beta ^5-6720 \beta ^4+6400 \beta ^3-7680 \beta ^2+8960 \beta -17920) \end{split} $

 [Mma polys for sigma_k (k=0..25)]

3. Three Variants to determine the concrete α=α0 and β=β0 if n=n0 and s=s0 are prescribed and thus creating Rn0,α0,β0(x)= Sn0,α0,β0(x)= Pn0,α0,β0(x) , i.e., solving ZFP algebraically, see [6].

3.1 Variant 1 (Malyshev polynomials and largest root index)
Degree m(n) of the n-th Malyshev Polynomials

n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
m(n)

1

2

4

6

8

12

16

18

24

30

32

42

48

48

64
$\left\lfloor\frac{n^2}{4}\right\rfloor$

1

2

4

6

9

12

16

20

25

30

36

42

49

56

64

The n-th Malyshev polynomials for n = 2,3,4,5 are to be found in [1]. For convenience, the n = 4 and n = 5 case is reproduced here in red font. For n = 6 and n = 7 they have been given in [2] and are reproduced here for convenience.
For n = 8,9,10,11,12 they have been announced in [2] and are given here, also for n = 13, for the first time.
Malyshev Polynomials

$\deg=4$ and $m(4)=4$
$ \color{red}{ \begin{split} \color{black}{F_{4,s}=}&\\ &(-13 - 136 s - 448 s^2 - 896 s^3 + 256 s^4) +\\ & (44 + 184 s + 128 s^2 - 640 s^3) \alpha +\\ & (-22 + 168 s + 576 s^2) \alpha^2 +\\ & (-36 - 216 s) \alpha^3 +\\ & 27 \alpha^4 \end{split} } $
$ \color{red}{ \begin{split} \color{black}{G_{4,s}=}&\\ &(-13 + 136 s - 448 s^2 + 896 s^3 + 256 s^4) +\\ & (-44 + 184 s - 128 s^2 - 640 s^3) \beta +\\ & (-22 - 168 s + 576 s^2) \beta^2 +\\ & (36 - 216 s) \beta^3 +\\ & 27 \beta^4 \end{split} } $
$\deg=5$ and $m(5)=6:$
$ \color{red}{ \begin{split} \color{black}{F_{6,s}=}&\\ &(-16 + 160 s^2 + 4000 s^4 - 3125 s^6) +\\ & (-320 s - 1400 s^3 + 6250 s^5) \alpha +\\ & (64 - 1160 s^2 - 3875 s^4) \alpha^2 +\\ & (600 s + 100 s^3) \alpha^3 +\\ & (-72 + 765 s^2) \alpha^4 -\\ & 270 s \alpha^5 +\\ & 27 \alpha^6 \end{split} } $
$ \color{red}{ \begin{split} \color{black}{G_{6,s}=}&\\ &(-16 - 800 s^2 - 137500 s^4 - 15625 s^6) +\\ & (-480 s + 113000 s^3 - 31250 s^5) \beta +\\ & (160 - 46000 s^2 + 118125 s^4) \beta^2 +\\ & (10200 s - 114500 s^3) \beta^3 +\\ & (-900 + 48825 s^2) \beta^4 - \\ & 9450 s \beta^5 +\\ & 675 \beta^6 \end{split} } $

$\deg=6$ and $m(6)=8:$
$ \begin{split} F_{8,s}=&\\ &(-59 + 2000 s - 34688 s^2 - 16128 s^3 - 318816 s^4 - 3960576 s^5 - 2861568 s^6 - 1492992 s^7 + 186624 s^8) +\\ & (-376 + 16976 s + 57792 s^2 - 220800 s^3 + 3162240 s^4 + 3117312 s^5 - 995328 s^6 - 995328 s^7) \alpha +\\ & (-2564 - 18672 s + 284160 s^2 - 637440 s^3 - 1873728 s^4 + 4154112 s^5 + 2280960 s^6) \alpha^2 +\\ & (1816 - 89584 s - 39296 s^2 + 1278720 s^3 - 2888064 s^4 - 2923776 s^5) \alpha^3 +\\ & (8558 + 27248 s - 589440 s^2 + 705792 s^3 + 2282400 s^4) \alpha^4 +\\ & (-2312 + 121584 s - 22080 s^2 - 1104000 s^3) \alpha^5 +\\ & (-8932 - 11600 s + 320000 s^2) \alpha^6 +\\ & (1000 - 50000 s) \alpha^7 +\\ & 3125 \alpha^8 \end{split} $
$ \begin{split} G_{8,s}=&\\ &(-59 - 2000 s - 34688 s^2 + 16128 s^3 - 318816 s^4 + 3960576 s^5 - 2861568 s^6 + 1492992 s^7 + 186624 s^8) +\\ & (376 + 16976 s - 57792 s^2 - 220800 s^3 - 3162240 s^4 + 3117312 s^5 + 995328 s^6 - 995328 s^7) \beta +\\ & (-2564 + 18672 s + 284160 s^2 + 637440 s^3 - 1873728 s^4 - 4154112 s^5 + 2280960 s^6) \beta^2 +\\ & (-1816 - 89584 s + 39296 s^2 + 1278720 s^3 + 2888064 s^4 - 2923776 s^5) \beta^3 +\\ & (8558 - 27248 s - 589440 s^2 - 705792 s^3 + 2282400 s^4) \beta^4 +\\ & (2312 + 121584 s + 22080 s^2 - 1104000 s^3) \beta^5 +\\ & (-8932 + 11600 s + 320000 s^2) \beta^6 +\\ & (-1000 - 50000 s) \beta^7 + \\ & 3125 \beta^8 \end{split} $
 [Mma poly input F8s]  [Mma coeff input F8s]
 [Mma poly input G8s]  [Mma coeff input G8s]

$\deg=7:$ and $m(7)=12:$
$ \begin{split} F_{12,s}=&\\ &(1792 + 163072 s^2 + 8410752 s^4 - 376438384 s^6 + 2733221568 s^8 + 2029209952 s^{10} - 282475249 s^{12}) +\\ & (64512 s - 1436288 s^3 + 447392736 s^5 - 6100537632 s^7 - 322828856 s^9 + 1129900996 s^{11}) \alpha +\\ & (-19712 + 1728384 s^2 - 223227536 s^4 + 5813877440 s^6 - 7355415480 s^8 - 1395081842 s^{10}) \alpha^2 +\\ & (-868224 s + 78809024 s^3 - 3231784416 s^5 + 12151057632 s^7 - 668716916 s^9) \alpha^3 +\\ & (100864 - 23298576 s^2 + 1235270400 s^4 - 9953009360 s^6 + 4106538345 s^8) \alpha^4 +\\ & (4101216 s - 351315552 s^3 + 5195725584 s^5 - 5802179768 s^7) \alpha^5 +\\ & (-279792 + 70263872 s^2 - 1874768224 s^4 + 4661407044 s^6) \alpha^6 +\\ & (-8268960 s + 469393568 s^3 - 2424683464 s^5) \alpha^7 +\\ & (410688 - 77006160 s^2 + 843673425 s^4) \alpha^8 +\\ & (7295400 s - 194631500 s^3) \alpha^9 +\\ & (-297000 + 28428750 s^2) \alpha^{10} -\\ & 2362500 s \alpha^{11} + \\ & 84375 \alpha^{12} \end{split} $
$ \begin{split} G_{12,s}=&\\ &(565504 - 102271232 s^2 + 3016577984 s^4 + 196082294128 s^6 - 158647323520 s^8 + 571729903976 s^{10} + 13841287201 s^{12}) +\\ & (74400256 s - 6755815808 s^3 - 186738408864 s^5 + 874115128544 s^7 - 1168317629864 s^9 + 55365148804 s^{11}) \beta +\\ & (-10513664 + 4103042048 s^2 + 22357074768 s^4 - 1148314476288 s^6 + 1805996857280 s^8 - 485292477782 s^{10}) \beta^2 +\\ & (-953905792 s + 34684357568 s^3 + 570329544736 s^5 - 2201060316896 s^7 + 1228202382652 s^9) \beta^3 +\\ & (78266944 - 16893148272 s^2 - 43921473792 s^4 + 1570190613600 s^6 - 1640714247809 s^8) \beta^4 +\\ & (3041893344 s - 65016346976 s^3 - 537505339280 s^5 + 1310441386632 s^7) \beta^5 +\\ & (-196530768 + 26344802176 s^2 + 14921350640 s^4 - 627807535124 s^6) \beta^6 +\\ & (-4059208608 s + 52325237216 s^3 + 152679225048 s^5) \beta^7 +\\ & (229065984 - 17822979720 s^2 + 4354873775 s^4) \beta^8 +\\ & (2480095800 s - 14912439500 s^3) \beta^9 +\\ & (-129654000 + 4425986250 s^2) \beta^{10} -\\ & 578812500 s \beta^{11} +\\ & 28940625 \beta^{12} \end{split} $
 [Mma poly input F12s]  [Mma coeff input F12s]
 [Mma poly input G12s]  [Mma coeff input G12s]

$\deg=8$ and $m(8)=16:$
$ \begin{split} F_{16,s}=&\\ & (-226207279 + 3808114592 s - 109775676416 s^2 + 1600912840704 s^3 - 6466135932928 s^4 + 35732817838080 s^5 - 311599898296320 s^6 - 1053891332407296 s^7 + 6934274460614656 s^8 + \\ & 20846439278051328 s^9 + 9898370608922624 s^{10} + 22849226014720 s^{11} + 56609180789768192 s^{12} - 27868221717610496 s^{13} - 10907155347537920 s^{14} - 4362862139015168 s^{15} + 281474976710656 s^{16}) +\\ & (-2064295792 + 46555718432 s - 970639472640 s^2 + 1943290472448 s^3 + 25429755625472 s^4 + 227094333882368 s^5 + 1248306600607744 s^6 - 14181100997836800 s^7 - 45368610620702720 s^8 + \\ & 12979843750690816 s^9 + 69853451181359104 s^{10} - 110138938747781120 s^{11} + 77642013595402240 s^{12} + 56468718179319808 s^{13} + 703687441776640 s^{14} - 2955487255461888 s^{15}) \alpha + \\ & (-8028082664 + 144597989280 s + 1005138621440 s^2 - 39221169762304 s^3 - 51294239096832 s^4 - 78754892873728 s^5 + 11542910663131136 s^6 + 40941852622323712 s^7 - 63172707801366528 s^8 - \\ & 187294245038587904 s^9 + 134629958961070080 s^{10} - 28300535945756672 s^{11} - 142575322041155584 s^{12} + 38326776321015808 s^{13} + 14425592556421120 s^{14}) \alpha^2 + \\ & (-7133218192 - 559381970144 s + 14216741507072 s^2 + 19786696159232 s^3 - 506929965367296 s^4 - 5073011378618368 s^5 - 17783916437962752 s^6 + 75978033642602496 s^7 + 217747504224010240 s^8 - \\ & 179808256628097024 s^9 - 164289233580720128 s^{10} + 240221781472837632 s^{11} - 73866290665619456 s^{12} - 43424112227385344 s^{13}) \alpha^3 + \\ & (62536537084 - 2195637150176 s - 12302923711488 s^2 + 308298127558656 s^3 + 1492052338425856 s^4 + 1975492595154944 s^5 - 48401335724277760 s^6 - 138942048060309504 s^7 + 202934622990893056 s^8 + \\ & 321874433918631936 s^9 - 305765404771352576 s^{10} + 22812564173881344 s^{11} + 90177270785769472 s^{12}) \alpha^4 + \\ & (130310242832 + 2925786752928 s - 80649712109568 s^2 - 369264401172480 s^3 + 1671384814714880 s^4 + 19789175884873728 s^5 + 48701984886226944 s^6 - 157752564895449088 s^7 - 295412288125403136 s^8 + \\ & 306244405830877184 s^9 + 91549564376449024 s^{10} - 136897066038198272 s^{11}) \alpha^5 + \\ & (-220239705496 + 10216050752544 s + 75823521821696 s^2 - 888258901581824 s^3 - 5774702268317696 s^4 - 6019685750669312 s^5 + 84386120183316480 s^6 + 158562080146849792 s^7 - 241048019229736960 s^8 - \\ & 154284613533958144 s^9 + 157033642248372224 s^{10}) \alpha^6 + \\ & (-510519433104 - 9614111273568 s + 196990683144192 s^2 + 1256000315039744 s^3 - 2212432519757824 s^4 - 32178208012238848 s^5 - 49273528951767040 s^6 + 147001720201805824 s^7 + 123908592892903424 s^8 -\\ & 138719624135966720 s^9) \alpha^7 + \\ & (506862648774 - 21471067730208 s - 193658297781248 s^2 + 1167959320127488 s^3 + 8906873428000768 s^4 + 6254948116463616 s^5 - 68430095399780352 s^6 - 59519587640672256 s^7 + 95272133696946176 s^8) \alpha^8 + \\ & (934066095408 + 18192372666720 s - 234711598991360 s^2 - 1764063379367936 s^3 + 1303523206168576 s^4 + 23940264280326144 s^5 + 17021817197690880 s^6 - 50977319082262528 s^7) \alpha^9 + \\ & (-754649268696 + 23053515352800 s + 236311010196480 s^2 - 720361609801728 s^3 - 6164281194086400 s^4 - 2177753178832896 s^5 + 21147604276477952 s^6) \alpha^{10} + \\ & (-908324190000 - 18980478640800 s + 136013623971840 s^2 + 1129623660032000 s^3 - 279650931507200 s^4 - 6717285047861248 s^5) \alpha^{11} + \\ & (680894653500 - 12446879882400 s - 138906380160000 s^2 + 170070973184000 s^3 + 1597593222758400 s^4) \alpha^{12} + \\ & (457448958000 + 10216262700000 s - 30901328640000 s^2 - 274113833728000 s^3) \alpha^{13} +\\ & (-336422025000 + 2692172700000 s + 31849937280000 s^2) \alpha^{14} +\\ & (-93999150000 - 2223566100000 s) \alpha^{15} +\\ & 69486440625 \alpha^{16} \end{split} $
$ \begin{split} G_{16,s}=&\\ & (-226207279 - 3808114592 s - 109775676416 s^2 - 1600912840704 s^3 - 6466135932928 s^4 - 35732817838080 s^5 - 311599898296320 s^6 + 1053891332407296 s^7 + 6934274460614656 s^8 - \\ & 20846439278051328 s^9 + 9898370608922624 s^{10} - 22849226014720 s^{11} + 56609180789768192 s^{12} + 27868221717610496 s^{13} - 10907155347537920 s^{14} + 4362862139015168 s^{15} + 281474976710656 s^{16}) + \\ & (2064295792 + 46555718432 s + 970639472640 s^2 + 1943290472448 s^3 - 25429755625472 s^4 + 227094333882368 s^5 - 1248306600607744 s^6 - 14181100997836800 s^7 + 45368610620702720 s^8 + \\ & 12979843750690816 s^9 - 69853451181359104 s^{10} - 110138938747781120 s^{11} - 77642013595402240 s^{12} + 56468718179319808 s^{13} - 703687441776640 s^{14} - 2955487255461888 s^{15}) \beta + \\ & (-8028082664 - 144597989280 s + 1005138621440 s^2 + 39221169762304 s^3 - 51294239096832 s^4 + 78754892873728 s^5 + 11542910663131136 s^6 - 40941852622323712 s^7 - 63172707801366528 s^8 + \\ & 187294245038587904 s^9 + 134629958961070080 s^{10} + 28300535945756672 s^{11} - 142575322041155584 s^{12} - 38326776321015808 s^{13} + 14425592556421120 s^{14}) \beta^2 + \\ & (7133218192 - 559381970144 s - 14216741507072 s^2 + 19786696159232 s^3 + 506929965367296 s^4 - 5073011378618368 s^5 + 17783916437962752 s^6 + 75978033642602496 s^7 - 217747504224010240 s^8 - \\ & 179808256628097024 s^9 + 164289233580720128 s^{10} + 240221781472837632 s^{11} + 73866290665619456 s^{12} - 43424112227385344 s^{13}) \beta^3 + \\ & (62536537084 + 2195637150176 s - 12302923711488 s^2 - 308298127558656 s^3 + 1492052338425856 s^4 - 1975492595154944 s^5 - 48401335724277760 s^6 + 138942048060309504 s^7 + 202934622990893056 s^8 - \\ & 321874433918631936 s^9 - 305765404771352576 s^{10} - 22812564173881344 s^{11} + 90177270785769472 s^{12}) \beta^4 + \\ & (-130310242832 + 2925786752928 s + 80649712109568 s^2 - 369264401172480 s^3 - 1671384814714880 s^4 + 19789175884873728 s^5 - 48701984886226944 s^6 - 157752564895449088 s^7 + 295412288125403136 s^8 + \\ & 306244405830877184 s^9 - 91549564376449024 s^{10} - 136897066038198272 s^{11}) \beta^5 + \\ & (-220239705496 - 10216050752544 s + 75823521821696 s^2 + 888258901581824 s^3 - 5774702268317696 s^4 + 6019685750669312 s^5 + 84386120183316480 s^6 - 158562080146849792 s^7 - 241048019229736960 s^8 + \\ & 154284613533958144 s^9 + 157033642248372224 s^{10}) \beta^6 + \\ & (510519433104 - 9614111273568 s - 196990683144192 s^2 + 1256000315039744 s^3 + 2212432519757824 s^4 - 32178208012238848 s^5 + 49273528951767040 s^6 + 147001720201805824 s^7 - 123908592892903424 s^8 -\\ & 138719624135966720 s^9) \beta^7 + \\ & (506862648774 + 21471067730208 s - 193658297781248 s^2 - 1167959320127488 s^3 + 8906873428000768 s^4 - 6254948116463616 s^5 - 68430095399780352 s^6 + 59519587640672256 s^7 + 95272133696946176 s^8) \beta^8 + \\ & (-934066095408 + 18192372666720 s + 234711598991360 s^2 - 1764063379367936 s^3 - 1303523206168576 s^4 + 23940264280326144 s^5 - 17021817197690880 s^6 - 50977319082262528 s^7) \beta^9 + \\ & (-754649268696 - 23053515352800 s + 236311010196480 s^2 + 720361609801728 s^3 - 6164281194086400 s^4 + 2177753178832896 s^5 + 21147604276477952 s^6) \beta^{10} + \\ & (908324190000 - 18980478640800 s - 136013623971840 s^2 + 1129623660032000 s^3 + 279650931507200 s^4 - 6717285047861248 s^5) \beta^{11} + \\ & (680894653500 + 12446879882400 s - 138906380160000 s^2 - 170070973184000 s^3 + 1597593222758400 s^4) \beta^{12} + \\ & (-457448958000 + 10216262700000 s + 30901328640000 s^2 - 274113833728000 s^3) \beta^{13} + \\ & (-336422025000 - 2692172700000 s + 31849937280000 s^2) \beta^{14} + \\ & (93999150000 - 2223566100000 s) \beta^{15} +\\ & 69486440625 \beta^{16} \end{split} $
 [Mma poly input F16s]  [Mma coeff input F16s]
 [Mma poly input G16s]  [Mma coeff input G16s]

$\deg=9$ and $m(9)=18:$
$ \begin{split} F_{18,s}=&\\ & (-6230016 + 2236502016 s^2 - 910058747904 s^4 - 52670745331200 s^6 + 451322485390080 s^8 + 3465165381120 s^{10} + 5458957181785584 s^{12} - 8814814119929664 s^{14} - 1545454423624032 s^{16} + 68630377364883 s^{18}) + \\ & (-779010048 s + 973688242176 s^3 + 27520406885376 s^5 - 1170469980919296 s^7 + 3293733364419456 s^9 - 10490817884012352 s^{11} + 30054519457208352 s^{13} - 1321770230731080 s^{15} - 411782264189298 s^{17}) \alpha + \\ & (-8073216 - 357847289856 s^2 + 9702634968576 s^4 + 914011199995392 s^6 - 8291330789917824 s^8 + 17969272250983776 s^{10} - 49173350731931520 s^{12} + 21338367388689384 s^{14} + 651564940661667 s^{16}) \alpha^2 + \\ & (55308828672 s - 10555506468864 s^3 - 232355480256000 s^5 + 8081351260171776 s^7 - 30054161506894272 s^9 + 61360438087525824 s^{11} - 54567032210366472 s^{13} + 1736000217569880 s^{15}) \alpha^3 + \\ & (-3357075456 + 3134739322368 s^2 - 49871187717120 s^4 - 3780475618798080 s^6 + 31947848535551760 s^8 - 71632905758428608 s^{10} + 82526739123034008 s^{12} - 10568847986421516 s^{14}) \alpha^4 + \\ & (-421762968576 s + 45310663319040 s^3 + 675964424966400 s^5 - 19875696060516480 s^7 + 70103471795227872 s^9 - 94391000177292072 s^{11} + 25643858065135920 s^{13}) \alpha^5 + \\ & (22174161408 - 11559607414272 s^2 + 142758363499776 s^4 + 6907950238089024 s^6 - 49342895863818624 s^8 + 87316581695897448 s^{10} - 39381913096193052 s^{12}) \alpha^6 + \\ & (1360780604928 s - 102312938764800 s^3 - 927183293557632 s^5 + 22972153720634496 s^7 - 63361284710779368 s^9 + 42649674690228120 s^{11}) \alpha^7 + \\ & (-62647029504 + 22762213208064 s^2 - 234462446593776 s^4 - 6368605605220032 s^6 + 34042404752162712 s^8 - 33932732567701086 s^{10}) \alpha^8 + \\ & (-2388733186176 s + 131686105093056 s^3 + 633820991604192 s^5 - 12683443234481496 s^7 + 20078539379213148 s^9) \alpha^9 + \\ & (99201446784 - 25962880125600 s^2 + 215320823192448 s^4 + 2885326729943544 s^6 - 8747719176134862 s^8) \alpha^{10} + \\ & (2482108948800 s - 97141850284608 s^3 - 198627995959704 s^5 + 2687547014649288 s^7) \alpha^{11} + \\ & (-95317674000 + 17289420857280 s^2 - 102137203006200 s^4 - 506851108240428 s^6) \alpha^{12} +\\ & (-1535962917600 s + 38297388717000 s^3 + 19969259885568 s^5) \alpha^{13} + \\ & (55575024000 - 6268834845000 s^2 + 19446183281700 s^4) \alpha^{14} + \\ & (526256325000 s - 6269498403000 s^3) \alpha^{15} + \\ & (-18187575000 + 960236431875 s^2) \alpha^{16} - \\ & 77207156250 s \alpha^{17} + \\ & 2573571875 \alpha^{18} \end{split} $
$ \begin{split} G_{18,s}=&\\ & (-719097856 - 80999092224 s^2 - 477079953408 s^4 + 447169162857984 s^6 - 16520620834379520 s^8 + 251570486894504832 s^{10} - \\ & 1894988492721762192 s^{12} + 370570146225990480 s^{14} - 225199144926636084 s^{16} - 1853020188851841 s^{18}) + \\ & (-49470504960 s - 3631982837760 s^3 - 303263355773952 s^5 + 27994969349494272 s^7 - 626799457596428928 s^9 + 5738882266926279360 s^{11} -\\ & 3700242664178800032 s^{13} + 980651836569328200 s^{15} - 11118121133111046 s^{17}) \beta + \\ & (11818659840 - 887801536512 s^2 + 41294973657600 s^4 - 18766681376140800 s^6 + 695830836195484416 s^8 - 8352808627165631136 s^{10} + \\ & 12264721229122230096 s^{12} - 3386160684685301400 s^{14} + 197358088508948547 s^{16}) \beta^2 + \\ & (779494883328 s - 28156142204928 s^3 + 6814224888712704 s^5 - 448823265749604864 s^7 + 7654764891954805056 s^9 - 21632991781598184384 s^{11} + 9647273044795358088 s^{13} - 1016481977285453784 s^{15}) \beta^3 + \\ & (-88349847552 + 23239270973952 s^2 - 1974818932337664 s^4 + 189512385867991296 s^6 - 4851998384930030448 s^8 + 24066692476917591312 s^{10} - 18902286665993919672 s^{12} + 2994825189219081876 s^{14}) \beta^4 + \\ & (-5472104408064 s + 640640444613120 s^3 - 58998167670461184 s^5 + 2232755284542500736 s^7 - 18409588092859010400 s^9 + 25173937756021583016 s^{11} - 5942916449226056592 s^{13}) \beta^5 + \\ & (392537083392 - 163026601026048 s^2 + 15463437920206848 s^4 - 779150374234570944 s^6 + 10218938154524561808 s^8 - 23725823505788526552 s^{10} + 8547194281777783380 s^{12}) \beta^6 + \\ & (21944269232640 s - 3465813353292288 s^3 + 216175920333342336 s^5 - 4283022723053664384 s^7 + 16457109600676876776 s^9 - 9271274238702943128 s^{11}) \beta^7 + \\ & (-1132847209728 + 566043640131456 s^2 - 48914305398001008 s^4 + 1397898886819994352 s^6 - 8656187736720317184 s^8 + 7760921327796553146 s^{10}) \beta^8 + \\ & (-54106896703104 s + 8699052825233856 s^3 - 361456880202319968 s^5 + 3523093456271027544 s^7 - 5077154731497177708 s^9) \beta^9 + \\ & (2179452213504 - 1095515733301920 s^2 + 73459012715441136 s^4 - 1120341814938678024 s^6 + 2609145514640565090 s^8) \beta^{10} + \\ & (83066700225600 s - 11247767848262592 s^3 + 277442793078898968 s^5 - 1052086154070875976 s^7) \beta^{11} + \\ & (-2768715162000 + 1196872637523120 s^2 - 52469138935787400 s^4 + 330431182304137236 s^6) \beta^{12} + \\ & (-77159595146400 s + 7281070037955000 s^3 - 79666658300551392 s^5) \beta^{13} + \\ & (2226538206000 - 691970720805000 s^2 + 14394586615283700 s^4) \beta^{14} + \\ & (39755389275000 s - 1875608409213000 s^3) \beta^{15} + \\ & (-1028318287500 + 165332066169375 s^2) \beta^{16} - \\ & 8755291518750 s \beta^{17} + \\ & 208459321875 \beta^{18} \end{split} $
 [Mma poly input F18s]  [Mma coeff input F18s]
 [Mma poly input G18s]  [Mma coeff input G18s]

$\deg=10$ and $m(10)=24:$
$ \begin{split} F_{24,s}=&\\ & (1567760865637 + 72509360360304 s + 1779352005408128 s^2 + 32587232931254016 s^3 + 390454835284672992 s^4 + 2719779471117131520 s^5 + 13219980581130560000 s^6 + \\ & 29563226986125568000 s^7 - 679077568547370400000 s^8 - 6170351518954009600000 s^9 - 2421260235737600000000 s^{10} + 102341474426511360000000 s^{11} + 8505648648224000000000 s^{12} - \\ & 205901659025664000000000 s^{13} + 998751565760000000000000 s^{14} + 2488060146022400000000000 s^{15} + 9873835852080000000000000 s^{16} + 8686805134080000000000000 s^{17} + 2284728320000000000000000 s^{18} + \\ & 839516672000000000000000 s^{19} + 2225835200000000000000000 s^{20} - 767808000000000000000000 s^{21} - 176000000000000000000000 s^{22} - 38400000000000000000000 s^{23} + 1600000000000000000000 s^{24}) + \\ & (-25308274594152 - 915928964442128 s - 18313909320044224 s^2 - 233301337997494144 s^3 - 964679652105060480 s^4 + 6237369084091825920 s^5 + 69893903375961907200 s^6 + 1134684931318268416000 s^7 + \\ & 10465608020926574080000 s^8 - 4936738030453888000000 s^9 - 323032112831660544000000 s^{10} - 126312405714887680000000 s^{11} + 1632784983919078400000000 s^{12} - 2361810978140416000000000 s^{13} - \\ & 8239298439485440000000000 s^{14} - 25478197074739200000000000 s^{15} - 21907550215296000000000000 s^{16} + 11802941510400000000000000 s^{17} + 9965716608000000000000000 s^{18} - \\ & 5480993536000000000000000 s^{19} + 3324307200000000000000000 s^{20} + 1274867200000000000000000 s^{21} - 25600000000000000000000 s^{22} - 25600000000000000000000 s^{23}) \alpha + \\ & (149792841475892 + 3359727810443952 s + 28062229846732288 s^2 - 470285981880512000 s^3 - 14010168532734266560 s^4 - 87565898421351791360 s^5 - 450007062578896217600 s^6 - \\ & 5822713654536738816000 s^7 + 13751219702159970560000 s^8 + 416096636657104153600000 s^9 + 194985832922557952000000 s^{10} - 4138255981153607680000000 s^{11} + 1534146266028044800000000 s^{12} + \\ & 17133665138608384000000000 s^{13} + 28817668477539840000000000 s^{14} + 16208899873792000000000000 s^{15} - 61402313504192000000000000 s^{16} - 47042890164480000000000000 s^{17} + \\ & 15348081203200000000000000 s^{18} - 1335493632000000000000000 s^{19} - 4878243200000000000000000 s^{20} + 1011264000000000000000000 s^{21} + 195200000000000000000000 s^{22}) \alpha^2 + \\ & (-290214332581752 + 5517749532287664 s + 323800773131391360 s^2 + 6075640341599520000 s^3 + 28191676645649110400 s^4 - 159129857986763406080 s^5 + 583779190936807833600 s^6 - \\ & 8818273118617202688000 s^7 - 282335444551091392000000 s^8 - 115062164277934412800000 s^9 + 5354757074809650176000000 s^{10} + 723935795726182400000000 s^{11} - 28027772212075443200000000 s^{12} - \\ & 21774061574563584000000000 s^{13} + 18833958915645440000000000 s^{14} + 115776942263500800000000000 s^{15} + 84500775363200000000000000 s^{16} - 55620209035520000000000000 s^{17} - \\ & 21943340288000000000000000 s^{18} + 14233360896000000000000000 s^{19} - 2827244800000000000000000 s^{20} - 943756800000000000000000 s^{21}) \alpha^3 + \\ & (-833206135224198 - 67493471176516080 s - 1107297628030294400 s^2 + 781790088513949440 s^3 + 186801316860173388640 s^4 + 602953494311830952960 s^5 - 29773117920576870400 s^6 + \\ & 105794030896471780352000 s^7 + 39423405716187233920000 s^8 - 4134220713832952857600000 s^9 - 1770657651422784000000000 s^{10} + 33573642190100275200000000 s^{11} + 16913316812070931200000000 s^{12} - \\ & 67236716493916672000000000 s^{13} - 132597181466301440000000000 s^{14} - 69126710749491200000000000 s^{15} + 130125443988000000000000000 s^{16} + 69070048743680000000000000 s^{17} - \\ & 35230223769600000000000000 s^{18} + 742230528000000000000000 s^{19} + 3247364800000000000000000 s^{20}) \alpha^4 + \\ & (4815995963023944 + 92476875917082768 s - 2144175609805242304 s^2 - 65530149944403613568 s^3 - 221479450225543133696 s^4 + 2617038137789013990400 s^5 - 18995107572683531059200 s^6 - \\ & 31212133760451610624000 s^7 + 2022745826310245588480000 s^8 + 1395633912600012774400000 s^9 - 28050428017383284224000000 s^{10} - 14785744810313354240000000 s^{11} + 99195068314802329600000000 s^{12} + \\ & 113482161551242752000000000 s^{13} - 11081705220444160000000000 s^{14} - 197716468621312000000000000 s^{15} - 103096442100608000000000000 s^{16} + 72112613996800000000000000 s^{17} + \\ & 11754804352000000000000000 s^{18} - 8461025024000000000000000 s^{19}) \alpha^5 + \\ & (-3159540766278844 + 410031704029437264 s + 11701682897735025152 s^2 + 13466638552193185792 s^3 - 1452595413293301715200 s^4 - 170960650946013199360 s^5 + 31976426622638994073600 s^6 - \\ & 630044914321678927872000 s^7 - 778345147168524099840000 s^8 + 16271992750483980672000000 s^9 + 11166068079740833280000000 s^{10} - 95374133474232647680000000 s^{11} - 82629249204980147200000000 s^{12} + \\ & 103389251488944640000000000 s^{13} + 216559846867942400000000000 s^{14} + 78571555535052800000000000 s^{15} - 117577752012736000000000000 s^{16} - 31009891111680000000000000 s^{17} + \\ & 17340240947200000000000000 s^{18}) \alpha^6 + \\ &(-24323877667280616 \ - 1096622315721854896 s + 5925939789194605056 s^2 + 398503139391282007040 s^3 + 679067442820544325120 s^4 - 18996047671365534120960 s^5 + 116835090840103786086400 s^6 + \\ & 399783686087775547392000 s^7 - 6567226263420163796480000 s^8 - 6620310245692024857600000 s^9 + 64760729507198257152000000 s^{10} + 53639776266610667520000000 s^{11} - \\ & 145198552081586636800000000 s^{12} - 185184755172085248000000000 s^{13} - 1323457962270720000000000 s^{14} + 151275746369536000000000000 s^{15} + 40449267142016000000000000 s^{16} - \\ & 28663143269120000000000000 s^{17}) \alpha^7 + \\ & (43319350052004571 \ - 1158970194084484512 s - 61929506028250824960 s^2 - 85627019346155430400 s^3 + 6689576376902596836800 s^4 - 9239838686546240519680 s^5 - 187755352086533492761600 s^6 + \\ & 1821476444523977770496000 s^7 + 3231467827063532868160000 s^8 - 31808058068083224550400000 s^9 - 30581982905554816512000000 s^{10} + 124174005147713740800000000 s^{11} + \\ & 130211330629440480000000000 s^{12} - 72145727884391680000000000 s^{13} - 155007184152998400000000000 s^{14} - 28061038009600000000000000 s^{15} + 38877214104112000000000000 s^{16}) \alpha^8 + \\ & (62135988266423664 + 5219336102432496480 s - 4288587668699498880 s^2 - 1455518553370447937280 s^3 - 780636153420074653440 s^4 + 69663964532455575421440 s^5 - 331034046522064867737600 s^6 - \\ & 1369031183615396967424000 s^7 + 11345316662941671610880000 s^8 + 15054683007324059059200000 s^9 - 74753292699863440896000000 s^{10} - 77530013004701875200000000 s^{11} + \\ & 96764631488259865600000000 s^{12} + 128597520598272768000000000 s^{13} + 1856189780879360000000000 s^{14} - 43786686614323200000000000 s^{15}) \alpha^9 + \\ & (-185560874469412632 + 1566076508103703776 s + 193089419839656748032 s^2 + 233647547737639209984 s^3 - 18484690604504906620032 s^4 + 33640112729465969640960 s^5 + 496464934985049412275200 s^6 - \\ & 2900773919284206757888000 s^7 - 6403591215712051656960000 s^8 + 33052327602331733836800000 s^9 + 39686231291752700416000000 s^{10} - 76520346306822594560000000 s^{11} -\\ & 87966384819073216000000000 s^{12} + 19063314282829056000000000 s^{13} + 41283974753431040000000000 s^{14}) \alpha^{10} + \\ & (-86498702589995568 - 14326070893112139552 s - 14605455002127501056 s^2 + 3320922829615344274944 s^3 - 391185528034204302080 s^4 - 144454426483866531985920 s^5 + 510542273824397973094400 s^6 + \\ & 2346144831609288945664000 s^7 - 10838458173783340587520000 s^8 - 17587097600714746854400000 s^9 + 42586645461424012288000000 s^{10} + 50461976537904424960000000 s^{11} - \\ & 24374198146352972800000000 s^{12} - 32752532372445952000000000 s^{13}) \alpha^{11} + \\ &(454959556509133548 - 383601294173832672 s - 381360880565432187648 s^2 - 358174136863658682880 s^3 + 31504905680637280216000 s^4 - 55168883883223303459840 s^5 - 721257678448150250649600 s^6 + \\ & 2607130757488966077440000 s^7 + 6736213801692441482880000 s^8 - 17543216897217294412800000 s^9 - 24588600780915752448000000 s^{10} + 18181206175088762880000000 s^{11} + \\ & 21922583333297747200000000 s^{12}) \alpha^{12} + \\ & (53186235275403792 \ + 25156868469494450976 s + 43319029456679470720 s^2 - 4851329065651269917440 s^3 + 2043095571116522257920 s^4 + 177908828141668865305600 s^5 - 443311624305735731609600 s^6 - \\ & 2199075020087847815168000 s^7 + 5417455490050247252480000 s^8 + 10233242892622133785600000 s^9 - 9574809691056816640000000 s^{10} - 12385642939506938880000000 s^{11}) \alpha^{13} + \\ & (-722416785320474712 - 1944231421475729760 s + 492289152178180193280 s^2 + 329467915722013440000 s^3 - 33469611834074799857920 s^4 + 48335228103867178193920 s^5 + 595933051638296550246400 s^6 -\\ & 1242211172449379022848000 s^7 - 3623268414583208011520000 s^8 + 3751496806161357337600000 s^9 + 5896413515431291392000000 s^{10}) \alpha^{14} + \\ & (21492348534215856 \ - 29323502077852561248 s - 53721946209837436416 s^2 + 4546854670820755729408 s^3 - 2265330424433143017984 s^4 - 129455857791454847134720 s^5 + 204253101513831881932800 s^6 + \\ & 1077115177098072002560000 s^7 - 1107958714564340984320000 s^8 - 2356048675165557427200000 s^9) \alpha^{15} + \end{split} $ $ \begin{split} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &(772364198299163499 + 3140963888051620656 s - 415644570436281478272 s^2 - 180568268913265642752 s^3 + 21643319823974681749600 s^4 - 22040156620943257790720 s^5 - 263340014586898315046400 s^6 + \\ & 244409236954339223808000 s^7 + 785033207588675403360000 s^8) \alpha^{16} + \\ & (-66068071482756744 + 22693988562209639856 s + 36394058215594465344 s^2 - 2650653457208203708800 s^3 + 1128323373873084044160 s^4 + 51553891950756856559360 s^5 - 38945705027064579072000 s^6 - \\ & 216044041189063343616000 s^7) \alpha^{17} + \\ & (-555446853689861916 \ - 2282413124066655888 s + 221840255466906693120 s^2 + 54347708654787555840 s^3 - 7814825675336965055680 s^4 + 4126347815135613984000 s^5 + 48445747112873265523200 s^6) \alpha^{18} + \\ & (52170773732773416 - 11248236100220207760 s - 13201484528521457280 s^2 + 876942459040314553600 s^3 - 219062328670733143680 s^4 - 8684181856938993672960 s^5) \alpha^{19} + \\ & (258677302586303034 + 845814336597993168 s - 68079772163543462784 s^2 - 6904855688966178048 s^3 + 1210777556951196559584 s^4) \alpha^{20} + \\ & (-19567289474742744 + 3240646099728511824 s + 2016811292438198592 s^2 - 126034590502821264768 s^3) \alpha^{21} + \\ & (-70668562986638316 - 129621057110271216 s + 9171351455934398976 s^2) \alpha^{22} + \\ & (2972617389024696 - 413498431577194992 s) \alpha^{23} +\\ & 8614550657858229 \alpha^{24} \end{split} $
 [Mma poly input F24s]  [Mma coeff input F24s]
 [Mma poly input G24s]  [Mma coeff input G24s]

$\deg=11$ and $m(11)=30:$

 [Mma poly input F30s]
 [Mma poly input G30s]

$\deg=12$ and $m(12)=32:$

 [Mma poly input F32s]
 [Mma poly input G32s]

$\deg=13$ and $m(13)=42:$

 [Mma poly input F42s]
 [Mma poly input G42s]

Algorithmic generation of Malyshev polynomials by means of Resultant

 [MalyshevPolynomialsbyResultants]

3.2 Variant 2 (System of two determinant equations)

Variant 2 is inferior to the Variant 3, see [6], and is therefore skipped here. However, the Formula 23 in [6] is given here exemplarily for n= 2,..,6:

sn(σk) (n=2,...,6)
 [s_n (n=2,..,4) (txt)]  [s_n (n=2,...,4) (nb)]
 [s_5 (nb)]  [s_6 (nb)]
3.3 Variant 3 with the reduced relation curves Hm(n)(α,β)=0 and algebraic terms sn(α, β) (n=4,...,13).

$\deg=2\, (g=0):$
$H^{2}_{m(2)}=H^{2}_{1}=2+\alpha-\beta$

$\deg=3\, (g=0):$
$H^{3}_{m(3)}=H^{3}_{2}=4+\alpha^2+2\alpha\beta-3\beta^2$

$\deg=4\, (g=0):$
$ H^{4}_{m(4)}=H^{4}_{4}= 16 - 8 \alpha^2 + 8 \alpha^3 + \alpha^4 + 16 \alpha \beta + 8 \alpha^2 \beta - 4 \alpha^3 \beta - 8 \beta^2 - 8 \alpha \beta^2 + 6 \alpha^2 \beta^2 - 8 \beta^3 - 4 \alpha \beta^3 + \beta^4 $

$\deg=5\, (g=1):$
$ H^{5}_{m(5)}=H^{5}_{6}= 64 - 80 \alpha^2 + 44 \alpha^4 + \alpha^6 + 96 \alpha \beta - 16 \alpha^3 \beta + 6 \alpha^5 \beta - 16 \beta^2 + 104 \alpha^2 \beta^2 - 29 \alpha^4 \beta^2 - 80 \alpha \beta^3 + 36 \alpha^3 \beta^3 - 52 \beta^4 - 9 \alpha^2 \beta^4 - 10 \alpha \beta^5 + 5 \beta^6 $

$\deg=6\, (g=1):$
$ \begin{split} H^{6}_{m(6)}=H^{6}_{8}=&\\ &256 - 256 \alpha^2 - 256 \alpha^3 + 96 \alpha^4 + 256 \alpha^5 - 80 \alpha^6 + 16 \alpha^7 + \alpha^8 + 512 \alpha \beta - 256 \alpha^2 \beta - 384 \alpha^3 \beta + 256 \alpha^4 \beta - 32 \alpha^5 \beta + 80 \alpha^6 \beta - 8 \alpha^7 \beta - 256 \beta^2 + 256 \alpha \beta^2 + 576 \alpha^2 \beta^2 - 176 \alpha^4 \beta^2 - 112 \alpha^5 \beta^2 + 28 \alpha^6 \beta^2 + 256 \beta^3 - \\ & 384 \alpha \beta^3 + 576 \alpha^3 \beta^3 - 176 \alpha^4 \beta^3 - 56 \alpha^5 \beta^3 + 96 \beta^4 - 256 \alpha \beta^4 - 176 \alpha^2 \beta^4 + 176 \alpha^3 \beta^4 + 70 \alpha^4 \beta^4 - 256 \beta^5 - 32 \alpha \beta^5 + 112 \alpha^2 \beta^5 - 56 \alpha^3 \beta^5 - 80 \beta^6 - 80 \alpha \beta^6 + 28 \alpha^2 \beta^6 - 16 \beta^7 - 8 \alpha \beta^7 + \beta^8 \end{split} $

$\deg=7\, (g=4):$
$ \begin{split} H^{7}_{m(7)}=H^{7}_{12}=&\\ & 4096 - 2048 \alpha^2 - 8448 \alpha^4 + 7424 \alpha^6 - 1040 \alpha^8 + 184 \alpha^{10} + \alpha^{12} + 12288 \alpha \beta - 17408 \alpha^3 \beta + 1536 \alpha^5 \beta + 4992 \alpha^7 \beta - 144 \alpha^9 \beta + 12 \alpha^{11} \beta - 10240 \beta^2 + 25088 \alpha^2 \beta^2 - 8448 \alpha^4 \beta^2 - 4288 \alpha^6 \beta^2 + 1976 \alpha^8 \beta^2 - \\ & 118 \alpha^{10} \beta^2 - 13312 \alpha \beta^3 + 33792 \alpha^3 \beta^3 - 10624 \alpha^5 \beta^3 - 4032 \alpha^7 \beta^3 + 364 \alpha^9 \beta^3 + 14080 \beta^4 - 29952 \alpha^2 \beta^4 + 23712 \alpha^4 \beta^4 - 3472 \alpha^6 \beta^4 - 441 \alpha^8 \beta^4 + 5632 \alpha \beta^5 - 19328 \alpha^3 \beta^5 + 11680 \alpha^5 \beta^5 - 168 \alpha^7 \beta^5 - 9984 \beta^6 + \\ & 10560 \alpha^2 \beta^6 - 7632 \alpha^4 \beta^6 + 1260 \alpha^6 \beta^6 - 5760 \alpha \beta^7 + 3648 \alpha^3 \beta^7 - 1800 \alpha^5 \beta^7 + 1776 \beta^8 - 3624 \alpha^2 \beta^8 + 1311 \alpha^4 \beta^8 + 1136 \alpha \beta^9 - 484 \alpha^3 \beta^9 + 280 \beta^{10} + 42 \alpha^2 \beta^{10} + 28 \alpha \beta^{11} - 7 \beta^{12} \end{split} $


$\deg=8\, (g=5):$
$ \begin{split} H^{8}_{m(8)}=H^{8}_{16}=&\\ & 65536 - 131072 \alpha^2 + 131072 \alpha^3 + 114688 \alpha^4 - 327680 \alpha^5 - 90112 \alpha^6 + 253952 \alpha^7 + 50688 \alpha^8 - 57344 \alpha^9 - 15872 \alpha^{10} - 512 \alpha^{11} + 6592 \alpha^{12} + 768 \alpha^{13} - 160 \alpha^{14} + 32 \alpha^{15} + \alpha^{16} + 262144 \alpha \beta + 131072 \alpha^2 \beta - \\ & 458752 \alpha^3 \beta - 196608 \alpha^4 \beta + 278528 \alpha^5 \beta - 106496 \alpha^6 \beta - 143360 \alpha^7 \beta + 221184 \alpha^8 \beta + 60416 \alpha^9 \beta - 45568 \alpha^{10} \beta + 2816 \alpha^{11} \beta - 768 \alpha^{12} \beta + 1216 \alpha^{13} \beta + 160 \alpha^{14} \beta - 16 \alpha^{15} \beta - 131072 \beta^2 - 131072 \alpha \beta^2 +\\ & 688128 \alpha^2 \beta^2 + 131072 \alpha^3 \beta^2 - 827392 \alpha^4 \beta^2 + 122880 \alpha^5 \beta^2 + 370688 \alpha^6 \beta^2 - 262144 \alpha^7 \beta^2 - 124416 \alpha^8 \beta^2 + 170496 \alpha^9 \beta^2 + 41856 \alpha^{10} \beta^2 - 16896 \alpha^{11} \beta^2 - 4320 \alpha^{12} \beta^2 - 864 \alpha^{13} \beta^2 + 120 \alpha^{14} \beta^2 - 131072 \beta^3 - \\ & 458752 \alpha \beta^3 - 131072 \alpha^2 \beta^3 + 1277952 \alpha^3 \beta^3 + 483328 \alpha^4 \beta^3 - 1003520 \alpha^5 \beta^3 - 65536 \alpha^6 \beta^3 + 331776 \alpha^7 \beta^3 - 185856 \alpha^8 \beta^3 - 123136 \alpha^9 \beta^3 + 66048 \alpha^{10} \beta^3 + 11136 \alpha^{11} \beta^3 - 1504 \alpha^{12} \beta^3 - 560 \alpha^{13} \beta^3 + 114688 \beta^4 +\\ & 196608 \alpha \beta^4 - 827392 \alpha^2 \beta^4 - 483328 \alpha^3 \beta^4 + 1451008 \alpha^4 \beta^4 + 475136 \alpha^5 \beta^4 - 777216 \alpha^6 \beta^4 - 62464 \alpha^7 \beta^4 + 117312 \alpha^8 \beta^4 - 86784 \alpha^9 \beta^4 - 27040 \alpha^{10} \beta^4 + 12192 \alpha^{11} \beta^4 + 1820 \alpha^{12} \beta^4 + 327680 \beta^5 + 278528 \alpha \beta^5 -\\ & 122880 \alpha^2 \beta^5 - 1003520 \alpha^3 \beta^5 - 475136 \alpha^4 \beta^5 + 1050624 \alpha^5 \beta^5 + 338944 \alpha^6 \beta^5 - 338432 \alpha^7 \beta^5 - 11520 \alpha^8 \beta^5 + 59200 \alpha^9 \beta^5 - 19424 \alpha^{10} \beta^5 - 4368 \alpha^{11} \beta^5 - 90112 \beta^6 + 106496 \alpha \beta^6 + 370688 \alpha^2 \beta^6 + 65536 \alpha^3 \beta^6 -\\ & 777216 \alpha^4 \beta^6 - 338944 \alpha^5 \beta^6 + 585984 \alpha^6 \beta^6 + 156672 \alpha^7 \beta^6 - 99552 \alpha^8 \beta^6 + 32 \alpha^9 \beta^6 + 8008 \alpha^{10} \beta^6 - 253952 \beta^7 - 143360 \alpha \beta^7 + 262144 \alpha^2 \beta^7 + 331776 \alpha^3 \beta^7 + 62464 \alpha^4 \beta^7 - 338432 \alpha^5 \beta^7 - 156672 \alpha^6 \beta^7 +\\ & 119040 \alpha^7 \beta^7 + 32160 \alpha^8 \beta^7 - 11440 \alpha^9 \beta^7 + 50688 \beta^8 - 221184 \alpha \beta^8 - 124416 \alpha^2 \beta^8 + 185856 \alpha^3 \beta^8 + 117312 \alpha^4 \beta^8 + 11520 \alpha^5 \beta^8 - 99552 \alpha^6 \beta^8 - 32160 \alpha^7 \beta^8 + 12870 \alpha^8 \beta^8 + 57344 \beta^9 + 60416 \alpha \beta^9 -\\ & 170496 \alpha^2 \beta^9 - 123136 \alpha^3 \beta^9 + 86784 \alpha^4 \beta^9 + 59200 \alpha^5 \beta^9 - 32 \alpha^6 \beta^9 - 11440 \alpha^7 \beta^9 - 15872 \beta^{10} + 45568 \alpha \beta^{10} + 41856 \alpha^2 \beta^{10} - 66048 \alpha^3 \beta^{10} - 27040 \alpha^4 \beta^{10} + 19424 \alpha^5 \beta^{10} + 8008 \alpha^6 \beta^{10} + 512 \beta^{11} + 2816 \alpha \beta^{11} + \\ & 16896 \alpha^2 \beta^{11} + 11136 \alpha^3 \beta^{11} - 12192 \alpha^4 \beta^{11} - 4368 \alpha^5 \beta^{11} + 6592 \beta^{12} + 768 \alpha \beta^{12} - 4320 \alpha^2 \beta^{12} + 1504 \alpha^3 \beta^{12} + 1820 \alpha^4 \beta^{12} - 768 \beta^{13} + 1216 \alpha \beta^{13} + 864 \alpha^2 \beta^{13} - 560 \alpha^3 \beta^{13} - 160 \beta^{14} - 160 \alpha \beta^{14} + 120 \alpha^2 \beta^{14} -\\ & 32 \beta^{15} - 16 \alpha \beta^{15} + \beta^{16} \end{split} $

 [Mma H-poly input deg2-10]
 [Mma H-poly input deg11]  [Mma H-poly input deg12]  [Mma H-poly input deg13]
Auxiliary algebraic terms sn(α, β) (n=4,...,13).
 [Mma s_n, n=5,7,9,11,13]  [Mma s_n, n=4,6,8,10,12]
3.4 Hands-on calculations for n = 5 as referenced to after Formula 10 in [6]:

 [R5, F6, G6, α5(β, s), β5(α, s) granulated creation]
3.5 Canonical Solutions

 [Canonical Sols]
3.6 Miscellaneous

Figures




Fig 1. - Probably the first symbolically computed particular proper monic quintic Zolotarev polynomial CM51 with 2nd lcf=a4=1, see the Reference [10] (Collins) in [6].





Fig 2. - Proper monic normalized Zolotarev polynomials with negative 2nd lcf of degree 3,4,5,6.
The list of monic normalized (i.e., norm = 1) proper Zolotarev polynomials of degree n (with negative 2nd lcf).
Notation. $P_{n,a_j}$ denotes the defining polynomial of the coefficient $a_j$ of the monic normalized proper Zolotarev polynomials of degree n
n=3:
$P_{3,a_0}=-11 + 12 a_0 - 2 a_0^2 + 2 a_0^3 + a_0^4, {\rm root\,} 2\approx 0.881649$,
$P_{3,a_1}=a_1+1, {\rm root\, } 1\approx-1.00000$,
$P_{3,a_2}=-26 - 18 a_2 - 2 a_2^2 + 2 a_2^3 + a_2^4, {\rm root\,} 1\approx -1.88165$.

n=4:
$P_{4,a_0}=-19343 + 61944 a_0 + 76936 a_0^2 - 29116 a_0^3 - 42196 a_0^4 + 26332 a_0^5 - 5176 a_0^6 + 292 a_0^7 - 12 a_0^8 + 8 a_0^9 + 4 a_0^{10}, {\rm root\,} 2\approx0.246155$,
$P_{4,a_1}=52877 - 23430 a_1 + 8773 a_1^2 + 11116 a_1^3 + 2982 a_1^4 - 932 a_1^5 - 314 a_1^6 - 152 a_1^7 - 3 a_1^8 + 6 a_1^9 + a_1^{10}, {\rm root\,} 1\approx2.93751$,
$P_{4,a_2}=-49247 - 154396 a_2 - 436428 a_2^2 - 517592 a_2^3 - 262724 a_2^4 - 64192 a_2^5 - 7388 a_2^6 - 196 a_2^7 + 96 a_2^8 + 32 a_2^9 + 4 a_2^{10}, {\rm root\,} 1\approx-1.24615$,
$P_{4,a_3}=77708 + 23328 a_3 + 864 a_3^2 + 8620 a_3^3 + 7496 a_3^4 + 1568 a_3^5 + 372 a_3^6 + 32 a_3^7 - 12 a_3^8 + 4 a_3^9 + a_3^{10}, {\rm root\,} 1\approx-3.93751$.
n=5:
$ \begin{split} P_{5,a_0}=& 49399619050433 + 144024581179752 a_0 + 42390202098736 a_0^2 - 160120948449896 a_0^3 + 14187234446720 a_0^4 + 52084960180736 a_0^5 - 45660203338528 a_0^6 + 22821007971840 a_0^7 - 5035238644944 a_0^8-\\ & 365164694816 a_0^9 + 942959227616 a_0^{10} - 350581788672 a_0^{11} + 69307827200 a_0^{12} - 7386803328 a_0^{13} + 390016768 a_0^{14} + 92469760 a_0^{15} - 20865536 a_0^{16} + 965632 a_0^{17} - 4608 a_0^{18} + 512 a_0^{19} + 256 a_0^{20}, {\rm root\,} 4\approx-0.992218, \end{split} $
$ \begin{split} P_{5,a_1}=&218859405947089 - 1044670380960544 a_1 + 2357240777558160 a_1^2 - 3159190724102048 a_1^3 + 1774807744447632 a_1^4 + 256619353711632 a_1^5 - 789605814774496 a_1^6 + 281647414281408 a_1^7 + 32201164426208 a_1^8-\\ &40354530720960 a_1^9 + 9305238171488 a_1^{10} - 198167934720 a_1^{11} - 321469371136 a_1^{12} + 73321644288 a_1^{13} - 7331407616 a_1^{14} + 291084544 a_1^{15} + 1456896 a_1^{16} + 45056 a_1^{17} + 7680 a_1^{18} + 3072 a_1^{19} + 256 a_1^{20}, {\rm root\,} 3\approx0.499023, \end{split} $
$ \begin{split} P_{5,a_2}=&24348153990155617081 - 3792309151242446288 a_2 - 2475014306861899160 a_2^2 - 1114721260854855808 a_2^3 + 441490175671697984 a_2^4 + 90417498002924832 a_2^5 + 16416470976606848 a_2^6 - 2447052939695680 a_2^7 - 576532226982832 a_2^8-\\ &85321982823840 a_2^9 + 2665696273632 a_2^{10} + 1266685678848 a_2^{11} + 166073764992 a_2^{12} + 1755589120 a_2^{13} - 857873920 a_2^{14} - 106505728 a_2^{15} - 3022336 a_2^{16} - 118784 a_2^{17} - 10240 a_2^{18} + 2560 a_2^{19} + 256 a_2^{20}, {\rm root\,} 2\approx7.96097, \end{split} $
$ \begin{split} P_{5,a_3}=&7308553139410433 + 15054806052718864 a_3 + 4915180002600368 a_3^2 - 11709669091277696 a_3^3 - 12071073757084576 a_3^4 - 1948954442609936 a_3^5 + 3129609514216480 a_3^6 + 2203306702704960 a_3^7 + 569461517900160 a_3^8\\+ & 5039259532672 a_3^9 - 38903859069344 a_3^{10} - 12438654221312 a_3^{11} - 2071859535360 a_3^{12} - 205814301696 a_3^{13} - 11555784192 a_3^{14} - 275572992 a_3^{15} + 129536 a_3^{16} - 140288 a_3^{17} - 2048 a_3^{18} + 2048 a_3^{19} + 256 a_3^{20}, {\rm root\,} 2\approx-1.49902, \end{split} $
$ \begin{split} P_{5,a_4}=&-3061811343439 - 1226305456200 a_4 - 194624790282 a_4^2 - 123031965120 a_4^3 - 27493526803 a_4^4 - 21849416464 a_4^5 - 7265383512 a_4^6 - 777178704 a_4^7 - 1330816702 a_4^8-\\ &288348856 a_4^9 - 7276796 a_4^{10} - 24779448 a_4^{11} - 3845102 a_4^{12} - 64832 a_4^{13} - 198648 a_4^{14} - 19264 a_4^{15} + 2013 a_4^{16} - 176 a_4^{17} - 42 a_4^{18} + 8 a_4^{19} + a_4^{20}, {\rm root\,} 1\approx-7.96875. \end{split} $
 [Mma Coeff of Monic Normalized Z-Polys]
n=6:
 [Mma Coeff of Monic Normalized Z-Polys]
n=7:
 [Mma Coeff of Monic Normalized Z-Poly, degree 7]
The following files are referenced to in the paper: H.-J. Rack, R. Vajda. Explicit solution, for n = 7, to a Markov-type extremal problem initiated by Schur. Annales Univ. Sci. Budapest. Sect. Math. 64 (2021), pp. 179-201

 [Septic Extremizers k=2 (txt)]  [Septic Extremizers k=2 (nb)]
 [Septic Extremizers k=3 (txt)]  [Septic Extremizers k=3 (nb)]
 [Septic Extremizers k=4 (txt)]  [Septic Extremizers k=4 (nb)]

Auxiliary algebraic terms α=αn(β,s) and β=βn(α,s) (n=6,7), see p.87 in [6]
 [alpha6 and beta6]    [alpha7 and beta7]
Worked-out Example of Z4,s via T-Polynomials (referenced to in Remark 6 (v) of [6])
 [Z4s along Section 7 of Peherstorfer and Schiefermayr (1999)]
Worked-out Example of Z3,s via recurrent Formula by Vlcek and Unbehauen (referenced to in Remark 6 (vi) of [6])
 [Z3s by recursion]
Worked-out Example of 2.6 related to Proposition 2.5
 [Example 2.6 in detail]
Algebraic solution of ZFP if n = 6 and s = 2 and coefficients and least deviation expressed in terms of root objects, see Example 5.1 in [6]
 [Explicit Z62 with root objects]
Algebraic solution of ZFP if n = 8 and s = 3 and coefficients expressed in terms of root objects, see Appendix 3 in [6]
 [Explicit Z83 with root objects]
Algebraic solution of ZFP if n = 9 and s = 4 and coefficients expressed in terms of root objects, see Appendix 3 in [6]
 [Explicit Z94 with root objects]
Algebraic solution of ZFP if n = 10 and s = 5 and exemplarily one coefficient expressed in terms of root objects, see Appendix 3 in [6]
 [Explicit Z105 with root objects]
Algebraic solution of ZFP if n = 11 and s = 6 and exemplarily one coefficient expressed in terms of root objects, see Appendix 3 in [6]
 [Explicit Z116 with root objects]
Algebraic solution of ZFP if n = 12 and s = 7 and exemplarily one coefficient expressed in terms of root objects, see Appendix 3 in [6]
 [Explicit Z127 with root objects]
Algebraic solution of ZFP if n = 13 and s = 8 and exemplarily one coefficient expressed in terms of root objects, see Appendix 3 in [6]
 [Explicit Z138 with root objects]



Robert Vajda