12 13573 454292 369811 231038 color colorblue Roots of P2 x0000 x0000 x0000 x0000 x0000 x0000 1228122812131213x0000 x0000 1222122213113 ID: 824738
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(1.1.1)(1.1.1)� � (1.1.3)(
(1.1.1)(1.1.1)� � (1.1.3)(1.1.3)� � � � (1.1.2)(1.1.2)�
� � � (1.1.6)(1.1.6
� � � (1.1.6)(1.1.6)(1.1.4)(1.1.4)(1.1.5)(1.1.5)restart;The series U in the paramatr
izationPositivity of U 12 13573 454292
izationPositivity of U 12 13573 454292 369811 231038,color=color=blueRoots of P2:� � &
#x0000; � � � (1.2.
#x0000; � � � (1.2.2.8)(1.2.2.8)(1.2.1.3)(1.2.1.3)� � (1.2.2.2)(1.2.2.
2)(1.3.1)(1.3.1)� � (1.3.2
2)(1.3.1)(1.3.1)� � (1.3.2)(1.3.2)123450.00500.0050.0100.0150.020Computation of U(rho)The c
aracteristic equation:PhiUUwUnu;eqUrhofa
aracteristic equation:PhiUUwUnu;eqUrhofactornumerPhiUUdiffPhiU,U7 214 plotUrho2implicitploteqUrho2plot
Urho3implicitploteqUrho3 3 3 3 372 47
Urho3implicitploteqUrho3 3 3 3 372 472 31/2 8 3 3 3 372 472 3 3 372 472 31/2 8 3 3 18 3 18 234
3 1 372 90 _Z5 3 Uw21sing33 3 1 36 6:p
3 1 372 90 _Z5 3 Uw21sing33 3 1 36 6:plotUw21singplotUw21sing,nu=c..3,color=yellow:plotsdisplayplotUrho
3,plotUw21sing,3.877093669,1.270337153,
3,plotUw21sing,3.877093669,1.270337153,3.270337153,1.877093669,3.877093669We have three possible values
for nu:nu1132 3:evalf%;nu2129 13610 3.5
for nu:nu1132 3:evalf%;nu2129 13610 3.5980762123.2703371533.877093669First value is when one of the root
s of P1 is 0, which is not singular for
s of P1 is 0, which is not singular for Uevalfsolvesimplifysubsnu=nu1 105640239942016 w32140229112368301
00657738663418262152583755330558830.0016
00657738663418262152583755330558830.001674427933,0.001242162516,0.001242162516We could try to check the s
ingular behavior at -0.0012... with puis
ingular behavior at -0.0012... with puiseux or algeqtoseries but Maple does not handle it well ... Instea
d we can see that our branch of U is not
d we can see that our branch of U is not singular directly:implicitplotfactorsubsnu=puiseuxsubswithgfun
:algeqtoseries,0.11548793340.1270358602
:algeqtoseries,0.11548793340.1270358602 xOx,0.48812853480.008985460244 I0.00074145229;131072 w3We have
to write an algebraic equation for (w-w1
to write an algebraic equation for (w-w1i) and (U-U(w1i))First, an equation for U(w1i)eqUw1i factorresult
antalgU,P1 U7 2127176 U6 313548they do
antalgU,P1 U7 2127176 U6 313548they do meet at nu_cfactorsubs;47775744 7 214 6 22 72 13 37 1131,1,1,3,
3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,12
3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,12 2,12 2,12 2,12 2,177,1772048 U9 510240 U9 45376 U8 520480 U9 33
4560 U8 45472 U7 520480 U9 284480 U8 348
4560 U8 45472 U7 520480 U9 284480 U8 348480 U7 42972 U6 510240 U9 99840 U8 2142656 U7 335332 U6 41428 U5
52048 U957600 U8 191808 U7 2127176 U6 31
52048 U957600 U8 191808 U7 2127176 U6 313548 U U5 211610 U4 31076 U3 448 U2 531236 U672084 U5 28470 U4 2
3760; fsolve%; od;9191102976 13573 4542
3760; fsolve%; od;9191102976 13573 454292 369811 231038 67482 12 22 72 7 214 62 39 1101.828427125,1.8284
27125,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,0.62
27125,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,0.6220355270,0.6220355270,1.,1.,1.377964473,1.377964473,3. 102548413
070 93210739098 81764577920 710717931194
070 93210739098 81764577920 710717931194 63195829884 525842319124 435230532064 323728532391 219564752366
14587990002 12 17 15 277982901 1233357
14587990002 12 17 15 277982901 123335794812 1116122301322 1038910536780 936257967195 856689258248 72205
15526388 6263055018888 5116228389445 453
15526388 6263055018888 5116228389445 453162531700 381758241270 212201729732 179748005013 12 12 52192248
08 4105745968 32509110 2415307412 116574
08 4105745968 32509110 2415307412 116574633 12 361.643450035,1.,0.2823650610 28478 127710 12 117 U7 21
27176 U6 313548 U U5 211610 U4 31076 U3
27176 U6 313548 U U5 211610 U4 31076 U3 448 U2 531236 U672084 U5 28470 U4 23760 3 3 U2 3 U23 U 3 U 4 U3
28 U3 3 U2 24 U312 U2 9 369811 231038
28 U3 3 U2 24 U312 U2 9 369811 2310385 8 U1Starting from eqUw1i2, we write an equation for w and UU wi
th U=Uw1i-UU (Maple does not factorize i
th U=Uw1i-UU (Maple does not factorize it)eqUUw1i2 resultanteqUw1i2,subsU:indets%;UU,,wThen an equation
for UU and WW with w=w1i - WWeqWW1iUU2re
for UU and WW with w=w1i - WWeqWW1iUU2resultantP1,subsw=wWW,eqUUw1i2,w5There are two factorsop11827eqWW
1iUU21collectop4,eqWW1iUU2,UU,WW643210
1iUU21collectop4,eqWW1iUU2,UU,WW64321013759414272816511297126461981355655168423776267862016 3 U2 3 U23
U 3 U2eqUrho;128 12 U 3 3 U2 3 U23 172
U 3 U2eqUrho;128 12 U 3 3 U2 3 U23 1728 16 36 � � (1.4.3.1)(1.4.3.1)� � (1
.4.5.6)(1.4.5.6)(1.5.1.14)(1.5.1.14)�
.4.5.6)(1.4.5.6)(1.5.1.14)(1.5.1.14)� � UU,,wThen an equation for UU and WW with w=w2i - W
WeqWW2iUU2resultantP2,subsw=wWW,eqUUw2i2
WeqWW2iUU2resultantP2,subsw=wWW,eqUUw2i2,w5There are two factorsop1(1.4.3.1)(1.4.3.1)(1.4.5.6)(1.4.5.6)
(1.5.1.14)(1.5.1.14)� � &#
(1.5.1.14)(1.5.1.14)� � � � (1.2.2.8)(1.2.2.8)(1.4.1.1)(1.4.1.1)(1.5.1.3)(1.5
.1.3)(1.4.5.11)(1.4.5.11)(1.4.7.23)(1.4.
.1.3)(1.4.5.11)(1.4.5.11)(1.4.7.23)(1.4.7.23)(1.4.7.16)(1.4.7.16)(1.5.2.3)(1.5.2.3)� � �
000; � � � (1.5.2.1
000; � � � (1.5.2.12)(1.5.2.12)(1.4.7.6)(1.4.7.6)� � (1.5.2.13)(1.5.2.
13)(1.5.2.10)(1.5.2.10)(1.4.5.7)(1.4.5.7
13)(1.5.2.10)(1.5.2.10)(1.4.5.7)(1.4.5.7)(1.4.4.1)(1.4.4.1)� � (1.4.5.9)(1.4.5.9)(1.3.2)(1.
3.2)� � � �
3.2)� � � � (1.4.7.13)(1.4.7.13)(1.4.5.1)(1.4.5.1)(1.4.7.10)(1.4.7.10)(1.2.1.
3)(1.2.1.3)(1.4.7.2)(1.4.7.2)� &#
3)(1.2.1.3)(1.4.7.2)(1.4.7.2)� � (1.5.1.10)(1.5.1.10)13 350The first factor is the good one
013 35215552 3 Again a generic square
013 35215552 3 Again a generic square root singularity except maybe at 1-sqrt(7)/7 and nu_ceqWW2iUU21ba
dcollectfactorsubs,U,0;130.2615831876,
dcollectfactorsubs,U,0;130.2615831876,0.9394189531,0.09121213316 1025.652231 w13.22875656130.2615831876