C(i)
P1
Ψ
∆I1 . . . ∆Ir OoA11 . . . OoAss Ck Cn HPnr
HPnr
Gk [Gk ]
Gk
CH(Ck )
Gk
A(Ck )
i) ⇒ ii)
(r − 2) gdr
r
C(d−2r+1) C(i) ˜ d, g) C(r,
2r − 2 gdr
C(i+1) C(r, d, g) P1 Ig Ig
P1 gd2
Ig
g ≤ 10 P3
gd3
g ≤ 10
Ig
Q
k∗
k∗
k : x $→ kx k∗
k∗
Q k∗
k : x $→ kx k∗
k∗
k∗
C
g n
Cn
JC
un : Cn −→ JC k
Cn JC
k
Z
k : JC −→ JC x $−→ kx A(JC)
CH(JC) Q
A(JC) p A (JC) = p
p !
Ap(i) (JC)
i=p−g
α Ap(i) (JC) k∗ α = k 2g−2p+i α
k
C Ag−1 (JC)
k ∗ α = k 2p−i α
JC
C C = C(0) + . . . + C(g−1)
C(i) ∈ Ag−1 (i) (JC) A(JC)
R Ai (X) i
CH (X)
C k∗
CHi (X)
k∗
R Ai (X)
C(i)
R R=
Q[C(0) , . . . , C(g−1) ] R
R
C(i) Q[C(0) , . . . , C(g−1) ]
C(i) C(1)
d 0≤i≤d−3
i
C(2)
C(i)
Ig g
P1
C i≥d−1
d
C(i)
G ⊂ Cd 1≤n≤d
n Gn = {D ∈ Cn | ∃ E ∈ Cd−n , D + E ∈ G} Gn
C # $ n " (−1)i−1 d un ∗ [Gn ] = i∗ C i n−i i=1
Gd ud
r
ud ∗ [Gd ]
An−r (Cn )
gdr "
1≤i1 ≤...≤ir
C
gdr
d
C [Gn ] =
G
Gd
#
n−
d %r
u=1 iu
$& ' r
v=1
" (−1)iv −1 ( [δi1 ,...,ir + (n − iu )o] iv r
u=1
o
C
CP iu
δi1 ,...,ir
δi1 ,...,ir = {i1 x1 + . . . + ir xr | xi ∈ C}
[Gd ] R
gdr
C
Ag−r (s) (JC) "
0≤a1 ,...,ar a1 +...+ar =s
β(d, a1 + 1, . . . , ar + 1) C(a1 ) ∗ . . . ∗ C(ar ) = 0
β(d, a1 , . . . , ar ) =
d "
...
i1 =1
[HPnr ]
Φ×n
s
d "
(−1)i1 +...+ir
ir =1
#
$ d i1 a1 . . . ir ar i1 + . . . + ir [Gn ]
HPnr (Pr )n gdr
Φ : C −→ Pr [Gn ] [Gk ] k0
R ∩ A(i) =
g−1 !
d=i+1
d R(i)
⊂
d+e R(i+j)
pi i pa1 . . . pat d R(i)
R d R(i)
-g−1
i
*
d=i+1
i+1 R(i)
)) )) )) )) ' )) ×p2 i+2 i+2 )) R(i) )) R(i+1) * )) )) ) )) ×p1 ))) )) ×p1 )) ' ( ' ) ))×p2 i+3 i+3 R(i) R )) (i+1) )) * )) )) ×p1 ×p1 )( ' '
d R(i)
×p1
i+4 R(i)
F
F
$
×p1
i+4 R(i+2)
×p1
i+4 R(i+3)
×p1
×p1
. .' .
. .' .
. .' .
. .' .
...
...
...
...
.) . .
...
...
...
'
×p1
'
×p1
'
g−2 R(i)
$
'
i+4 R(i+1)
×p1
F
i+3 R(i+2)
×p1
'
g−1 R(i)
×p1
'
g−2 R(i+1) ×p1
'
g−1 R(i+1)
×p1
'
g−2 R(i+2)
g−2 R(i+3)
×p1
'
g−1 R(i+2)
×p1
g−1 R(i+3)
pi d R(i) i+1 R(i)
i > 0
⊂
d+1 R(i)
d=i+1
1 d R(i)
d R(i) d R(i)
pa1
j≥i j
d=i+1
d ⊂ Rd+k pk R(i) (i+k−1)
p1
p1
e
-g−1
p1
k>1 -g−1
g−1 R(i)
g−1 pi R(i)
i+1 R(i) d p1 R(i)
F
g−d+i R(i)
-g−1
d R(i)
d=i+k+1
a1 + . . . + as − s = k
pa s
e ≥ j +1 d − 2r + 2 ≤ j ≤ e − 1
d R(i+k)
d R(i) gdr
e R(j) e R(j)
r (r − 2)
2r − 2 gdr
C d−r+1 R(d−2r+1)
a R(i)
Ag−r (Cd ) a
d − 2r + 2 ≤ i ≤ a − 1
i
a ≥ i+1 C(d−2r+1)
p1 C(r, d, g) C(d−2r+1) = 0 C(r, d, g)
Q[d, g]
r
C(r, d, g) =
# $# $# $ r−1 " (−1)i d − r − i + 1 d − r − i g i=0
r−1−i
r−i
Ig
i
C(i)
C
C(i) = 0
C(d−2r+1)
Ag−1 (i)
i ≥ d − 2r + 1. C(r, d, g)
Φ(C)
2r − 2
r−2
Pr
C(d−2r+1)
C(d−2r+1)
2r − 2
1 gd−2r+2
C(i)
C(i+1)
gdr
g
C(r, d, g) /= 0
1 gd−2r+2
r−1−i
r−2 C(d−2r+1)
D G(−D)
(r − 2)
r
2r − 2
r−2
C(r, d, g) 2r − 2 A(Cd )
gdr
gdr G p1 , . . . , pl
gdr
V p
G(−p1 . . . − pl ) r−2
r−2
Pr
r−1 C G(−p) gd−1 r−l gd−l k = r−1 P1
G(r − 1, r + 1)
Pr G(−p1 . . . − pl )
k ≤ r−1 C r−k−1 gd−l
l
l
1 gd−r+1
2r − 2 1
2r−2 # $# $# $ r−1 " (−1)i d − r − i + 1 d − r − i g C(r, d, g) = r−i r−1−i r−1−i i i=0
Cd gdr
1 gd−2r+2
gdr
gdr C(d−2r+1)
r−2 C(j)
C(d−2r+1) Ad−r+1 (d−2r+1) (JC)
gdr "
1≤a1 ≤...≤ar a1 +...ar =d−r+1
a1 ! . . . ar ! pa . . . par γ(a1 , . . . , ar ) 1
2r − 2 j≥i
C(i)
gdr
pg+r−d−2 1
Ag−1 d−2r+1 (JC)
"
1≤a1 ≤...≤ar a1 +...ar =d−r+1
t=0
p1 a1 = . . . = at = 1
1
ai γ(a1 , . . . , at ) = t! r−1 "
a1 ! . . . ar ! pg+r−d−2 pa1 . . . par γ(a1 , . . . , ar ) 1
"
1 1
(−1)r+t
10 g≥0 6 d g √ 24g − 15 + 3 ≤d 2 d
g
gd3
Ig
g d
g F
p d!
d R!I
I
Q[p1 , . . . , pg ]
I
d−4
g √
C
P d−3
Ig
d−2 R(d−5)
d%
d
24g − 15 + 3 g ≤d≤ +3 2 2
g ≤ 10
gd3
g 3 g+3≤d 4 P3 3
0
6
gd3 g=6 g=7 g=8 g=9
g = 10
gd1!
g ≤ 10
gd3
gd3
g73 g83 g83
g31 g41 g41
p1 p22 = 0 9p1 p2 p3 + p32 = 0 9p1 p2 p3 + p32 = 0
g93
g51
8p1 p2 p4 + 3p1 p23 +2p22 p3 = 0
g83
g41
9p1 p2 p3 + 2p32 = 0
3 g10
g61
g93
g61
20p1 p2 p5 + 12p1 p3 p4 +4p22 p4 + 3p2 p23 = 0 10p21 p5 + 8p1 p2 p4 +3p1 p23 + 2p22 p3 = 0
g
p2 p3 = 0 = p1 p2 p3 = 0 p1 p2 p4 = p1 p23 = p22 p3 = 0 p2 p4 = p23 = 0 p2 p3 = 0 pm p p = 0 m ∈ {0 . . . 2} 1 2 3 n 3 p1 p2 = 0 n ∈ {0, 1 p32
m 2 20pm+1 p5 + 4pm 1 1 p2 p4 + p1 p3 = 0 m ∈ {0 . . . 3} n+1 n+1 2 10pn+2 p + 8p p p p3 5 2 4 + 3p1 1 1 2 n +2p1 p2 p3 = 0 n ∈ {0 . . . 2}
10 gd3
Ig
gd3
gd3
Ig
gd1!
1
theorem1.nb
Computer Theorem Theorem Let
# a " i # 1 &( #% "a # g # i " 2 &( #% a " n " 1 &( (% (% ( !"1"i %% i $ "i # n # 1 ' $ '$ n"i ' F!i, n" ! $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ a"n#1
and SUM!n" ! ) F!i, n". n
i%0
If ‘n’ is a natural number, then:
!a " 3 n " 6" !2 a " g " 2 n " 1" !2 a " g " 2 n" !a " n # 1" SUM!n" # !"a4 # 6 n a3 # 6 a3 " 16 n2 a2 # 4 g a2 # 2 g n a2 " 36 n a2 " 19 a2 # 26 n3 a " 8 g n2 a # 104 n2 a " 20 g a " 26 g n a # 136 n a # 58 a " 15 n4 # 6 g n3 " 78 n3 # 24 g n2 " 141 n2 # 12 g # 30 g n " 102 n " 24" SUM!n # 1" # !a " 3 n " 3" !a " n " 2" !n # 2" !n # 3" SUM!n # 2" ! 0.
Proof
Let &i !." denote the forward difference operator in i and define R!i, n" ! !i !"a # i " 2" !"a " i # 2 n # 1" !!a " n " 2" !a " n # 1" !3 a3 " g a2 " 18 n a2 " 27 a2 # 33 n2 a # 7 g a # 4 g n a # 89 n a # 52 a " 18 n3 " 3 g n2 " 66 n2 " 6 g " 9 g n " 72 n " 24" i2 # !a " n " 2" !a " n # 1" 4 !a " 16 n a3 " 29 a3 # 68 n2 a2 # 5 g a2 # 2 g n a2 # 222 n a2 # 174 a2 " 104 n3 a " 8 g n2 a " 461 n2 a " 35 g a " 34 g n a " 645 n a " 278 a # 51 n4 # 6 g n3 # 276 n3 # 33 g n2 # 531 n2 # 30 g # 57 g n # 426 n # 120" i " !a " n " 2" !a " n # 1" !36 n5 " 81 a n4 # 3 g n4 # 261 n4 # 62 a2 n3 " 508 a n3 " 4 a g n3 # 24 g n3 # 726 n3 " 19 a3 n2 # 319 a2 n2 " 1143 a n2 # a2 g n2 " 27 a g n2 # 69 g n2 # 963 n2 # 2 a4 n " 73 a3 n # 521 a2 n " 1088 a n # 5 a2 g n " 59 a g n # 84 g n # 606 n # 5 a4 " 64 a3 # 273 a2 " 366 a # 6 a2 g " 42 a g # 36 g # 144""" * !!"a # n # 1" !"a # n # 2" !"i # n # 1" !"i # n # 2"2 !"i # n # 3"".
Then the Theorem follows from summing the equation
!a " 3 n " 6" !2 a " g " 2 n " 1" !2 a " g " 2 n" !a " n # 1" F!i, n" # !"a4 # 6 n a3 # 6 a3 " 16 n2 a2 # 4 g a2 # 2 g n a2 " 36 n a2 " 19 a2 # 26 n3 a " 8 g n2 a # 104 n2 a " 20 g a " 26 g n a # 136 n a # 58 a " 15 n4 # 6 g n3 " 78 n3 # 24 g n2 " 141 n2 # 12 g # 30 g n " 102 n " 24" F!i, n # 1" # !a " 3 n " 3" !a " n " 2" !n # 2" !n # 3" F!i, n # 2" ! &i !F!i, n" R!i, n""
over i from 0 to n # 2.
theorem1.nb
This equation is routinely verifyable by dividing both sides by F and checking the resulting rational equation.
2
1
'#("()*+!,
Computer Theorem Theorem Let
# $ & # % " " &( #% % " " # 1 &( (% ( !"1"" %% (( %% $ " '$#""'$ #"" ' !!", #" ! " $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ""#"1
and SUM!#" ! ) !!", #". #
"%0
If ‘n’ is a natural number, then:
!% " 3 # " 6" !2 % " $ " 2 # " 1" !2 % " $ " 2 #" !% " # # 1" SUM!#" # !"%4 # 6 # %3 # 6 %3 " 16 #2 %2 # 4 $ %2 # 2 $ # %2 " 36 # %2 " 19 %2 # 26 #3 % " 8 $ #2 % # 104 #2 % " 20 $ % " 26 $ # % # 136 # % # 58 % " 15 #4 # 6 $ #3 " 78 #3 # 24 $ #2 " 141 #2 # 12 $ # 30 $ # " 102 # " 24" SUM!# # 1" # !% " 3 # " 3" !% " # " 2" !# # 2" !# # 3" SUM!# # 2" ! 0.
Proof
Let &" !." denote the forward difference operator in " and define 1 &!", #" ! $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ !" !"% # " " 2" !"% # " " 1" !"" # # # 1" !"" # # # 2"2 !"" # # # 3" !!% " 3 # " 6" !2 % " $ " 2 # " 1" !% " # # 1" "2 # !% " 3 # " 6" !% " # # 1" !%2 " 8 # % " 13 % # 7 #2 # 5 $ # 2 $ # # 15 # # 5" " " !% " # # 1" !"18 #4 # 30 % #3 " 3 $ #3 " 108 #3 " 14 %2 #2 # 154 % #2 # % $ #2 " 21 $ #2 " 222 #2 # 2 %3 # " 53 %2 # # 251 % # # 5 % $ # " 48 $ # " 174 # # 5 %3 " 45 %2 # 132 % # 6 % $ " 36 $ " 36""".
Then the Theorem follows from summing the equation
!% " 3 # " 6" !2 % " $ " 2 # " 1" !2 % " $ " 2 #" !% " # # 1" !!", #" # !"%4 # 6 # %3 # 6 %3 " 16 #2 %2 # 4 $ %2 # 2 $ # %2 " 36 # %2 " 19 %2 # 26 #3 % " 8 $ #2 % # 104 #2 % " 20 $ % " 26 $ # % # 136 # % # 58 % " 15 #4 # 6 $ #3 " 78 #3 # 24 $ #2 " 141 #2 # 12 $ # 30 $ # " 102 # " 24" !!", # # 1" # !% " 3 # " 3" !% " # " 2" !# # 2" !# # 3" !!", # # 2" ! &" !!!", #" &!", #""
over " from 0 to # # 2. This equation is routinely verifyable by dividing both sides by ! and checking the resulting rational equation.
C
g
g
C G
r G
d Cn n σn
σn : C n → Cn
un
D
un : Cn → JC
Cn
gdr G Gn Φ
d
Φ : C → Pr
gdr
G
Gn = {D ∈ Cn | ∃ E ∈ Cd−n , D + E ∈ G}
k
r≤n≤d
Z
x %→ kx
k
JC A1 (JC)
θ D ( = CHi (X) )
0
CH 1 (JC) D
X
JC g
i CH(X)
gdr
G
Θ
CH g−i (X)
G
Φ : C → Pr
k
D
r
X
Q
i (X) CH(s)
CH i (X) k ∗ α = k 2p−i α
α
Ag−i (X)
i (X) CH(s)
( = Ai (X) )
k k∗ α =
k 2g−2p+i α
X
Q
g i
A(X)
X Q
Ai(s) (X) α Ai (X) k ∗ α = k 2p−i α
Ai(s) (X)
k k∗ α =
[V ]
k 2g−2p+i α
CH(X) V
A(X)
X A = I1 & . . . & In
&
A
R
A(JC) CH(JC) k∗
k∗ p R(i)
R
Ap(i)
Ig
R
∆I1 . . . ∆Ir . . . OoAss
Cn j
δi1 ,...,ir
x
It xi = xj n = i1 + . . . ir
t A(Cn )
t {1, . . . , s} a
CH(Cn )
ΨP
At
{1, . . . , r} i xt = ot
{i1 x1 +. . .+ir xr | xi ∈ C}
ΨP : C k → C n (Pn )r
HPnr JC
g
Pr C
∗ F
CH(X) S
A(X
g−1 CH(s) (J)
C(i) C
k∗ C(s) = k 2g−2−s C(s) pi
[a]
pi
Ni
i≥0
Ag−1 (s) (J)
k ∗ C(s) = k 2+s C(s)
pi+1
[a]
pi
pi a!
C(i) [a]
a ≥ 0 pi
Ni = −F(C(i−1) ) N i = −pi
i≥1
0
a