global b local b (D[s] D alg [s] ) local b f f local b s + 1 ( ) b ( [19]) b global b ( [2], [11], [12]) Kan/sm1 ([21]) bfunction.sm1 global b ( [10])

Transcription

1 Mora local b Risa/Asir Introduction x = (x 1,, x n ), = ( 1,, n ), s = (s) D[s] = a k,β (x)s k β a k,β (x) C[x] k N,β N { n } f(x) C[x] x = f(x), g(x) C[x], g(0) 0 g(x) D alg [s] = a k,β (x)s k β a k,β (x) C[x] x k N,β N n D[s] = a k,β (x)s k β a k,β (x) C[[x]] k N,β N n f C[x] f global b f local b f global b P f s+1 = b(s)f s P D[s] monic s b(s) C[s] f local b P f s+1 = b(s)f s P D alg [s] monic s b(s) C[s] Example 1.1. (f = x 1 (x 1 + x 2 + 1) b ) global b b(s) = (s + 1) 2 ( )f s+1 = (s + 1) 2 f s local b b(s) = s x 1 + x 2 1 f s+1 = (s + 1)f s b Bernstein global b ([17], [18]) 1

2 global b local b (D[s] D alg [s] ) local b f f local b s + 1 ( ) b ( [19]) b global b ( [2], [11], [12]) Kan/sm1 ([21]) bfunction.sm1 global b ( [10]) Risa/Asir ([22]) bfct local b ([2], [11], [12], [13]) D Mora ([5], [6]) D alg local b Risa/Asir [9] 2 b 2.1 Global b global b ([2], [11], [12]) Algorithm 2.1. ( f global b ) 1. f s D[s] Ann D[s] f s G ( P f s = 0 P D[s] ) 2. J G {f} D[s] J C[s] f global b (1.) Ann D[s] f s D n+1 ([2], [11], [12]) (2.) J C[s] 2a. J s < ( x i, i > s ) 2b. J < G NF(s i, G, <) (s i G ) a l NF(s l, G, <) + + a 0 NF(1, G, <) = 0 l a l,, a 0 C a l s l + + a 0 J C[s] ([10]) D[s] global b Example 2.2. (global b ) f = x 2 + y 2 global b Ann D[s] f s {x x + y y 2s, y x x y } 2

3 f D[s] I I C[s] ( (2a.)) x, y, x, y I G { s 2 2s 1, ( s 1)y, ( s 1)x, y x x y, x x + y y 2s, x 2 + y 2, y x 2 y y 2 } + 2s y I C[s] G C[s] = { s 2 2s 1 } global b (s + 1) 2 f = n i=1 x2 i global b (s + 1)(s + n 2 ) P = n i=1 2 i 2.2 Local b local b ([11], [12]) K[x, s] D[s], D n+1 [y] Algorithm 2.3. (local b 1 [11], [12]) y { } 1. I = D n+1 [y] t yf, i + y f x i t (i = 1,, n) I y w G w x 1 x n t y ξ 1 ξ n t ψ(g) = {ψ(p (1)) P (y) G} P D n+1, ord w (P ) = m (ord w (P ) P w P w ) ψ(p ) in w (t m P ) (m 0) ψ(p )(s) = ψ(p )(t t ) = in w ( t m P ) (m < 0) ( in w (P ) P w P w ) 3. D[s] ψ(g) i G 1 J = K[x, s] (G 1 K[x, s]) 4. local b b( s 1) K[[x]][s]J K[s] monic global b b( s 1) J K[s] monic K[[x]][s]J K[s] 5. K[[x]][s]J K[s] J {f 1 (x, s),, f k (x, s)} (a) J(0) = K[s] {f 1 (0, s),, f k (0, s)} f 0 (s) (b) f 0 (s) Q f 0 (s) = g 1 (s) l1 g d (s) l d (c) i = 1,, d J : g i (s) l (l = l i, l i + 1, ) g i (s) a i (x, s) J : g i (s) l l l i (d) b(s) = g 1 (s) l 1 gd (s) l d 3

4 Ann D[s] f s Algorithm 2.4. (local b 2 [12]) 1. Ann D[s] f s D[s] f D[s] J 2. J i G J = K[x, s] (G K[x, s]) local b b(s) K[[x]][s]J K[s] monic 2.3 Risa/Asir 3 Local b D[y] Mora (Algorithm 3.5) local b (Algorithm 3.13) 3.1 D[y] Mora D[s] s y D[y] s [5],[6] Mora ( Mora Mora y ) D[y] Mora D alg [y] D[y] D alg [y] D[y] < 1 x 1 x n y ξ 1 ξ n ( < ) < 1 x 1,, x n Mora s ( 1, 1)- < 1 D[y][s] < s 1 s 1 < 1 ( 1, 1)- x i 1 y ξ i 1 1 s Definition 3.1. (( 1, 1)- ) P D[y] P = a αβγ x α β y γ (α (Z 0 ) n, β (Z 0 ) n, γ Z 0 ) m = min { β α + γ a αβγ 0} ( v v ) 4

5 P ( 1, 1) - P (s) P (s) = a αβγ x α β y γ s α + β + γ m P D[y][s] P = a αβγν x α β y γ s ν a αβγν 0 (α, β, γ, ν) d = α + β + γ ν P d ( 1, 1)- < 1 D[y][s] < s 1 Definition 3.2. (< 1 ) D[y][s] < s 1 x 1 x n y s ξ 1 ξ n ( < ) x s < 1 x s ( 1, 1)- < 1 leading monomial < s 1 leading monomial Lemma 3.3. (< 1 < s 1 LM) P D[y][s] P ( 1, 1)- LM <1 (P s=1 ) = (LM < s 1 (P )) s=1 P, Q D[y][s] P, Q ( 1, 1)- LM < s 1 (P ) < s 1 LM < s 1 (Q) LM <1 (P s=1 ) < 1 LM <1 (Q s=1 ) P, Q D[y][s] ( 1, 1)- P Q LM < s 1 (P ) LM < s 1 (Q) R P ( 1, 1)- LM < s 1 (R) < s 1 LM < s 1 (P ) LM <1 (R s=1 ) < 1 LM <1 (P s=1 ) D[y] Mora Theorem 3.4. (D[y] Mora, [5] [6]) P, P 1,, P m D[y] a(x) C[x], Q 1,, Q m D[y], R D[y] a(x)p = Q 1 P Q m P m + R a(0) 0 ( a(x) ) Q i 0 LM <1 (Q i P i ) 1 LM <1 (P ) R 0 LM <1 (R) LM <1 (P i ) a(x), Q 1,, Q m, R 5

6 Algorithm 3.5. (Mora, [5],[6]) MoraDivision(P, {P 1,, P m }, < 1 ) input P, P 1,, P m D[y] output a(x) C[x], Q 1,, Q m, R D[y] 3.4 G [P (s) 1,, P (s) m ] R P (s) A 1 Q [0,, 0] (Q m ) if (R 0) F { P G l Z 0 s.t. LM < s 1 (P ) LM < s 1 (s l R) } else H [] F φ while (F = φ){ } P (F l (G i )) if (l > 0){ } G append (G, [R]) H append (H, [[A, Q]]) R (s l R P ) U ( P ) if (i m) { Q[i] Q[i] + U } else if (i > m) { } [A, Q ] H[i m] A A UA for (j 1; j m; j j + 1) if (s R) { } Q[j] Q[j] UQ [j] ν (s k R k) R R/s ν F { P G l Z 0 s.t. LM < s 1 (P ) LM < s 1 (s l R) } for (j 1; j m; j j + 1) Q[j] Q[j] s=1 R R s=1 a A s=1 return [a, Q, R] { } G reducer set G = P (s) 1,, P m (s) s l reduce G 6

7 Example 3.6. (Mora ) x 1 + x Mora 1 + x D alg [y] x = 1 (1 + x) + 0 ( (1 + x) x = (1 + x) + 0) 1 + x Mora 1 + x ( 1, 1) s + x Mora x s,x x 2 x 0 s+x x reducer s reducer reducer set G s x = x (s + x) x 2 x 2 = x x + 0 (s + x) x = x (s + x) + 0 (1 + x) x = x (1 + x) + 0 D[y] Mora C[x] Mora D[y] Mora C[x] Mora Example 3.7. (Mora ) 1 + x 1 + x D alg [y] 0 Mora 1 + x ( 1, 1) s + x s, x 1 x s+x 1 s, 1 s+x x x 1 0 s = (s + x) x 1 x 1 = x 1 s ( 1) = ( 1) (s + x) + x x = ( x) ( 1) + 0 (s + x) 2 = (s + x 1) (s + x) + 0 7

8 (1 + x) 2 = ( + x 1) (1 + x) + 0 Mora Example 3.8. (Mora ) P = x 1 x P 1 = x x 1 x 2 2, P 2 = x x 1 x 2 1 ([5], [6] ) Mora P (s) = x 1 x 2 1 2, P (s) 1 = sx x 1 x 2 2, P (s) 2 = sx x 1 x 2 1 s,x 2 2 P 3 = x 1 x P (s) 1 P 5 = x 2 s, x 1 x 2 1x 2 1 P (s) 1 P 4 = x 1 x s, x 1 x 2 2 x 1 x 2 2 P (s) 2 P 6 = x 2 1x 2 s,x x 2 2 P (s) 2 x 3 1x 2 x 1 x P5 x 2 1x x 2 x 1x 2 1 x 2 1 P (s) (1 x 1 x 2 ) 2 P = (x 2 2 x 1 x x 1 x 2 )P 1 + ( x 1 x x 2 1x x 2 1x 2 )P P P 1, P 2 (1 + x 1 ) n P P 1, P 2 n n ( ) D[y] Mora 2( ) section < 1 < 1 Mora Mora < 2 x 1 x n y ξ 1 ξ n ( < ) 8

9 < 1 y < 2 ξ 1, ξ n > y ξ 1,, ξ n ( 1, 1)- < 1 < 1 v 1 s x 1 x n y s ξ 1 ξ n < 2 v 2 s x 1 x n y s ξ 1 ξ n < 2 D[y][s] < s 2 x 1 x n y s ξ 1 ξ n ( < ) < 2 < < 3 x 1 x n y ξ 1 ξ n ( < ) ( 1, 1)- v 3 < 3 < s 3 x 1 x n y s ξ 1 ξ n x 1 x n y s ξ 1 ξ n ( < ) < 3 Mora 3.3 D[y] Mora 9

10 reducer 3.5 G reducer P Example 3.9. reducer ) reducer sugar degree (min sugar) (max sugar) sugar degree Buchberger algorithm sugar S ([20]) P = (1 + x 1 ) 50 x 1 x P 1 = x x 1 x 2 2, P 2 = x x 1 x 2 1 < 1 Mora reducer ( ) G min sugar max sugar min sugar (P A Q) max sugar modular < 1 < 1 < 2, < D alg [y] D[y] D alg [y], D[y] I 1. D[y] Mora Buchberger 2. ( 1, 1) < s 1 Example (Mora ) I I = { P 1 =, P 2 = 1 + x 3} P 1 = 1 + x 3 D alg [y] I = D alg [y] Mora sp <1 (P 1, P 2 ) {P 1, P 2 } < 1 0 {P 1, P 2 } 10

11 { } I (s) = P (s) 1 =, P (s) 2 = s 3 + x 3 < s 1 { } P (s) 1 =, P (s) 2 = s 3 + x 3, 3x 2, 6x, 6 {1} local b Section Local b I Ann D[s] f s f D alg [s] I C[s] local b D alg [s] local x 1,, x n (< 1) global s(> 1) x 1,, x n < 1 < s section I D alg [s] G G D alg [s] 0 I local b b(s) global b b(s) b(s) I local b b(s) Algorithm (local b )) Input : f C[x] Output : f local b H Ann Dn[s]f s I (H {f} D alg [s] ) G I BF f global b L BF LF L G Mora 0 return LF Example ( f = x 2 1(x 1 + 1) 3 local b ) f = x 2 1(x 1 + 1) 3 global b b(s) = 1 18 (s + 1)(2s + 1)(3s + 1)(3s + 2) Ann D[s]f s { g 1 = 2s + 5sx 1 x 1 1 x } J Dalg [s] {f, g 1 } J G {f, g 1 } b(s) G Mora (s + 1)(2s + 1)(3s + 1)(3s + 2) 1 6 (s + 1)(2s + 1)(3s + 1) 1 6 (s + 1)(2s + 1)(3s + 2) 1 2 (s + 1)(2s + 1) 11

12 1 2 (s+1)(2s+1) f local b 1 2 (s+1)(2s+1) G 1 2 (s + 1)(2s + 1)f s = (x 1 + 1) 2 (5x 1 + 2) 3 ( x x x 1 ) 1 f s+1 global b g(s) I C[s] (g(s) ) I C[s] g(s) / I g(s) global b b(s) = b1 (s) e 1 b 2 (s) e2 b l (s) e l (b i (s) = s + a i a i Q >0 ) f i = b(s)/b i (s) J G Mora f i1 (s),, f im (s) I f j1,, f jl m / I f i I I C[s] b(s) m > 0 g(s) = gcd(f i1 (s),, f im (s)) g(s) I b(s) g(s) Algorithm (local-b ( )) localb-rr(f) input f C[x] output f local b H Ann Dn [s]f s I (H {f} D alg [s] ) G I BF f global b LF BF while (true) { LF = b 1 (s) e1 b l (s) e l (b i (s) = s + a i, a i Q >0 ) (LF ) f i LF/b i (s) (1 i l, b i (s) LF i ) f i I (f i G Mora 0 ) if (f i / I(1 i l)) return LF f i1,, f im I LF gcd(f i1,, f im ) 12

13 } Example (f = (x 1) 3 + (y + 1) 2 local b ) f = (x 1) 3 + (y + 1) 2 global b b(s) = (s + 1)(s )(s ) f local b s + 1 Ann D[s] f s { g 1 = 6s 2 x + 3 y + 2x x + 3y y, g 2 = 2 x 3 y + 2y x + 6x y 3x 2 y } I D alg [s] {f, g 1, g 2 } I G {f, g 1, g 2 } f G g f G Mora g b(s) (s + 1) = gcd(f 1, f 2 ) b(s) = (s + 1)(s )(s ) G 0 f 1 = (s + 1)(s ) G 0 f 2 = (s + 1)(s ) G 0 f 3 = (s )(s ) G non-zero s + 1 G 0 local b b(s) = s Local b 2 D[s] (Theorem 4.5) (Algorithm 4.6) local b (Algorithm 4.10) [9] Section LE < (f) f Exps(f) f Mono(A) (A = {α 1,, α n α i (Z 0 ) n }) A Mono(A) = n i=1 (α i + (Z 0 ) n ) P e ord e (P ) P e P e in e (P ) P e P e 4.1 C[[x]] D C[[x]] Weierstrass- Hironaka (WH ) C[[x]] [15] ( M-reduction, M-Gröbner basis ) {x α } (α (Z 0 ) n ) < r 13

14 x 1 x n 1 1 ( < ) < r C[x] < r f C[[x]] < r Lemma 4.1. ( ) { x α(1),, x α(s)} (α(i) (Z 0 ) n ) f C[[x]] q 1,, q s, r C[[x]] f = q 1 x α(1) + + q s x α(s) + r q i 0 LM <r (q i x α(i) ) r LM <r (f) r 0 Exps(r) Mono({α(1),, α(s)}) = φ Theorem 4.2. (Weierstrass-Hironaka,[3]) f C[[x]], f 0 G = {g 1,, g s } C[[x]] q 1,, q s, r C[[x]] f = q 1 g q s g s + r q i 0 LM <r (q i g i ) r LM <r (f) Exps(r) Mono({LE <r (g 1 ),, LE <r (g s )}) = φ Mora r reducer Mora WH Algorithm 4.3. (WH ) Input f C[x], G = {g 1,, g s } C[x], N Z >0 Output Q 1,, Q s, R, F C[x] f = Q 1 g Q sg s + R + F Q i 0 LM < r (Q i g i) r LM <r (f) Exps(R ) Mono({LE <r (g 1 ),, LE <r (g s )}) = φ Q i N LE < r (g i ) 1 R N 1 WH-approximate-division(f, G) F f, Q i 0, R 0, β 0 while (F 0 and β < N) { [Q, R] mono-div(f, G) (Lemma 4.1 ) β max <r ({LE <r (Q 1 g 1 ),, LE <r (Q s g s ), LE <r (R)}) 14

15 } Q i Q i + Q i R R + R F s i=1 Q irest <r (g i ) return [Q, R, F ] D (Algorithm 4.6) Example 4.4. (WH ) f = x, G = 1 + x, N = 5 (WH-approximate-division(x, {1 + x}, 5)) mono-div Q R Q R F β 0 0 x 0 x = x x 0 x 0 x x LE(x) x 2 = x x 2 0 x x 2 0 ( x 2 ) x LE( x 2 ) x 3 = x x 3 0 x x 2 + x 3 0 x 3 x LE(x 3 ) x 4 = x x 4 0 x x 2 + x 3 x 4 0 ( x 4 ) x LE( x 4 ) x 5 = x x 5 0 x x 2 + x 3 x 4 + x 5 0 x 5 x LE(x 5 ) x = (x x 2 + x 3 x 4 + x 5 ) (1 + x) x 6 WH x = x (1 + x) x = (x x 2 + x 3 x 4 + x 5 ) (1 + x) D[y] Castro [4] D WH y = (y) < 1 section 3.1 { x α ξ β y γ} { x α ξ β y γ} < r < r e x 1 x n y ξ 1 ξ n ( < 1 < ) x 1 x n y ξ 1 ξ n

16 < r β + γ = β + γ ( e ), x α ξ β y γ < 1 x α ξ β y γ x α ξ β y γ < r x α ξ β y γ LM <1 (P ) = LM <r (in e (P )) D[y] D[y] D[y] D[y] < 1 Theorem 4.5. ( D[y],[4]) P, P 1,, P s D[y] Q 1,, Q s, R D[y] P = Q 1 P Q s P s + R (1) Exps(R) Mono({LE <1 (P 1 ),, LE <1 (P s )}) = φ (2) Q k 0 LM <1 (Q k P k ) LM <1 (P ) (3) D[y] WH (Algorithm 4.3) Algorithm 4.6. ( D[y], [9]) input P, P 1,, P s D[y], N Z >0 output Q 1,, Q s, R D[y] s.t. P = Q 1 P Q s P s + R Q i 0 LM <1 (Q i P i ) LM <1 (P ) { α α Exps(R), α < N } Mono({LE<1 (P 1 ),, LE <1 (P s )}) = φ Q i N LE <1 (P i ) R N D-approximate-division(P, {P 1,, P s }, N) Q 1 0,, Q s 0, R P m 0 ord e (R) for (k 0; k m 0 ; k k + 1) M k max({ LE <1 (P i ) + 2(max(k ord e (P i ), 0) 1 i s}) Bound N + m 0 i=0 M i for (k m 0 ; k 0; k k 1) { } r (R k Bound ) [q i, r ] WH-approximate-division(r, {in e (P 1 ),, in e (P s )}, < r, Bound) Q i (q i ξ ) R R Q i P i Q i Q i + Q i Bound Bound M k return [Q, R] Example 4.7. (D-approximate-division ) P = 2, P 1 = (1 + x) + x D-approximate-division(P, {P 1 }, 5) 16

17 2 1 = ( 1 + x 1 x 1 )((1 + x) + x) x 1 + x x x = (1 x + x2 x 3 + x 4 )( 1) x x = 1 + 2x 2x2 + 2x 3 2x 4 + 2x 5 D-approximate-division(P, {P 1 }, 5) R P, m 0 ord e (R) = 2 M 0 1, M 1 1, M 2 3 Bound 5 + ( ) = 10 r (R 2 Bound ) = ξ 2 [q 1, r ] WH-approximate-division( r, in e (P 1 ) = (1 + x)ξ, < r,bound) (ξ 2 = (1 x + x 2 + x 8 )ξ (1 + x)ξ + x 9 ξ 2 q 1 = (1 x + x2 + x 8 )ξ, r = x 9 ξ 2 ) Q 1 (1 x + x2 + x 8 ) R R Q 1 P 1 = x 9 2 x x x x 7 x 8 Bound 10 3 = 7 r (R 1 Bound ) = ξ [q 1, r ] WH-approximate-division( r, in e (P 1 ) = (1 + x)ξ, < r, Bound) ( ξ = (1 x+x 2 x 3 + +x 6 ) (1+x)ξ +x 7 ξ q 1 = (1 x+x2 x 3 + +x 6 ), r = x 7 ξ ) Q 1 (1 x + x2 x x 6 ) R R Q 1 P 1 = x 9 ξ + x 7 ξ x 9 ξ 1 + 2x 2x 2 + 2x 3 + 2x 7 x 8 Bound 7 1 = 6 r (R 0 Bound ) = 1 + 2x 2x 2 + 2x 3 2x 4 + 2x 5 [q 1, r ] WH-approximate-division( r, in e (P 1 ) = (1 + x)ξ, < r, Bound) (r = 0 (1 + x)ξ + r q 1 = 0, r = r ) Q 1 0 R R = x 9 2 x x x x 7 x 8 2 = { (1 x + x x 8 ) (1 x + x 2 + x 6 ) } ((1 + x) + x)+ x 9 2 x 9 + x x 2x 2 + 2x 3 2x x 7 x x 2x 2 + 2x 3 2x 4 17

18 4.3 I C[s] D[s] I I C[s] P G (Theorem 4.5) NF(P, G, < 1 ) g(s) = a l s l + a l 1 s l a 0 I G I global b (Algorithm 2.1(2a.)) g(s) I NF(g(s), G, < 1 ) = 0 a l NF(s l, G, < 1 ) + a l 1 NF(s l 1, G, < 1 ) + + a 0 NF(1, G, < 1 ) = 0 NF(s i, G, < 1 ) a l NF(s l, G, < 1 ) + a l 1 NF(s l 1, G, < 1 ) + + a 0 NF(1, G, < 1 ) = 0 l a l,, a 0 C I C[s] a l s l + + a 0 D[s] NF(s i, G, < 1 ) Definition 4.8. ( ) P D[s] D[s] I G 4.6 P G N 1 R P G < 1 N 1 NF(P, G, < 1, N) NF(P, G, < 1 ) NF(P, G, < 1, N) N 1 I C[s] a l NF(s l, G, < 1, N) + a l 1 NF(s l 1, G, < 1, N) + + a 0 NF(1, G, < 1, N) = 0 a l s l + a l 1 s l a 0 I D[s] P I Mora (3.4) 0 P I P / I I C[s] L d,n = { (a 0,, a d ) C d+1 a d NF(s d, G, < 1, N) + + a 0 NF(1, G, < 1, N) = 0 } L d = { (a 0,, a d ) C d+1 a d NF(s d, G, < 1 ) + + a 0 NF(1, G, < 1 ) = 0 } I < 1 G d (d I C[s] ) ) N L d,n L d,n+1 L d,n+2 L d L d 1,N L d 1,N+1 L d 1,N+2 L d

19 I C[s] L i {0}, L i 1 = {0} i L i (a 0,, a i ) a i s i + + a 0 Algorithm 4.9. (I C[s], [9]) 1. L d,n, L d 1,N, L i,n {0}, L i 1,N = {0} i ( L i 1 = L i 2 = = {0} ) 2. (a i,, a 0 ) L i,n g(s) = a i s i + + a 0 3. g(s) G < 1 Mora R R = 0 g(s) I (a 0,, a i ) L i g(s) I C[s] R 0 N 1 1. N N = d + 1 N 0 N, n N 0 s.t. L j,n = L j (j = 0,, d) ( N N 0 ) 4.4 local b Algorithm ( local b, [9]) f C[x] local b 1. Ann bd[s] f s H 2. H f D[s] I I C[s] local b b(s) 1. Ann bd[s] f s Ann D[s] f s local b global b global b I C[s] Example (f = x 2 (y + 1) 2 z 2 local b ) global b (s + 1) 3 (s + 1/2) 3 local b (s + 1) 2 (s + 1/2) 2 Ann bd[s] f s H {P 1 = 2s + z z, P 2 = x x z z, P 3 = y y y + z z } J = D[s] (H {f}) < 1 G {f, P 1, P 2, P 3, P 4 = z 4 z 2 2yz 4 z 2 y 2 z 4 z 2 4z 3 z 8yz 3 z 4y 2 z 3 z 2z 2 4yz 2 2y 2 z 2, P 5 = xz 3 z + 2xyz 3 z + xy 2 z 3 z + 2xz 2 + 4xyz 2 + 2xy 2 z 2 } 19

20 d = 6, N = 7 NF(s i, G, < 1, N) (i = 0,, d) NF(1, G, < 1, 7) = 1 NF(s, G, < 1, 7) = 1/2z z NF(s 2, G, < 1, 7) = 1/4z 2 z 2 + 1/4z z NF(s 3, G, < 1, 7) = 1/8z 3 z 3 + 3/8z 2 z 2 + 1/8z z NF(s 4, G, < 1, 7) = 3/8z 3 z 3 31/16z 2 z 2 31/16z z 1/4 NF(s 5, G, < 1, 7) = 23/32z 3 z /32z 2 z /32z z + 3/4 NF(s 6, G, < 1, 7) = 9/8z 3 z 3 447/64z 2 z 2 555/64z z 23/16 L d,n = { (a 0,, a d ) C d a d NF(s d, G, < 1, N) + + a 0 NF(1, G, < 1, N) = 0 } L d = { (a 0,, a d ) C d a d NF(s d, G, < 1 ) + + a 0 NF(1, G, < 1 ) = 0 } L 6,7, L 5,7, L 6,7 ={c 0 (1/4, 3/2, 13/4, 3, 1, 0, 0) + c 1 ( 3/4, 17/4, 33/4, 23/4, 0, 1, 0) + c 2 (23/16, 63/8, 231/16, 9, 0, 0, 1) c i C} L 5,7 = {c 0 (1/4, 3/2, 13/4, 3, 1, 0, 0) + c 1 ( 3/4, 17/4, 33/4, 23/4, 0, 1, 0) c i C} L 4,7 = {c 0 (1/4, 3/2, 13/4, 3, 1, 0, 0) c i C} L 3,7 = {0} L 3 = L 2 = = L 0 = {0} local b 3 (1/4, 3/2, 13/4, 3, 1, 0, 0) L 4,7 g(s) = s 4 + 3s /4s 2 + 3/2s + 1/4 local b g(s) G < 1 Mora 0 g(s) J g(s) J s g(s) local b 5 Timing data(local b ) f local b 1. Ann bd[s] f s Ann D[s] f s 2. I = Ann bd[s] f s + D[s]f 3. I C[s] 1. Ann D[s] f s ([12], [2] ) I 3. 3a. (Algorithm 3.13 ) 3b. (Algorithm 4.9) 20

21 (3a),(3b) localb-rr (round-robin), localb-nf (normal form) 2. (Section 3.4) 2a. D[s] Mora Buchberger 2b. GB (2a.) (Mora GB ) machine CPU : Athlon MP (1533MHz) 2, Memory : 3GB, OS : FreeBSD 4.8 f I GB GB localb-rr localb-nf deg locdeg x + y 2 + z x 2 + y 2 + z x 3 + y 2 + z x 4 + y 2 + z x 5 + y 2 + z x 6 + y 2 + z x 7 + y 2 + z x 3 + xy 2 + z x 4 + xy 2 + z x 5 + xy 2 + z x 3 + y 4 + z x 3 + xy 3 + z x 3 + y 5 + z GB (2b.) ( GB ) f I GB GB localb-rr localb-nf deg locdeg x + y 2 + z x 2 + y 2 + z x 3 + y 2 + z x 4 + y 2 + z x 5 + y 2 + z x 6 + y 2 + z x 7 + y 2 + z x 3 + xy 2 + z x 4 + xy 2 + z x 5 + xy 2 + z x 3 + y 4 + z x 3 + xy 3 + z x 3 + y 5 + z

22 Mora GB GB local b GB [12] (2a.)(Mora GB ) f I GB GB localb-rr localb-nf deg locdeg x 5 + x 3 y 3 + y x 4 + x 2 y 2 + y x 3 + x 2 y 2 + y x 3 + y 3 + z x 6 + y 4 + z x 3 + y 2 z (x 3 y 2 z 2 ) x 5 y 2 z x 5 y 2 z x 3 + y 3 3xyz x 3 + y 3 + z 3 3xyz y(x 5 y 2 z 2 ) y(x 3 y 2 z 2 ) y((y + 1)x 3 y 2 z 2 ) (2b.)( GB ) f I GB GB localb-rr localb-nf deg locdeg x 5 + x 3 y 3 + y x 4 + x 2 y 2 + y x 3 + x 2 y 2 + y x 3 + y 3 + z x 6 + y 4 + z x 3 + y 2 z (x 3 y 2 z 2 ) x 5 y 2 z x 5 y 2 z x 3 + y 3 3xyz x 3 + y 3 + z 3 3xyz y(x 5 y 2 z 2 ) y(x 3 y 2 z 2 ) y((y + 1)x 3 y 2 z 2 ) (2a.) (2b.) (2b.) (2b.) 22

23 local b (Algorithm 2.3) localb quo localb rr, localb nf f localb quo localb rr localb nf locdeg x + y 2 + z x 2 + y 2 + z x 3 + y 2 + z x 4 + y 2 + z x 5 + y 2 + z x 6 + y 2 + z x 7 + y 2 + z x 3 + xy 2 + z x 4 + xy 2 + z x 5 + xy 2 + z x 3 + y 4 + z x 3 + xy 3 + z x 3 + y 5 + z x 5 + x 3 y 3 + y x 4 + x 2 y 2 + y x 3 + x 2 y 2 + y x 3 + y 3 + z x 6 + y 4 + z x 3 + y 2 z (x 3 y 2 z 2 ) x 5 y 2 z x 5 y 2 z x 3 + y 3 3xyz x 3 + y 3 + z 3 3xyz y(x 5 y 2 z 2 ) y(x 3 y 2 z 2 ) y((y + 1)x 3 y 2 z 2 ) localb quo C[x, s] localb rr, localb nf D[s] Mora D[s] 23

24 6 localb.rr 6.1 localb.rr Asir localb.rr localb.rr ord -norc ord ( Asir ) VL3 A3 < 1 nakayama@orange2(91)=> asir -norc This is Risa/Asir, Version (Kobe Distribution). Copyright (C) , all rights reserved, FUJITSU LABORATORIES LIMITED. Copyright , Risa/Asir committers, GC 6.5 copyright , H-J. Boehm, A. J. Demers, Xerox, SGI, HP. PARI , copyright , C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier. [0] load("localb.rr"); 1 [679] VL3; [x,y,z,ss,s,dx,dy,dz,dss,ds] [680] A3; [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [681] Mora (Algorithm 3.5) weyl mora Mora [2018] weyl_mora(x*y*dx*dy,[x*dx+x*y*dy,y*dy+x*y*dx],vl2,a2s,v2); [0,x^2*y^2-2*x*y+1,[(-x*y^2+y)*dy+x*y,(x^2*y^2-x*y)*dy-x^2*y]] xy x y {x x + xy y, y + y + xy x } VL2 < s 1 A2S (-1,1)- V2 x, y VL2, A2S, V2 0 24

25 1 2xy + x 2 y 2 (1 2xy + x 2 y 2 )xy x y = {( xy 2 + y) y + xy}(x x + xy y ) + {(x 2 y 2 xy) y x 2 y}(y y + xy x )

26 D alg [s] mora gr Mora [2075] J; [x*y,-y*dy+x*dx,x*dx-ss] [2076] mora_gr(j,vl2,a2,a2s,v2); Count : crib:0 crif:1 crim:0 --> criterion [x*y,-y*dy+x*dx,x*dx-ss,y^2*dy+y] {xy, y y + x x, x x s} D alg [s] { xy, y y + x y, x x s, y 2 y + y } tangentcone gr mora gr tangentcone gr [2026] J=jf(x^3+x*y+y^3); [y^3+x*y+x^3,(y+3*x^2)*dy+(-3*y^2-x)*dx,(-y^3-x*y-x^3)*dx+(y+3*x^2)*ss, (-9*x*y^2+2*y)*dy+(-3*y^2-9*x^2*y+x)*dx+(27*x*y-3)*ss, (-3*y^3-2*x*y)*dy+(-3*x*y^2-x^2)*dx+(9*y^2+3*x)*ss] 0.01sec( sec) [2027] tangentcone_gr(j, VL2, A2S, V2); [y^3+x*y+x^3,(y+3*x^2)*dy+(-3*y^2-x)*dx,(-y^3-x*y-x^3)*dx+(y+3*x^2)*ss, (-9*x*y^2+2*y)*dy+(-3*y^2-9*x^2*y+x)*dx+(27*x*y-3)*ss, (-3*y^3-2*x*y)*dy+(-3*x*y^2-x^2)*dx+(9*y^2+3*x)*ss, (-y^2-3*x^2*y)*dy+(2*y^3-x^3)*dx-y-3*x^2, (-y^4+2*x^3*y)*dy+(-2*x*y^3+x^4)*dx-3*y^3+3*x^3, (-6*y^3+3*x^3)*ss-6*y^3+3*x^3,-9*y^3*ss-9*y^3, (27*x^2*y^2*ss-9*y^3)*dy+(-18*y^4+9*x^3*y)*ss*dx+(-18*y^2+27*x^2*y)*ss-27*y^2, (9*x*y^3-18*x^4)*ss*dy+(27*x^2*y^2*ss-9*x^3)*dx+(27*x*y^2-18*x^2)*ss-27*x^2, (-9*y^5-9*x^3*y^2)*ss+9*x*y^3,(-9*x^2*y^3-9*x^5)*ss+9*x^3*y, (27*x*y^4+27*x^4*y)*ss-27*x^2*y^2, (27*x^2*y^3*ss-9*y^4)*dy+(-27*y^5*ss+9*x*y^3)*dx, (81*y^5*ss-54*x*y^3)*dy+(81*x*y^4*ss-27*x^2*y^2)*dx+ (378*y^4+135*x^3*y)*ss-216*x*y^2] 0.01sec( sec) [2028] mora_gr(j, VL2, A2, A2S, V2); -----> mora gr Mora tangentcone gr 26

27 local b (Algorithm 3.13) localb rr [2011] localb_rr(x*(x+1)*(y+1),vl2, A2, A2S, V2); J : +x+x^2+x*y+x^2*y -ss+dy+y*dy +dy-x*dx+2*x*dy+y*dy-x^2*dx+2*x*y*dy Gr(J) : +x -ss+dy+y*dy +dy+y*dy+1 --> J (tangentcone_gr2 ) +y*dy^2+y^2*dy^2+2*y*dy +y^2*ss*dy^2-dy^2+2*y*ss*dy-2*y*dy^2-2*dy compute global-b. --> global b (bfct ) ok : the remainder of global-b is 0. 0th challenge Pick :ss^2+2*ss+1 length(genl):1 ss+1-->zero --> ss+1 Mora InL :1,NotInL :0 1th challenge Pick :ss+1 length(genl):1 1-->non-zero --> 1 Mora InL :0,NotInL :1 ss+1 [2012] x(x+1)(y +1) local b s+1 D alg [s] Mora global b 27

28 (Algorithm 4.10) localb nf [2470] localb_nf(x*(x+1)*(y+1), VL2, A2, A2S, V2, W2, 3, 10); J : +x+x^2+x*y+x^2*y -ss+dy+y*dy +dy-x*dx+2*x*dy+y*dy-x^2*dx+2*x*y*dy Gr(J) : +x -ss+dy+y*dy +dy+y*dy+1 <-- J +y*dy^2+y^2*dy^2+2*y*dy +y^2*ss*dy^2-dy^2+2*y*ss*dy-2*y*dy^2-2*dy NF : +1 <-- 1 Gr(J) 10-1 <-- ss Gr(J) <-- ss^2 Gr(J) 10-1 <-- ss^3 Gr(J) 10 sol_space(4) <-- L_{4,10} [ ] [ ] [ ] [ ] sol_space(3) <-- L_{3,10} [ ] [ ] [ ] sol_space(2) <-- L_{2,10} [ 0 0 ] [ 1 1 ] sol_space(1) <-- L_{1,10} [ 0 ] BSum: <-- local b ss+1 weyl_mora(bsum, G) <-- Mora 0 ss+1 [2471] x(x + 1)(y + 1) local b s + 1 localb nf s 28

29 local b (Algorithm 2.3) [1549] lb2(x*(x+1)*(y+1)); comp psi(i): <-- psi(i) 0sec( sec) comp local intersection 0.01sec( sec) -s-1 [1550] <-- K[[x]][s] J \cap K[s] s + 1 local b x(x + 1)(y + 1) local b lb2 29

30 6.2 ( order matrix localb.rr (VL) ss s ( 1, 1) dss, ds 1 VL1 = [x,ss,s,dx,dss,ds] 2 VL2 = [x,y,ss,s,dx,dy,dss,ds] 3 VL3 = [x,y,z,ss,s,dx,dy,dz,dss,ds] order matrix(a, B, C) D alg [s], D[s] < 1 (Section 3.1) < 2, < 3 (Section 3.2) < s 1 (3.2) < s 2, < s 3 (Section 3.2) A* < 1 B* < 2 C* < 3 A1 1 A2 2 A3 3 B1 B2 B3 C1 C2 C3 A*S < s 1 B*S <s 2 C*S <s 3 A1S 1 A2S 2 A3S 3 B1S B2S B3S C1S C2S C3S A1, A1S localb.rr VL1 = [ x,ss, s,dx,dss,ds]$ A1=newmat(5, 6, [ [ 0, 1, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0], [ 0, 1, 0, 0, 0, 0], [ 0, 0, 0, 1, 0, 0], [ 1, 0, 0, 0, 0, 0] ])$ A1S=newmat(6, 6, [ [ 1, 0, 1, 0, 0, 0], [ 0, 1, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0], [ 0, 1, 0, 0, 0, 0], [ 0, 0, 0, 1, 0, 0], [ 1, 0, 0, 0, 0, 0] ])$ < 1, < s 1 lex order 30

31 ( 1, 1) V* v 1 (Section 3.2) v 1 ( 1, 1) x 1 x n y s ξ 1 ξ n V1 1 V2 2 V3 3 VV* v 2 (Section 3.2) v 2 ( 1, 1) x 1 x n y s ξ 1 ξ n VV1 1 VV2 2 VV3 3 V* A*,A*S VV* B*, C*, B*S, C*S e W* e (Section 4.5) e D[s] x 1 x n y ξ 1 ξ n W1 1 W2 2 W Mora weyl mora(p, PL, VL, Ord, Vect) : P, PL, VL, Ord s order (< s 1, < s 2, < s 3 order), Vect ( 1, 1) : Q, A, R AP = Q[0]P L[0] + + Q[N 1]P L[N 1] + R Q[i] 0 LM < (Q[i]P L[i]) LM < (P ) R 0 LM < (R) LM < (P L[i]) < Ord s order Ord < s 1 < 1 < s 2 < 2 < s 3 < 3 N P L : Mora Algorithm 3.4) Ord < s 1, < s 2, < s 3 < 1, < 2, < 3 Mora : P = x 2 + x + 1 P L = [ x + x x + x] < 1 Mora [381] weyl_mora(dx^2+dx+1, [dx+x*dx+x], VL1, A1S, V1); [<0>3<[0,1,3]>+(2)2<1>3] [x,x+1,[dx]] 31

32 P = xy x y P L = [x + y + x, y + y 2 + y 3 ] < 1 Mora x, y x 1, x 2 y [392] weyl_mora(x*y*dx*dy, [x+y+dx, y+y^2+y^3], VL2, A2S, V2); [<[0,2,3]>+(3)1<[1,2,3]>+(4)3<1>9<3>13<3>10] [(3*y^2+2*y+1)*x^2+(2*y^3+y^2)*x,y^2+y+1,[(dy*y^3+dy*y^2+dy*y)*x,-dy*x^2-dy*y*x]] dp weyl mora(p, PL, VL, Ord, Vect) : P, PL, VL, Ord s order (< s 1, < s 2, < s 3 order), Vect ( 1, 1) : R AP = Q[0]P L[0] + + Q[N 1]P L[N 1] + R Q[i] 0 LM < (Q[i]P L[i]) LM < (P ) R 0 LM < (R) LM < (P L[i]) < Ord s order N P L : Mora Algorithm 3.4) weyl mora D alg [y] mora gr Mora weyl mora mod(p, PL, VL, Ord, VL, Prime) : P,PL,VL,Ord,VL weyl mora Prime GF ( Prime ) : weyl mora : weyl mora modular dp weyl mora mod(p, PL, VL, Ord, VL, Prime) : P,PL,VL,Ord,VL dp weyl mora Prime GF ( Prime ) : dp weyl mora : dp weyl mora modular mora gr mod 6.4 D alg mora gr(i, VL, Ord1, Ord2, Vect) : I, VL, Ord1 order, Ord2 Ord1 s order, Vect ( 1, 1) : I order Ord1 32

33 : Buchberger Mora D alg : D alg [ss] (ss ) J = D alg [ss] {x 2, x x 2ss } < 2 [395] J=jf(x^2); [x^2,dx*x-2*ss] [396] mora_gr(j, VL1, B1, B1S, VV1); [0,1]:Rest 0 index N:2,bits of Nf:4 S non-zero Rest:2 LM:(1)*<<1,1,0,0,0,0>>,2 Leading monomial [1,2]:Rest 1 N:3,bits of Nf:9 Rest:2 LM:(1)*<<0,2,0,0,0,0>>,2 [2,3]:Rest 1 [0,2]:Rest 0 Count : crib:0 crif:0 crim:2 criterion [x^2,dx*x-2*ss,(2*ss+2)*x,-2*dx*x-4*ss^2-2*ss-2] mora gr mod(i, VL, Ord1, Ord2, Vect, Prime) : I, VL, Ord1 order, Ord2 Ord1 s order, Vect ( 1, 1), Prime GF ( Prime ) : I order Ord1 : mora gr modular Mora modular tangentcone gr(i, VL, Ord, Vect) : I, VL, Ord s order (< s 1, < s 2, < s 3 order), Vect ( 1, 1) : I Ord order(< 1, < 2, < 3 ) : ( 1, 1) < s < ( ) : {x x + xy y, y y + xy x } < 1 33

34 [2599] I=[x*dx+x*y*dy,y*dy+x*y*dx]; [x*y*dy+x*dx,y*dy+x*y*dx] [2600] tangentcone_gr(i, VL2, A2S, V2); [0,1]: index N:2,bits of Nf:4 S non-zero Rest:1 [0,2]: N:3,bits of Nf:2 Rest:2 [2,3]: N:4,bits of Nf:5 Rest:3 [0,3]: N:5,bits of Nf:2 Rest:4 [3,5]: N:6,bits of Nf:3 Rest:4 [1,5]: [4,6]: N:7,bits of Nf:6 Rest:5 [3,6]: N:8,bits of Nf:1 Rest:8 [0,6]:... Count : crib:10 crif:73 crim:33 criterion +x*s*dx+x*y*dy +y*s*dy+x*y*dx -x^2*y*dx^2+x*y^2*dy^2-x*y*dx+x*y*dy +x^2*y*dx-x*y^2*dy -x*y^2*dx*dy+x*y^2*dy^2+x*y*dx+x*y*dy-y^2*dy +x^2*y^2*dy-x*y^3*dy... 34

35 [x*y*dy+x*dx,y*dy+x*y*dx,x*y^2*dy^2+x*y*dy-x^2*y*dx^2-x*y*dx,-x*y^2*dy+x^2*y*dx,... ] 6.5 D[y] whdiv(f, G, VL, Ord, N) : F, G, VL, Ord order < r, N : Q, R P = Q[0]G[0] + + Q[M 1]G[M 1] + R Q[i] 0 LM <r (Q[i]G[i]) LM <r (P ) Q[i] N LE <r (G[i]) 1 R N 1 M G : Weierstrass-Hironaka 4.3 d app div : x 1 + x WH N = 5 [377] whdiv(x, [1+x], VL1, A1, 5); [[ x^5-x^4+x^3-x^2+x ],-x^6,0] x = (x x 2 + x 3 x 4 + x 5 )(1 + x) x 6 (4 ) x x 2 + x 3 x 4 (4 ) 0 d app div(p, G, VL, W, Ord, N) : P, G, VL, W e Ord order < 1 N : Q, R P = Q[0]G[0] + + Q[M 1]G[M 1] + R Q[i] 0 LM <1 (Q[i]G[i]) LM <1 (P ) Q[i] N LE <1 (G[i]) 1 R N 1 M G : D[y] 4.6 local b localb nf : x 2 (1 + x) x + x N = 5 4.7) [2607] d_app_div(dx^2,[(1+x)*dx+x],vl1,w1,a1,5); M0 :2 M[0] :1 M[1] :1 35

36 M[2] :3 Bound :10 [2,3,4,5,6,7,8,9,10,][1,2,3,4,5,6,7,][0,] check Sum :+dx^2 P :+dx^2 Monos : ( ) [[ (x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1)*dx-x^6+x^5-x^4+x^3-x^2+x-1 ], -x^9*dx^2+(-x^9+x^7)*dx-x^8+2*x^7-2*x^6+2*x^5-2*x^4+2*x^3-2*x^2+2*x-1] 2 =((1 x + x x 8 ) (1 x + x 2 + x 6 )) ((1 + x) + x)+ x 9 2 x 9 + x x 2x 2 + 2x 3 2x x 7 x 8 ( 4 ) 1 + 2x 2x 2 + 2x 3 2x local b localb rr(f, VL, Ord1, Ord2, Vect) : F, VL, Ord1 order < 1 or < 2 or < 3, Ord2 Ord1 s order, Vect s, : F local b : global b local b 3.13 : x 2 (x + 1) 3 local b ( 3.12) [2598] localb_rr(x^2*(x+1)^3, VL2, A2, A2S, V2); J : +x^2+3*x^3+3*x^4+x^5 +2*ss+5*x*ss-x*dx-x^2*dx +2*ss*s^2+5*x*ss*s-x*s*dx-x^2*dx +x^2*s^3+3*x^3*s^2+3*x^4*s+x^5... J Gr(J) : +2*ss+5*x*ss-x*dx-x^2*dx +x^2+3*x^3+3*x^4+x^5 +2*x^3*dx+7*x^4*ss+3*x^4*dx+4*x^5*ss+x^5*dx 36

37 ... J compute global-b. global-b ok : the remainder of global-b is 0. global-b Gr(J) 0th challenge Pick :18*ss^4+45*ss^3+40*ss^2+15*ss+2 global-b length(genl):4 6*ss^3+11*ss^2+6*ss+1-->zero Gr(J) 0 6*ss^3+13*ss^2+9*ss+2-->zero 9*ss^3+18*ss^2+11*ss+2-->non-zero 18*ss^3+27*ss^2+13*ss+2-->non-zero InL :2,NotInL :2 1th challenge Pick :2*ss^2+3*ss+1 length(genl):2 ss+1-->non-zero 2*ss+1-->non-zero InL :0,NotInL :2 gcd(6*ss^3+11*ss^2+6*ss+1,6*ss^3+13*ss^2+9*ss+2) local-b 2*ss^2+3*ss+1 x 2 (x + 1) 3 local b s s localb nf(f, VL, Ord1, Ord2, Vect, W, Deg, N) : F, VL, Ord1 order < 1 or < 2 or < 3, Ord2 Ord1 s order, Vect s, W e, Deg (global b ), N ( Deg + 1) : F local b : local b 4.10 : x 2 (y + 1) 2 z 2 local b ( 4.11) [1991] localb_nf(x^2*(y+1)^2*z^2, VL3, A3, A3S, V3, W3, 6, 7); 0.02sec( sec) J : +x^2*z^2+2*x^2*y*z^2+x^2*y^2*z^2 -dy+x*dx-y*dy -x*dx+z*dz -2*ss+x*dx -x*dx+z*dz 37

38 +s*dy+y*dy-z*dz +2*ss*s-z*dz... J Gr(J) : -x*dx+z*dz +dy+y*dy-z*dz +2*ss-z*dz -2*y*ss*dy+2*z*ss*dz-z*dy*dz +x^2*z^2+2*x^2*y*z^2+x^2*y^2*z^2... J NF : s Gr(J) +1 +1/2*z*dz +1/4*z^2*dz^2+1/4*z*dz +1/8*z^3*dz^3+3/8*z^2*dz^2+1/8*z*dz -3/8*z^3*dz^3-31/16*z^2*dz^2-31/16*z*dz-1/4 +23/32*z^3*dz^3+135/32*z^2*dz^2+157/32*z*dz+3/4-9/8*z^3*dz^3-447/64*z^2*dz^2-555/64*z*dz-23/16 sol_space(7) L_{6,7} [ ] [ 1/4 3/2 13/ ] [ -3/4-17/4-33/4-23/ ] [ 23/16 63/8 231/ ] sol_space(6) L_{5,7} [ ] [ 1/4 3/2 13/ ] [ -3/4-17/4-33/4-23/4 0 1 ] sol_space(5) L_{4,7} [ ] [ 1/4 3/2 13/4 3 1 ] sol_space(4) [ ] L_{3,7} BSum: local-b ss^4+3*ss^3+13/4*ss^2+3/2*ss+1/4 [<2>18<2>20<2>25<2>30<[12,2,8]>+(20)38<19>90<19>103<2>118<2>118<2>117<2>122<2> 122<2>127<2>107<2>112<2>94<2>74<2>65<2>56<2>48<2>39<2>32<2>24<2>15<2>9] weyl_mora(bsum, G) 38

39 0 local-b ss^4+3*ss^3+13/4*ss^2+3/2*ss+1/4 x 2 (y + 1) 2 z 2 local b s 4 + 3s s s lb1(f), lb2(f), lb3(f) : F, : F local b : local b (Algorithm 2.3) local b lb : x 3 + y 3 + xyz local b [1549] lb3(x^3+y^3+x*y*z); comp psi(i): 0.09sec + gc : 0.03sec(0.1837sec) comp local intersection 0.05sec( sec) -9*s^5-54*s^4-128*s^3-150*s^2-87*s-20 [1550] 6.7 utility print poly(p, VL, Ord) : P, VL, Ord order : P Ord : [400] print_poly(1+x^2+(x+1)^3*dx, VL1, A1); +dx+3*dx*x+3*dx*x^2+dx*x^3+1+x^2 39

40 [1] : ( ),,No. 38, (1994) [2] : D,, (2002) [3], :,, (1981) [4] Castro,F.: Calculs effectifs pour les idéaux d opérateurs différentiels, Travaux en Cours 24, (1987), 1-19 [5] Granger,M.,Oaku,T.: Mimimal filtered free resolutions and division algorithms for analytic D-modules,Prépublications du départment de matheématiques, Univ.Angers, No.170 (2003) [6] Granger,M.,Oaku,T.,Takayama,N.:Tangent cone algorithm for homogenized differential operators, Journal of Symbolic Computation, 39, (2005) [7] Greuel,G.-M., Pfister,G.: Advances and improvements in the theory of standard bases and syzygies, Arch.Math., 66, (1995) [8] Mora,T.:An algorithm to compute the equations of tangent cones, EUROCAM 82, Springer Lecture Notes in Computer Science, (1982) [9] Nakayama,H.: Algorithms of computing local b-function by an approximate division algorithm, preprint [10] Noro,M.: An Efficient Modular Algorithm for Computing the Global b- function, Mathematical Software, (2002), [11] Oaku,T.: An algorithm of computing b-function, Duke Math. J., 87, (1997), no [12] Oaku,T.: Algorithm for the b-function and D-modules asscociated with a polynomial, J.Pure.Appl.Algebra 117/118, (1997), [13] Oaku,T., Takayama,N.:Algorithms for D-modules - restriction, tensor product, localization, and local cohomology groups, J.Pure.Appl.Algebra 156, (2001), [14] Saito, M., Sturmfels, B., Takayama,N.: Groebner Deformations of Hypergeometric Differential Equations, Springer, (2000) [15] Kobayashi,H., Furukawa,A., Sasaki,T.: POWER SERIES, RIMS, 581, (1986) GRÖBNER BASIS OF IDEAL OF CONVERGENT [16] Oaku,T., Shimoyama,T. : A Gröbner basis Method for Modules over Rings of Differential Operators, J.Symbolic Computation 18, , (1994) [17] Bernstein,I.N. : Modules over a ring of differential operators, Functional Anal. Appl., 5 (1971), [18] Bernstein,I.N. : The analytic continuation of generalized functions with respect to a parameter, Functional Anal. Appl., 6 (1972),

41 [19] Kashiwara,M. : B-functions and holonomic systems: rationality of roots of b-functions, Invent. Math. 38 ( ), [20] Giovini,A., Mora,T., Nielsi,G., Robbiano,L., Traverso,C.: one sugar cube, please OR Selection strategies in the Buchberger algorithm, Proc. Issac 91, [21] Takayama,N. : Kan A system for doing algebraic analysis by computer, [22] Noro,M. et al: Risa/Asir, 41