, 1993.,,., implement, ( )., 1970,,, I.N. Bernstein, ( D- ),., D-,., D-, ( ), ( ) (singular locus),, 3,. (Gröbner basis), 1960 B. Buchberger,.,,, impl

Transcription

1 No WEB i

2 , 1993.,,., implement, ( )., 1970,,, I.N. Bernstein, ( D- ),., D-,., D-, ( ), ( ) (singular locus),, 3,. (Gröbner basis), 1960 B. Buchberger,.,,, implement,.,, (standard base).,,,..,. ( ),,. (Weyl ) ( ),,,.,.,,.,, 1 2, 4 ii

3 . 5 6 D-. D- D-, 3,,,. 7 ( ). 1 5, 6 7.,. ( ),,,. (computational algebraic analysis)., Risa/Asir implement,., Weyl.. Risa/Asir Risa/Asir.,,,,, WEB WEB WEB iii

4

5 Tangent cone D Cauchy ( ) tangent cone D 0 filtration FD- ( ) FW- FR v

6 6.4 FW vi

7 1 1.1 K. n 1, x 1,...,x n R := K[x 1,..., x n ]. x = (x 1,..., x n ) R = K[x]. N (0 ), n N n α = (α 1,..., α n ) (multi-index) (exponent). α = (α 1,..., α n ), α := α α n α (length). x α α1 α = x 1 x n n., R f f = α c α x α (c α K) N n (total order) (term order, monomial order), (1), (2)., α,β,γ N n. α β α β α = β : (1) α β γ α + γ β + γ, (2) α 0 = (0,..., 0) α... α = (α 1,..., α n ), β = (β 1,..., β n ). (1) (lexicographic order) L : α L β j (α i = β i ( i < j), α j < β j ); (2) (total degree lexicographic order): α β α < β or ( α = β, α L β); (3) (total degree reverse lexicographic order): α β α < β or ( α = β, α L β);, α, β, α i β i i α β., N n (partial order). α β α β., (2) β α 0 (1) α = α + 0 α + (β α) = β 1

8 .,, f R leading exponent, leading coefficient, leading term f = α c α x α R \ {0}, {α c α 0} β, lexp(f) := β, lcoef(f) := c β, lterm(f) := c β x β f, g R \ {0}, f + g 0 lexp(f g) = lexp(f) + lexp(g), lcoef(f g) = lcoef(f)lcoef(g), lterm(f g) = lterm(f)lterm(g). :. lterm(f) = ax α, lterm(g) = bx β f := f lterm(f), g := g lterm(g) fg = abx α+β + ax α g + bx β f + f g., γ α δ β, γ α δ β γ, δ c R cx γ+δ., γ + δ γ + β α + β γ + δ α + β lterm(fg) = lterm(f)lterm(g) f, g R, f + g 0 lexp(f + g) max {lexp(f), lexp(g)}. (max.) : lexp(f) lexp(g). lexp(f) lexp(g) lexp(f + g) = lexp(f); lexp(f) = lexp(g) (leading term ) lexp(f + g) lexp(f) N n L (monoideal), α L β N n α + β L., N n S, mono(s) := {α + β α S, β N n } S. (. ), N n S, L = mono(s), L (Dickson ) (1) N n. 2

9 α 2 n = 2, S = {(2, 0), (1, 1), (0, 3)} mono(s) α 1 (2) L 1, L 2, L 3,... N n (i.e. L 1 L 2 L 3...), k, L j = L k j k. : (1) (2). (1) (2): L := j=1 L j, (1) L S. S L, k, S L k. j k L j S, L j = L k. (2) (1): L, L S 1 S 2 S 3... mono(s j ) mono(s j+1 ) ( j). (2). n. n = 1 : L N, d := min{α N α L}, L {α N α d} L., d. L {d}. n 1. L N n, j, L j = {(α 1,..., α n 1 ) N n 1 (α 1,..., α n 1, j) L}., {L j } N n 1., L j S j, k, j k L j = L k. k S := {(α, j) α S j } j=0. L S. (α, j) L. 0 j k, β S j γ N n 1 α = β + γ (α, j) = (β, j) + (γ, 0) mono(s). j > k L j = L k, (α, k) mono(s), (α, j) = (α, k) + (0, j k) mono(s). L = mono(s). 3

10 N n ; (well-order). : L N n. Dickson, mono(l) S. S α., α L., α S mono(l), β L α β. β L mono(s), β γ γ S. α γ, α γ, α α = γ, α = β L., α L. L β β α. L S, γ S γ β γ β. β α γ α, γ S, α L N n, ( ) S 0. : S 0 = {α L α β α β β L }. (1) S 0 L : α L, {β L β α} γ ( α )., γ γ S 0. L S 0. (2) S 0 : S N n L. α S 0, α β β S L. S 0, β = α. S 0 S. R S E(S) := {lexp(f) f S, f 0} I R, E(I) N n. : f I, β. x β f I lexp(x β f) = lexp(f)+β E(I), E(I) ( ) I R. I G I ( ) (Gröbner basis), : (1) I G ; (2) E(I) = mono(e(g))., E(G) E(I) (cf ), G (minimal Gröbner basis). 4

11 I I G, E(G) ( ). G R, f R, f G (reduction)., while ( ) { ;...} C ( ) Input: f R and a finite set G R; while (f 0 and lexp(f) mono(e(g))) { Choose g G such that lexp(f) lexp(g); f := f (lterm(f)/lterm(g))g; } Output: f; , output f f = 0 lexp(f) mono(e(g)). : lexp(f) lexp(g) lterm(f) = lterm((lterm(f)/lterm(g))g) f 1 = f (lterm(f)/lterm(g))g lexp(f 1 ) lexp(f). ( while ), f, f 1, f 2,... lexp(f) lexp(f 1 ) lexp(f 2 )..., output red(f, G), f G (reduction). g, f G. (g.), f G, f 0 lexp(f) mono(e(g)) f G (reducible), (irreducible) lexp(f) lexp(g) f := f (lterm(f)/lterm(g))g f g f, r = red(f, G) f G r f R R G = {g 1,..., g s },, r = red(f, G), (1) f r G I ; (2) r = 0 lexp(r) mono(e(g)); (3) q 1,...,q s R, f = s i=1 q i g i + r i q i = 0 lexp(q i g i ) lexp(f) ( r = 0 lexp(r) lexp(f)). 5

12 : (2). (1) (3) (3). f G r = f, q 1 =... = q s = 0. f, m 1,..., m N, {1,..., s} i(1),..., i(n), f = m 1 g i(1) + + m N g i(n) + r (1.1) lexp(f) = lexp(m 1 g i(1) ) lexp(m 2 g i(2) )... lexp(m N g i(n) ) lexp(r). (1.1) g 1,..., g s I R, G I mono(e(g)) = E(I), G I. : G I. G = {g 1,..., g s }. f I, r := red(f, G). r 0 lexp(r) mono(e(g)) = E(I), r = f (f r) I. r = 0, , q 1,..., q s R f = s j=1 q s g s. I G I R I. : Dickson, E(I) N n S. S α lexp(f) = α f I, I G {lexp(f) f G} = S. mono(e(g)) = mono(s) = E(I), G I. Hilbert : R (Noether)., R I, J R, I J E(I) = E(J), I = J. : G I, G J. I = J., I. S-. α = (α 1,..., α n ) β = (β 1,..., β n ),. α β := (max{α 1, β 1 },..., max{α n, β n }) 6

13 (S- ) f, g R, α := (lexp(f) lexp(g)) lexp(f), β := (lexp(f) lexp(g)) lexp(g), f g S- (S-polynomial) sp(f, g). sp(f, g) = lcoef(g)x α f lcoef(f)x β g lexp(sp(f, g)) lexp(f) lexp(g). key point G = {g 1,..., g s } R, I G R, (1) (3) : (1) G I ; (2) f I f G red(f, G) = 0. (3) g i, g j G, q ij1,..., q ijs R sp(g i, g j ) = q ij1 g q ijs g s k = 1,..., s q ijk = 0 lexp(q ijk g k ) lexp(g i ) lexp(g j ). : (1) (2): f I, r := red(f, G) 0. lexp(r) E(G) = E(I), (1) r I,. (2) (3): sp(g i, g j ) I, (2), red(sp(g i, g j ), G) = (3). (3) (1): G = {g 1,..., g s }. lcoef(g k ) = 1 (k = 1,..., s). (3), i j {1,..., s} (i, j), q ij1,..., q ijs, sp(g i, g j ) = q ijk g k, (1.2) q ijk 0 lexp(q ijk g k ) lexp(g i ) lexp(g j ). f I. lexp(f) mono(e(g))., q 1,..., q s R f = s q k g k k q k = 0 lexp(q k g k ) lexp(f)., k lexp(f) = lexp(q k g k ) mono(e(g)). 7

14 , f f = q k g k, (q 1,..., q s R) (1.3),, max {lexp(q k g k ) 1 k s, q k 0}, (1.3). (f I,,.), max. (1.3). α := max {lexp(q k g k ) 1 k s, q k 0}. α = lexp(f) (1). α lexp(f) ( α lexp(f)). g 1,..., g s, 1 k l lexp(q k g k ) = α, l < k s lexp(q k g k ) α q k = 0. q k := q k lterm(q k ), lterm(q k ) = c k x β(k), α (k) := lexp(g k ), : = l l f = lterm(q k )g k + q kg k + q k g k. (1.4) k=l+1 l l lterm(q k )g k = c k x β(k) g k l 1 (c c k )(x β(k) g k x β(k+1) g k+1 ) + (c c l )x β(l) g l. (1.5) 1 k l α (k) + β (k) = α, α (k) α (k+1) α, γ (k) := α α (k) α (k+1) (1.2) (1.5) l lterm(q k )g k = = l 1 l 1 (c c k )x γ(k) sp(g k, g k+1 ) + (c c l )x β(l) g l (c c k )x γ(k) q k,k+1,ν g ν + (c c l )x β(l) g l. (1.6) γ (k) + lexp(q k,k+1,ν g ν ) γ (k) + α (k) α (k+1) = α, c c l 0, (1.4) (1.6) lexp(f) = α,. c c l = 0. (1.4), (1.6) l 1 ν=1 f = (c c k )x γ(k) q k,k+1,ν g ν + ν=1 l q kg k + q k g k k=l+1. leading exponent α, (1.3). (1.3) α = lexp(f). (Buchberger ) : 8

15 ( ) Input: a finite set G R; while ( (f, g) G G such that r := red(sp(f, g), G) 0) G := G {r}; Output: G;, r,. sp(f, f) = 0 f g f, g , output G G I. : (1) : while G G 0, G 1, G 2,.... G 0 G 1 G {mono(e(g j ))}. Dickson k j k mono(e(g j+1 )) = mono(e(g j )). G j+1 = G j {r j }, r j 0 lexp(r j ) mono(e(g j )), mono(e(g j+1 )) mono(e(g j )). (2) output G I : r I G I,., f, g G, red(sp(f, g), G) = , G I ( ) Input: a finite set G R; G := the output of Algorithm with input G; while ( g G such that lexp(g) E(G \ {g}) ) G := G \ {g}; Output: G; , output G I. : G,. g G lexp(g) E(G \ {g}), mono(e(g \ {g})) = mono(e(g)) = E(I), G I. G output, g G lexp(g) E(G \ {g}) E(G \ {g}) E(I). E(G) E(I)., : 9

16 R f R I. G I, 3 : (1) f I; (2) red(f, G) = 0; (3) red(f, G) = 0., r := red(f, G), r = 0 f I; f 0 f I. : (1) (2) (2). (2) (3). (3) (1): r := red(f, G) f I r I, r = 0 f I., R I R/I G R. R f = α c α x α G (completely irreducible), f G, c α 0 α mono(e(g)). f G,. redlexp(f) := max {α c α 0, α mono(e(g))}, redlterm(f) := c α x α (α := redlexp(f)) ( ) Input: f R and a finite set G R; while (f is not completely irreducible with respect to G) { Choose g G such that redlexp(f) lexp(g); f := f (redlterm(f)/lterm(g))g; } Output: f; , output f G. :., while f f 0 = f, f 1, f 2,.... f j redlexp(f j ) lexp(g) g G, lterm((redlterm(f j )/lterm(g))g) = redlterm(f j ), redlterm(f j ) f j f j+1, redlexp(f j ) redlexp(f j+1 ). j,. redlexp(f) lexp(f), (2). 10

17 G R, f R G ( g ) : f G r 1,r 2. G I, r 1 r 2 I G., r 1 r 2 0 lexp(r 1 r 2 ) mono(e(g)) = E(I), r 1 r 2 I. r 1 = r I R S(I) := N n \ E(I), R/I K, K(S(I)) := α S(I) Kx α. : K- ϕ : K(S(I)) R/I f = α S(I) c α x α f R/I [f]. (K-.) G I. (1) ϕ : ϕ(f) = 0 f I. f 0 f G lexp(f) mono(e(g)) = E(I). f I. (2) ϕ : f R f G r. f r I r K(S(I)) ϕ(r) = [r] = [f] I R, G I (dim K K, ): (1) dim K (R/I) = S(I) < ; (2) i = 1,..., n α i N, (0,..., (i) α i,..., 0) E(G). : dim K (R/I) = S(I). (1) (2), i, k N (0,..., (i) k,..., 0) mono(e(g)) = E(I). S(I) (1). (2) β = (β 1,..., β n ) N n β i α i ( i) β E(I) E(I) N n, (1)., α, β S(I) x α+β G r(α, β), x α x β := r(α, β) K(S(I)) R/I. R/I. K (1), (2) {x = (x 1,..., x n ) K n f(x) = 0 ( f I)},, (resultant) ([CLO] ) n = 2 x = x 1, y = x 2. f 1 := xy 1, f 2 := x 2 y f 1, f 2 K[x, y] I. (K 0.) 11

18 (1) G = {xy 1, x 2 y, y 2 x} I E(G) = {(1, 1), (2, 0), (0, 2)} dim K K[x, y]/i = 3. (2) G = {x y 2, y 3 1} I E(G) = {(1, 0), (3, 0)} dim K K[x, y]/i = (1) (1),(2). (2) K[x, y]/i.,, α, β S(I) r(α, β) n = 3 x = x 1, y = x 2, z = x 3 f 1 := x 3 y 2, f 2 := y 3 z 2, f 3 := z 3 x 2 f 1, f 2, f 3 K[x, y, z] I (, K 0 ): (1), G := {f 1, f 2, f 3 } E(G) = {(3, 0, 0), (0, 3, 0), (0, 0, 3)}. (2),,. G := {x 2 z 3, xz 2 z 13, y 2 z 14, yz 2 z 9, z 21 z 2 } E(G) = {(2, 0, 0), (1, 0, 2), (0, 2, 0), (0, 1, 2), (0, 0, 21)} risa/asir (, ). load gr([f1,f2,f3],[x,y,z]); x, y, z F 1, F 2, F 3 ( ) ( ). Ord, Ord = 0 (default) ; Ord = 1 ; Ord = 2.. ( [Nor], [SN].) 12

19 2. (1) (1). (2) (2) dim K K[x, y, z]/i = 27. (3) (1), K(S(I)) K[x, y, z]/i. 3. G R. f, g G lexp(f) lexp(g) = lexp(f) + lexp(g), a, b R sp(f, g) = af + bg lexp(af), lexp(bg) lexp(f) lexp(g)., , f, g G S-. ( : fg gf = 0.) 1.2, R := K[x] n, R r R r R- N. r e i := (0,..., (i) 1,..., 0), R r f r r f = (f 1,..., f r ) = f i e i = c αi x α e i (c αi K) (2.1) i=1 i=1 α. N n, N n {1,..., r} r (α, β, γ N n, i, j {1,..., r} ): (r-1) (α, i) r (β, i) α β; (r-2) (α, i) r (β, j), γ (α + γ, i) r (β + γ, j) (1), (α, i) r (β, j) (i < j) or (i = j, α β) (2) (α, i) r (β, j) ( α < β ) or ( α = β, i < j) or ( α = β, i = j, α β) 13

20 (1), (2) r (r-1),(r-2).,., (r-1), (r-2) r, (2.1) f R r \ {0}, leading exponent lexp( f) := max r {(α, i) c αi 0} (α, i) := lexp( f), f leading point, leading term, leading coefficient lp( f) := i, lterm( f) := c αi x α, lcoef( f) = c αi., R R r. α, β N n i {1,..., r} (α, i) ± β = (α ± β, i), (α, i) ± (β, i) = (α ± β, i), (α, i) (β, i) = (α β, i) f R r a R ( f 0, a 0 ) f, g R r, lexp(a f) = lexp( f) + lexp(a), lcoef(a f) = lcoef(a)lcoef( f), lterm(a f) = lterm(a)lterm( f). lexp( f + g) r max r {lexp( f), lexp( g)} N n {1,..., r} L, i {1,..., r} L i := {α (α, i) L} N n. S N n {1,..., r} mono(s) := {(α, i) + β = (α + β, i) (α, i) S, β N n } S., N n {1,..., r} S, L = mono(s), L. Dickson N n {1,..., r} (r-1), (r-2) r. 14

21 : (α 1, ν 1 ) r (α 2, ν 2 ) r (α 3, ν 3 )..., i {1,..., r} ν k = i k. k k 1, k 2,..., (r-1) α k1 α k2.... R r S E(S) := {lexp( f) f S, f 0} : N R r R-, E(N) N n {1,..., r} ( ) N R r R-. N G N ( r ), : (1) N R G ; (2) E(N) = mono(e(g))., E(G) E(N), G. G R r, f 0 R r, f G ( (α, i) (β, j) (α β, i = j) ): ( ) Input: f R and a finite set G R r ; while ( f 0 and lexp( f) mono(e(g))) { Choose g G such that lexp( f) lexp( g); f := f (lterm( f)/lterm( g)) g; } Output: f; , output f f = 0 lexp( f) mono(e(g)). : lexp( f) lexp( g) f 1 := f (lterm( f)/lterm( g)) g (r-1), (r-2), 1.2.2, lexp( f 1 ) r lexp( f)., R r f, f 1, f 2,... lexp( f) r lexp( f 1 ) r lexp( f 2 ) r..., r output red( f, G), f G., f G, f 0 lexp( f) mono(e(g)) f G, f R r R r G = { g 1,..., g s },, r = red( f, G), 15

22 (1) f r G N ; (2) r = 0 lexp( r) mono(e(g)); (3) q 1,...,q s R, f = s i=1 q i g i + r i q i = 0 lexp(q i g i ) r lexp( f) ( r = 0 lexp( r) r lexp( f)). : : N R r R-, G N mono(e(g)) = E(N), G N N R r R- N N, M R r N = M. R-, N M E(N) = E(M), (S- ) f, g R, lexp( f) = (α, i), lexp( g) = (β, j), f, g S- ( ) sp( f, g), i = j i j sp( f, g) = 0. sp( f, g) = lcoef(g)x α β α f lcoef(f)x α β β g; lp( f) = lp( g) lexp(sp( f, g)) r lexp( f) lexp( g) G = { g 1,..., g s } R, N G R r R-, (1) (3) : (1) G N ; (2) f N f G red( f, G) = 0. (3) g i, g j G, lp( g i ) = lp( g j ), q ij1,..., q ijs R sp( g i, g j ) = q ij1 g q ijs g s k = 1,..., s, q ijk = 0 lexp(q ijk g k ) r lexp( g i ) lexp( g j ). : (1) (2) (2) (3) (3) (1) ,. lcoef( g k ) = 1 (k = 1,..., s). (3), i j lp( g i ) = lp( g j ) {1,..., s} (i, j), q ij1,..., q ijs, sp( g i, g j ) = q ijk g k, (2.2) 16

23 q ijk 0 lexp(q ijk g k ) r lexp( g i ) lexp( g j ). f N. lexp( f) mono(e(g))., q 1,..., q s R f = s q k g k k q k = 0 lexp(q k g k ) r lexp( f)., k lexp( f) = lexp(q k g k ) mono(e(g))., f f = q k g k, (q 1,..., q s R) (2.3),, max r {lexp(q k g k ) 1 k s, q k 0} r, (2.3). ( f N, r,.) (2.3). (α, i) = max r {lexp(q k g k ) 1 k s, q k 0}. (α, i) = lexp( f) (1). (α, i) lexp( f) ( (α, i) r lexp( f)). g 1,..., g s, 1 k l lexp(q k g k ) = (α, i), l < k s lexp(q k g k ) r (α, i) q k = 0. q k := q k lterm(q k ), lterm(q k ) = c k x β(k), (α (k), i) = lexp( g k ), : l l f = lterm(q k ) g k + q k g k + q k g k. (2.4) k=l+1 = l l lterm(q k ) g k = c k x β(k) g k l 1 (c c k )(x β(k) g k x β(k+1) g k+1 ) + (c c l )x β(l) g l. (2.5) 1 k l α (k) + β (k) = α, α (k) α (k+1) α, γ (k) := α α (k) α (k+1) (2.2) (2.5) l lterm(q k ) g k = = l 1 l 1 (c c k )x γ(k) sp( g k, g k+1 ) + (c c l )x β(l) g l (c c k )x γ(k) q k,k+1,ν g ν + (c c l )x β(l) g l. (2.6) ν=1 γ (k) + lexp(q k,k+1,ν g ν ) γ (k) + α (k) α (k+1) = α, c c l 0, (2.4) (2.6) lexp( f) = (α, i),. c c l = 0. (2.4), (2.6) l 1 f = (c c k )x γ(k) q k,k+1,ν g ν + ν=1 17 l q k g k + q k g k k=l+1

24 . leading exponent r (α, i), (2.3). (2.3) (α, i) = lexp( f). : ( ) Input: a finite set G R r ; while ( ( f, g) G G such that lp( f) = lp( g) and r := red(sp( f, g), G) 0) G := G { r}; Output: G; , output G G R r R- N. : R r f R r N. N G, 3 : (1) f N; (2) red( f, G) = 0; (3) red( f, G) = 0., r := red( f, G), r = 0 f N; f 0 f N G R r. R r f = r i=1 α c αi x α e i G, c αi 0 (α, i) mono(e(g)). f G,. redlexp( f) := max r {(α, i) c αi 0, (α, i) mono(e(g))}, redlterm( f) := c αi x α ((α, i) := redlexp( f)) ( ) Input: f R r and a finite set G R r ; while ( f is not completely irreducible with respect to G){ Choose g G such that redlexp( f) lexp( g); f := f (redlterm( f)/lterm( g)) g; } Output: f; 18

25 , output f G G R r, f R r, f G ( g ) N R r R- S(N) := N n {1,..., r} \ E(N), R r /N K, K(S(N)) := (α,i) S(N) Kx α e i. (syzygy) R r G := { g 1,..., g s }, R s R- G (1 ). S( g 1,..., g s ) := {(f 1,..., f s ) R s f k g k = 0} G = { g 1,..., g s } R r R- N. lp( g i ) = lp( g j ) i j i, j {1,..., s}, sp( g i, g j ) = q ijk g k lexp(q ijk g k ) r lexp( g i ) lexp( g j ) ( q ijk = 0) q ijk R (cf ). lexp( g i ) = (α (i), ν i ), s ij := lcoef( g j )x α(i) α (j) α (i), (j) v ij := (0,..., s (i) ij,..., s ji,..., 0) (q ij1,..., q ijs ) R s, S( g 1,..., g s ) R V := { v ij i < j, lp( g i ) = lp( g j )}. : v ij S( g 1,..., g s ). a i := lcoef( g i ) a i = 1. (f 1,..., f s ) S( g 1,..., g s ) (a 1 f 1,..., a s f s ) S((1/a 1 ) g 1,..., (1/a s ) g s ), v ij k a k {(1/a 1 ) g 1,..., (1/a s ) g s } v ij ( 1/a i a j )., S( g 1,..., g s ) V., (f 1,..., f s ) S( g 1,..., g s ), V (R- ) (α, i) := max r {lexp(f k g k ) 1 k s, f k 0} r., (f 1,..., f s ), (α, i). 19

26 g 1,..., g s, 1 k l lexp(f k g k ) = (α, i), l < k s lexp(f k g k ) r (α, i) ( f k = 0). lterm(f k ) = c k x β(k), f k := f k lterm(f k ), (α (k), i) = lexp( g k ) (1 k l), l l 0 = f k g k = lterm(f k ) g k + f k g k + f k g k. (2.7) k=l+1 : = l l lterm(f k ) g k = c k x β(k) g k l 1 (c c k )(x β(k) g k x β(k+1) g k+1 ) + (c c l )x β(l) g l. (2.8) 1 k l α (k) + β (k) = α, α (k) α (k+1) α, γ (k) := α α (k) α (k+1) (2.7), (2.8) l lterm(f k ) g k = = l 1 l 1 (c c k )x γ(k) sp( g k, g k+1 ) + (c c l )x β(l) g l (c c k )x γ(k) q k,k+1,ν g ν + (c c l )x β(l) g l. (2.9) ν=1 γ (k) + lexp(q k,k+1,ν g ν ) γ (k) + α (k) α (k+1) = α, c c l 0, lexp( s f k g k ) = (α, i), (2.7). c c l = 0. l 1 0 = (c c k )x γ(k) q k,k+1,ν g ν + ν=1 l f k g k + f k g k k=l+1 { l 1 h ν := (c c k )x γ(k) q k,k+1,ν + f ν (ν = 1,..., l) l 1 (c c k )x γ(k) q k,k+1,ν + f ν (ν = l + 1,..., s) (h 1,..., h s ) S( g 1,..., g s ), lexp(h k g k ) r (α, i) (h 1,..., h s ) V. l 1 w = (w 1,..., w s ) := (c c k )x γ(k) v k,k+1 1 i, j l s ij = x α(i) α (j) α (i) 1 ν l w ν = (c c ν )x γ(ν) s ν,ν+1 (c c ν 1 )x γ(ν 1) s ν,ν 1 l 1 (c c k )x γ(k) q k,k+1,ν 20

27 = (c c ν )x α α(ν) (c c ν 1 )x α α(ν) l 1 (c c k )x γ(k) q k,k+1,ν l 1 = c ν x β(ν) (c c k )x γ(k) q k,k+1,ν = lterm(f ν ) l 1 (c c k )x γ(k) q k,k+1,ν ( s 1,0 = s l,l+1 = 0 ), l < ν s l 1 w ν = (c c k )x γ(k) q k,k+1,ν. w = (f 1,..., f s ) (h 1,... h s ). w (h 1,..., h s ) V. (f 1,..., f s ) V,. 3. (1) x 3 y 2, y 3 z 2, z 3 x 2 K[x, y, z]., K 0. (2) xy 1, x 2 y, y 2 x K[x, y]., K (1),(2). 5. G R r. f, g, h G lp( f) = lp( g) = lp( h), lexp( f) lexp( h) lexp( g) red(sp( f, g), G) = red(sp( g, h), G) = 0, red(sp( f, h), G) = , red(sp( f, h), G). 21

28

29 2 2.1,,.,,,, ( ),,.,. K, n 1, x = (x 1,...,x n ) K[[x]] = K[[x 1,..., x n ]], K C, K{x} = K{x 1,..., x n } ( K ). R := K[[x]] R := K{x}., R f f = α c α x α (c α K). δ 1,..., δ n δ = (δ 1,..., δ n ), α = (α 1,..., α n ) N n n δ(α) = δ i α i. δ (weight). N n δ. L ( ) α δ β δ(α) > δ(β) or (δ(α) = δ(β), α L β) i=1 δ. N n δ. δ = (1,..., 1). δ,.,. x g := x x 2 ( ) x g x 2 g x 3 g., (Weierstrass- ). 23

30 f = α c α x α R \ {0}, {α c α 0} δ β, lexp(f) := β, lcoef(f) := c β, lterm(f) := c β x β f leading exponent, leading coefficient, leading term. 2 1 : f, g R, lexp(f g) = lexp(f) + lexp(g), lcoef(f g) = lcoef(f)lcoef(g), lterm(f g) = lterm(f)lterm(g) f, g R, lexp(f + g) δ (max δ δ.) max δ {lexp(f), lexp(g)}. R S E(S) := {lexp(f) f S, f 0} I R, E(I) N n. : f I, β. x β f I lexp(x β f) = lexp(f)+β E(I), E(I) ( ) I R. I G I ( δ ) (standard basis), : (1) I G ; (2) E(I) = mono(e(g))., E(G) E(I) (cf ), G., f = α c α x α R exps(f) := {α c α 0} α (1),..., α (s) N n f R, f = q i x α(i) + r, i=1 exps(r) mono({α (1),..., α (s) }) = i, q i = 0 lexp(q i ) + α (i) δ lexp(f) q i R r R. 24

31 : N n L 1,..., L s, L 0 L 1 := {α N n α α (1) }, L i := {α N n α α (i) } \ L 0 := s N n \ L j j=1 i 1 L j j=1 (2 i s),. q i := α L i c α x α α(i) (1 i s), r := α L 0 c α x α (Weierstrass- ) G := {g 1,..., g s } R \ {0}. 0 f R f = q i g i + r, i=1 exps(r) mono(e(g)) = i q i = 0 lexp(q i g i ) δ lexp(f) q i R r R. ( r, r red(f, G), f G WH-.) : (1), R. α lexp(0) δ α ( lexp(0) ). q i r. lterm(g i ) = x α(i) ( lcoef(g i ) = 1). g i := g i lterm(g i ) f = i=1 q (0) i lterm(g i ) + r 0, exps(r 0 ) mono(e(g)) = (1.1) lexp(q (0) i ) + α (i) δ lexp(f) q (0) i, r 0 R. k 1 i=1 lexp(q (k) i. q (k 1) i g i = i=1 g i ) δ max δ {lexp(q (k 1) q (k) i lterm(g i ) + r k, exps(r k ) mono(e(g)) = (1.2) j g j) 1 j s} q (k) i, r k R β (k) := max δ {lexp(r k ), lexp(q (k) 1 ) + α (1),..., lexp(q (k) s ) + α (s) }. lexp(f) = β (0) δ β (1) δ β (2) δ.... (1.2) k 1 β (k) δ max δ {lexp(q (k 1) j g j) j = 1,..., s} δ max δ {lexp(q (k 1) j g j ) j = 1,..., s} δ β (k 1). δ(β (0) ) δ(β (1) ) δ(β (2) )..., δ(α) = c α lim k δ(β (k) ) =. δ 0 := max{δ 1,..., δ n } α (1/δ 0 )δ(α), lim k β (k) =. 25

32 (1.1) (1.2) k 0, 1,..., k 1 f = = i=1 i=1 j=0 q (0) k 1 i x α(i) + r 0 + k 1 q (j) i g i + i=1 j=0 i=1 q (k) i x α(i) + q (j) i g i + k r j j=0 k j=1 i=1 q (j) i x α(i) + k r j j=1,, k f = ( q (k) i )g i + r k i=1 k=0 k=0. q i := k=0 q (k) i, r = k=0 r k. lexp(q i g i ) δ β (0) = lexp(f). (2) R = K{x} (K C) : ρ > 0 f = α c α x α f := α c α ρ δ(α). f < f x i ρ δ i. δ(lexp(g i)) < δ(lexp(g i ))., ε k δ (δ 1 + ε 1,..., δ n + ε n ), lterm(g i ) lterm(q i g i ),. d i := δ(α (i) ), f < ρ > 0, 2.1.8( ) q i ρ d i = c α ρ δ(α) f α L i. d i := δ(lexp(g i)), g i C i ρ d i ρ > 0 C i 0. C := max{c i i = 1,..., s}., ε := min{d i d i }, ε > 0. q (k) i. (1.1) q (0) i ρ d i f (i = 1,..., s). q (k 1) i ρ d i (Csρ ε ) k 1 f (i = 1,..., s) (1.2) q (k) i ρ d i j=1 j=1 q (k 1) j g j (Csρ ε ) k 1 f ρ d j Cρ d j (Csρ ε ) k f. ρ > 0, Csρ ε < 1/2 q (k) i 2 k ρ d i f (i = 1,..., s, k = 0, 1,...) 26

33 . i = 1,..., s q i k=0 q (k) i 2ρ d i f < q i. r = f s i=1 q i g i Weierstrass. R = C{x} g R x 2 =... = x n = 0 j g(x 1, 0,..., 0) = c j x 1 j=m (c m 0). δ = (1/2m, 1,..., 1) lexp(g) = (m, 0,..., 0), 2.1.9, f R, f = qg + r, r = m 1 j=0 r j (x 2,..., x n )x 1 j (r j C{x 2,..., x n }) q, r R. Weierstrass, Späth. 1. G, f R f G WH-. 2. R = C{x, y}, δ = (1, 1). G = {xy y 2 x 2 y}, f = xy f G WH I R, G I mono(e(g)) = E(I), G I. : G I. G = {g 1,..., g s }. f I, WH- r := red(f, G). r lexp(r) mono(e(g)) = E(I), r I. r = 0, q 1,..., q s R f = s j=1 q s g s. I G I R I. : Dickson, E(I) N n S. S α lexp(f) = α f I, I G {lexp(f) f G} = S. mono(e(g)) = mono(s) = E(I), G I R., R I, J R, I J E(I) = E(J), I = J. 27

34 : G I, G J. I = J (S- ) f, g R, α := (lexp(f) lexp(g)) lexp(f), β := (lexp(f) lexp(g)) lexp(g), f g S- (S-series) sp(f, g). sp(f, g) = lcoef(g)x α f lcoef(f)x β g lexp(sp(f, g)) δ lexp(f) lexp(g) G = {g 1,..., g s } R, I G R, (1),(2),(3) : (1) G I ; (2) f I f G WH- red(f, G) = 0 ; (3) g i, g j G, q ij1,..., q ijs R sp(g i, g j ) = q ij1 g q ijs g s k = 1,..., s, q ijk = 0 lexp(q ijk g k ) δ lexp(g i ) lexp(g j ). : (1) (2): f I r := red(f, G) I, r 0 lexp(r) mono(e(g)) = E(I) r = 0. (2) (3): sp(g i, g j ) I. (3) (1): G = {g 1,..., g s }. lcoef(g k ) = 1 (k = 1,..., s). (3), i j {1,..., s} (i, j), q ij1,..., q ijs R, sp(g i, g j ) = q ijk g k, (1.3) q ijk 0 lexp(q ijk g k ) δ lexp(g i ) lexp(g j ). f I. lexp(f) mono(e(g))., q 1,..., q s R f = s q k g k k q k = 0 lexp(q k g k ) δ lexp(f)., k lexp(f) = lexp(q k g k ) mono(e(g))., f f = q k g k, (q 1,..., q s R) (1.4) 28

35 ,, max δ {lexp(q k g k ) 1 k s, q k 0} δ, (1.4). f I, lexp(f) δ max δ {lexp(q k g k ) 1 k s, q k 0}, N n. ( α δ β δ(α) δ(β), α {β N n β δ α}.)., (1.4) α := max δ {lexp(q k g k ) 1 k s, q k 0} , α = lexp(f). lexp(f) = α mono(e(g)) G R, f R G WH- (G ) : f G WH- r 1,r 2. G I. exps(r 1 r 2 ) exps(r 1 ) exps(r 2 ), exps(r 1 r 2 ) mono(e(g)) =. r 1 r 2 I r 1 r 2 0 lexp(r 1 r 2 ) mono(e(g)) = E(I), lexp(r 1 r 2 ) exps(r 1 r 2 ), I R S(I) := N n \ E(I), R/I K, R S(I) := {f R exps(f) S(I)}. : K- ϕ : R S(I) R/I f = α S(I) c α x α f R/I [f]. (K-.) G I. (1) ϕ : ϕ(f) = 0 f I. f 0 exps(f) E(I) = lexp(f) E(I). f = 0. (2) ϕ : f R f G WH- r. f r I r R S(I) ϕ(r) = [r] = [f] I R, G I (dim K K, ): (1) dim K (R/I) = S(I) < ; (2) i = 1,..., n α i N, (0,..., (i) α i,..., 0) E(G) R/I K, Hilbert. Hilbert, I ( )..,. 29

36 f = α c α x α R d := min{δ(α) c α 0}, δ f initial part in(f) := c α x α K[x] δ(α)=d. f d δ- c α 0, δ(α) = d. R I {in(f) f I} K[x] in(i) δ α δ β δ(α) < δ(β) or (δ(α) = δ(β), α L β). I R, G = {g 1,..., g s } δ I, in(g) := {in(g 1 ),..., in(g s )} K[x] in(i) δ. : (1) in(g) , g i, g j G, q ij1,..., q ijs R sp(g i, g j ) = q ij1 g q ijs g s (1.5) k = 1,..., s, q ijk = 0 lexp(q ijk g k ) δ lexp(g i ) lexp(g j ). α (i,j) := lexp(g i ) lexp(g j ) lexp(g i ), s ij := lcoef(g j )x α(i,j), d ij := δ(lexp(g i ) lexp(g j )) (1.5), α = d j α s ij in(g i ) s ji in(g j ) = k L(d ij ) in(q ijk )in(g k ), (1.6) L(d ij ) := {k {1,..., s} δ(lexp(q ijk g k )) = d ij },. h R, in(h) δ leading term δ leading term, (1.6) in(g) K[x] δ. (2) in(g) in(i) : , f I f = q 1 g q s g s δ(lexp(q k g k )) δ(lexp(f)) q k = 0. initial part in(f) in(g) R G := {g 1,..., g s }, R s R- S(g 1,..., g s ) := {(f 1,..., f s ) R s f k g k = 0} G (1 ). 30

37 R = K[[x]]. G = {g 1,..., g s } R I. i, j {1,..., s}, sp(g i, g j ) = q ijk g k lexp(q ijk g k ) δ lexp(g i ) lexp(g j ) ( q ijk = 0) q ijk R (cf ). lexp(g i ) = α (i), s ij := lcoef(g j )x α(i) α (j) α (i), (j) v ij := (0,..., s (i) ij,..., s ji,..., 0) (q ij1,..., q ijs ) R s, S(g 1,..., g s ) R V := { v ij 1 i < j s}. : v ij S(g 1,..., g s ). f = (f1,..., f s ) S(g 1,..., g s ). d 0 := min{δ(lexp(f k g k )) k = 1,..., s}, L := {k {1,..., s} δ(lexp(f k g k )) = d 0 } in(f k )in(g k ) = 0 k L. k L f k := in(f k ), k L f k := 0, (f 1,..., f s) K[x] in(g 1 ),..., in(g s ). (j) v ij := (0,..., s (i) ij,..., s ji,..., 0) (q ij1,..., q ijs) R s,. δ(lexp(q ijk g k )) = δ(α (i) α (j) ) q ijk := in(q ijk ), q ijk := , u (0) ij K[x] (f 1,..., f s) = i<j u (0) ij v ij (1.7). d ij := δ(α (i) α (j) ) f k d 0 δ(α (k) ) δ-, v ij k d ij δ(α (k) ) δ-, u (0) ij d 0 d ij δ-. f (1) = (f (1) 1,..., f s (1) ) := f u (0) ij v ij i<j, (f (1) 1,..., f (1) ) S(g 1,..., g s ) (1.7) s d 1 := min{δ(lexp(f (1) ν g ν )) 1 ν s} > d 0. S(g 1,..., g s ) { f (k) } d k d ij δ-, u (k) ij f = f (k) + i<j 31 k ν=1 u (ν) ij v ij (1.8)

38 , {d k }. d k := min{δ(lexp(f (k) ν g ν )) 1 ν s}. k, (1.8), lim k f (k) = 0, f = i<j ν=1 u (ν) ij v ij R ,., K R := K[[x]] R := K{x} (K C ), δ = (δ 1,..., δ n ) (δ i > 0). R r R-. R r f r r f = (f 1,..., f s ) = f i e i = c αi x α e i (c αi K) (2.1) i=1 i=1 α. N n {1,..., r} δr (α, β N n, i, j {1,..., r} ): (α, i) δr (β, j) (δ(α) > δ(β)) or (δ(α) = δ(β), i < j) or (δ(α) = δ(β), i = j, α L β). (2.1) f R r, exps( f) := {(α, i) c αi 0}, f 0 f leading exponent lexp( f) := max δr exps( f). (α, i) := lexp( f), f leading point, leading term, leading coefficient lp( f) := i, lterm( f) := c αi x α, lcoef( f) = c αi f, g R r, lexp( f + g) δr max δr {lexp( f), lexp( g)}. 32

39 R r S E(S) := {lexp( f) f S, f 0} N R r R-, E(N) N n {1,..., r} ( ) N R r R-. N G N ( δr ) ( ), : (1) N R G ; (2) E(N) = mono(e(g))., E(G) E(N), G G = { g 1,..., g s } R r, f R r f = q 1 g q s g s + r, exps( r) mono(e(g)) = i, q i = 0 lexp(q i g i ) δr lexp( f) q 1,..., q s R r R r. r f G WH-, r = red( f, G) ( r ). : N R r R-, G N mono(e(g)) = E(N), G N (S- ) f, g R, lexp( f) = (α, i), lexp( g)) = (β, j), f, g S- ( ) sp( f, g), i = j i j sp( f, g) = 0. sp( f, g) = lcoef( g)x α β α f lcoef( f)x α β β g; lp( f) = lp( g) lexp(sp( f, g)) δr lexp( f) lexp( g). (α, i) N n {1,..., r} δ((α, i)) := δ(α) f = α,i c α,i x α e i R r d := δ(lexp( f)), f initial part in( f) r := c α,i x α e i. i=1 δ(α)=d G = { g 1,..., g s } R, N G R r R-, (1) (3) : 33

40 (1) G N ; (2) f N f G WH- red( f, G) = 0 ; (3) g i, g j G, lp( g i ) = lp( g j ), q ij1,..., q ijs R sp( g i, g j ) = q ij1 g q ijs g s k = 1,..., s, q k = 0 lexp(q ijk g k ) δr lexp( g i ) lexp( g j ). : (3) (1). (3), f N lexp( f) E(G)., f = q i g i (2.2) i=1 lexp( f) δr max δr {lexp(q i g i ) 1 i s, q i 0} (2.3), (2.2) (2.3) δr. (β, ν) {(α, µ) N n {1,..., r} (β, ν) δr (α, µ)}, N R r R- S(N) := N n {1,..., r} \ E(N), R r /N K, (R r ) S(N) := { f R r exps( f) S(N)} R r G := { g 1,..., g s }, R s R- G (1 ). S( g 1,..., g s ) := {(f 1,..., f s ) R s f k g k = 0} R = K[[x]], G = { g 1,..., g s } R r R- N. lp( g i ) = lp( g j ) i j i, j {1,..., s}, sp( g i, g j ) = q ijk g k lexp(q ijk g k ) δr lexp( g i ) lexp( g j ) ( q ijk = 0) q ijk R (cf ). lexp( g i ) = (α (i), ν i ), s ij := lcoef( g j )x α(i) α (j) α (i), (j) v ij := (0,..., s (i) ij,..., s ji,..., 0) (q ij1,..., q ijs ) R s, S( g 1,..., g s ) R V := { v ij i < j, lp( g i ) = lp( g j )}. 34

41 1.,,. 2. N n {1,..., r} δr (α, i) δr (β, j) (i < j) or (i = j, α δ β),,,. ( R r ( ).) 2.3 Tangent cone δ = (δ 1,..., δ n ). δ R n δ Q n, δ Z n,., R = K[[x]] R = K{x} I, I,. tangent cone. f 1,..., f s (tangent cone) f 1,..., f s R ( K[x]) I, δ = (1,..., 1) in(i) (cf ). G I, , in(i) in(g).,. Lazard [La] ([Mo2] ). Mora [Mo1], tangent cone algorithm,,. δ := (1, δ 1,..., δ n ) N n f = α c α x α 0, (homogenization) f h K[x 0, x] = K[x 0, x 1,..., x n ], d := deg δ (f) := max{δ(α) a α 0}, f h (x 0, x) := α c α x d δ(α) 0 x α. f h d δ-, f h (1, x) = f(x). N n+1 h : α, β N n, i, j N (i, α) h (j, β) (i + δ(α) < j + δ(β)) or (i + δ(α) = j + δ(β), i < j) or (i = j, δ(α) = δ(β), α L β). δ = (1,..., 1) h N n+1 -., n f R leading exponent lexp(f), n+1 g h leading exponent lexp h (g). 35

42 δ- n + 1 f = f(x 0, x) K[x 0, x], i lexp h (f) = (i, lexp(f(1, x))). : (i, α), (j, β) N n+1, i + δ(α) = j + δ(β), i < j δ(α) > δ(β) n f, g K[x], (fg) h = f h g h. : f = α a α x α, g = β b β x β d := deg δ (f), d := deg δ (g) deg δ (fg) = d + d (fg) h = d+d a α b β x δ(α+β) 0 x α+β α,β = α a α x 0 d δ(α) x α β b β x 0 d δ(β) x β = f h g h f 1,..., f m K[x 0, x] δ-, f 1,..., f m K[x 0, x] h δ-. : δ- S- δ-. δ- δ- δ input {f 1,..., f m } output δ f 1,..., f m K[x], (f 1 ) h,..., (f m ) h K[x 0, x] J. J h {g 1 (x 0, x),..., g s (x 0, x)}, G := {g 1 (1, x),..., g s (1, x)} R f 1,..., f m Ĩ δ ( )., g 1,..., g s δ-. : G Ĩ. f 1,..., f m K[x] I. Ĩ I R, G K[x] I. G I. g i J, q 1,..., q m K[x 0, x], x 0 = 1 G I. g i = q 1 (f 1 ) h + + q m (f m ) h g i (1, x) = q 1 (1, x)f 1 (x) + + q m (1, x)f m (x) I. 36

43 , f I q 1,..., q m K[x] f = q 1 f q m f m. d := deg δ (f), d i := deg δ (f i ), e i := deg δ (q i ), e := max{d i + e i 1 i m}, e d m x e d 0 f h e d = x i e i 0 (q i ) h (f i ) h J i=1. J {g 1,..., g s } p 1,..., p s K[x 0, x] x e d 0 f h = p k g k. x 0 = 1 f(x) = (f h )(1, x) = p k (1, x)g k (1, x), I G. {lexp(f) f I \ {0}} = mono(e(g)) (3.1). E(G) G δ leading exponents. f I., e N x 0 e f h J., {g 1,..., g s } h J, lexp h (x 0 e f h ) mono({lexp h (g 1 ),..., lexp h (g s )}) (3.2) 2.3.2, i N (i, lexp(f)) = lexp h (x 0 e f h ), (3.2) lexp(f) mono({lexp(g 1 (1, x)),..., lexp(g s (1, x))}) = mono(e(g)). (3.1). E(Ĩ) := {lexp(f) f Ĩ \ {0}} = mono(e(g)). f Ĩ \ {0} q 1,..., q s R f(x) = q 1 (x)g 1 (1, x) + + q s (x)g s (1, x). d := δ(lexp(f)), q i = α c iα x α q i := δ(α) d c iα x α K[x], f (x) := q 1(x)g 1 (1, x) + + q s(x)g s (1, x) I 37

44 f (x) := f(x) f (x) = (q 1 (x) q 1(x))g 1 (1, x) + + (q s (x) q s(x))g s (1, x) δ(α) d x α lexp(f) = lexp(f + f ) = lexp(f ), (3.1) lexp(f) = lexp(f ) E(I). E(Ĩ) E(I), E(Ĩ) E(I), (3.1), E(Ĩ) = E(I) = mono(e(g))., R,., f 1,..., f m K[x], dim K (R/Rf Rf m )., f K[x], Milnor µ := dim R/R( f/ x 1 ) + + R( f/ x n ). Milnor, ( [Kan] )., 2.3.5, f K[x], f Ĩ, f, f 1,..., f m R Ĩ, E(Ĩ ) = E(Ĩ)., ( ), ([KFS]) , i, j {1,..., s} sp(g i, g j ) = s ij (x 0, x)g i (x 0, x) s ji g j (x 0, x) = q ijk (x 0, x)g k (x 0, x) (3.3) lexp h (q ijk (x 0, x)g k (x 0, x)) h lexp h (g i (x 0, x)) lexp h (g j (x 0, x)) (3.4) δ- q ijk. G S(g 1 (1, x 0 ),..., g s (1, x 0 )) := {(h 1,..., h s ) R s h k (x)g k (1, x) = 0}, { v ij 1 i < j s} R.. (i) (j) {}}{{}}{ v ij := (0,..., s ij (1, x),..., s ji (1, x),..., 0) (q ij1 (1, x),..., q ijs (1, x)) : (3.4) lexp(q ijk (1, x)g k (1, x)) lexp(g i (1, x)) lexp(g j (1, x)). (3.3) x 0 =

45 δ = (1, 1, 1), f 1 := x y 2, f 2 := xy z 2 f 1, f 2 K[[x, y, z]] C{x, y, z} I. (K 0.) x 0 = t, (f 1 ) h = tx y 2, (f 2 ) h = xy z 2, K[t, x, y, z] -, {(f 1 ) h, (f 2 ) h, g 3 }, g 3 := tz 2 y 3. G := {f 1, f 2, z 2 y 3 } I ( ). lexp(g) = {(1, 0, 0), (1, 1, 0), (0, 0, 2)}, {x y 2, z 2 y 3 } I. f 1 = f 2 = 0 x = z 2 = : x = x 1, y = x 2, z = x 3 f 1 := x 3 y 2, f 2 := y 3 z 2, f 3 := z 3 x 2. δ = (1, 1, 1) I := Rf 1 + Rf 2 + Rf {f 1, f 2, f 3 } I. dim K R/I = K[x] r R r. 39

46

47 3 3.1 x = (x 1,..., x n ), K, R, K[x], K(x), K[[x]], K{x}, U C n, U O(U), U O alg (U) := O(U) C(x),. R K C ( ). R i := / x i (i = 1,..., n) (derivation). i : R R K- f, g R i (fg) = f( i g) + ( i f)g. R,, R,., K K p > 0 R = K[x], m p f R i m f = R K- ( )K- End K (R). R- n (ring of differential operators) R, R 1,..., n End K (R) K-. R R-., A n (K) := K[x], K Weyl. K = C Weyl A n := A n (C)., D 0 := C{x} (0 C n ). R R -. R, a R R i a = a i + a x i (i = 1,..., n), (1.1) i j = j i (i, j = 1,..., n) (1.2). R R K-, 1 (1.1) i x i = x i i + 1, R. R. 41

48 R P (1.1), (1.2) 1,..., n R ( ), P = P (x, ) = β N n a β β (a β R) (1.3). β = (β 1,..., β n ) β = 1 β1 n β n P R (1.3), P = 0, β a β = 0. : P End K (R) 0 P 0 l := min{ β a β 0} α N n, α = l β > l, β = l α β, β i > α i i β (x α ) = 0 0 = P x α = β l a β β (x α ) = α 1! α n!a α l P = 0 ( ), ξ = (ξ 1,..., ξ n ) (x ) R- P P (1.3) P (x, ξ) = β N n a β (x)ξ β, R- R[ξ] = R[ξ 1,..., ξ n ]. ( P (total symbol).), R K R[ξ]. P P (x, ) R, P (x, ξ) R[ξ]. m := max{ β a β 0} P (order) ord(p ). σ(p ) = σ m (P ) := a β ξ β R[ξ] β =m P ( m ) (principal symbol). l > m σ l (P ) = (Leibniz ) P, Q R, S = P Q, S(x, ξ) = ( ) ( ) 1 ν ν P (x, ξ) Q(x, ξ) ν N ν! ξν xν n (1.4). ν = (ν 1,..., ν n ) ν! = ν 1! ν n!. P (x, ξ) ξ, (1.4). 42

49 : (1) P = k i, Q = a R (1.4) k. k = 0 P = 1. k 1 (1.4), S k := ik a S k 1 (x, ξ) = k 1 ν=0 S k (x, ) = i S k 1 (x, ) = = k 1 ν=0 k 1 ν=0 1 ν! (k 1) (k ν)ξ i k 1 ν ν a x ν i ( (k 1) (k ν) ν a i ν! (k 1) (k ν) ν! ) x ν i k 1 ν i ( ν a x ν i k ν + ν+1 a i x ν+1 i k 1 ν i ), S k (x, ξ) = = k 1 ν=0 k 1 ν=0 = aξ i k + = k ν=0 (k 1) (k ν) ν! (k 1) (k ν) ν! k 1 ν=1 1 ν k ξ i ν a ν ν! ξ i x ν i ( ν ) a x ξ ν i k ν + ν+1 a i x ξ ν+1 i k 1 ν i ν a x ξ ν i k ν + i k(k 1) (k ν + 1) ν! k ν=1 (k 1) (k ν + 1) (ν 1)! ν a x ξ ν i k ν + k a i x k i ν a x ξ ν i k ν i, (1.4). (2) P = β, Q = a R, (1.4) : S β := β a (1) S β = β 1 β2 β 2 n n ν 1 =0 β 2 β 1 (β 1 1) (β 1 ν 1 + 1) ν 1! β 1 β3 β = 3 n β 2 (β 2 1) (β 2 ν 2 + 1) n ν 2 =0 ν 1 =0 ν 2! ν ) 1+ν 2 a β x ν 1 x2 ν 2 2 ν2 β 2 1 ν = = ν=(ν 1,,ν n ) β 1 β 1 ν1 n β n ν n ν 1 a β x ν 1 1 ν β 1 (β 1 1) (β 1 ν 1 + 1) ν 1! β 1 (β 1 1) (β 1 ν 1 + 1) β n (β n 1) (β n ν n + 1) ν!. total symbol, (1.4). 43

50 (3), P = β N n a β β, Q = β N n b β β S = a α α b β β α,β, S αβ := α b β (2) (1.4). S αβ (x, ξ) = ν N n 1 ν! ( ν ξ α ξ ν ) ( ν ) b β x ν S(x, ξ) = a α S αβ (x, ξ)ξ β α,β = ( 1 ν ξ α ) ( ν ) b β a α ξ β α,β ν ν! ξ ν x ν = ( 1 ν (a α ξ α ) ( ) ν (b β ξ β ) ) ν ν! ξ ν x ν α,β P, Q R ord(p Q) = ord(p ) + ord(q) σ(p Q) = σ(p )σ(q) (R[ξ] ). : ord(p ) = m, ord(q) = l, α P (x, ξ)/ ξ α ξ m α, ord(p Q) m + l, (1.4) σ m+l (P Q) = σ(p )σ(q), R 0. ord(p Q) = m + l σ(p Q) = σ m+l (P Q) = σ(p )σ(q). 2. k i = 1,..., n, R. x i k i k = x i i (x i i 1) (x i i k + 1). x = (x 1,..., x n ) K n, y = (y 1,..., y n ) ( ) x y i = F i (x) (i = 1,..., n), x j = G j (y) (j = 1,..., n). R : 44

51 (1) R = K[x], K(x) F i, G i R (i = 1,..., n); (2) R = K[[x]], K{x} F i, G i R F i (0) = G i (0) = 0( 0) (i = 1,..., n); (3) R = O(U) ( R = O alg (U)) F i R V G i O(V ) ( G i O alg (V )) (i = 1,..., n). (1),(2) R := R ( R x y ), (3) R := O(V ) R := O alg (V ). F := (F 1,..., F n ) G := (G 1,..., G n ) F : R g(y) g(f (x)) R, G : R f(x) f(g(y)) R,. P R g(y) R F (P ) End K (R ) F (P )g(y) := (F ) 1 P (F (g)(x)) = G (P g(f (x))) R. = ( 1,..., n) = ( / y 1,..., / y n ) F R R. : F R End K (R ). R. P = a(x) R, g(y) R F (a)g(y) = G (a(x)g(f (x))) = a(g(y))g(y) F (a) = a(g(y)) R R. g(y) R F ( i )g(y) = G ( i g(f (x))) n = G ( jg)(f (x)) F j (x) j=1 x i n = ( jg)(y) F j (G(y)) j=1 x i F ( i ) = n j=1 F j x i (G(y)) j R (1.5). R R, F R. G, F G,. η = (η 1,..., η n ) y. 45

52 P R σ(f (P ))(y, η) = σ(p ) ( G(y), F ) x (G(y))η. F n F (x)η := j n F j (x)η j,..., (x)η j x j=1 x 1 j=1 x n. : P R (1.3). m := ord(p ) (1.5) σ(f (P ))(y, η) = F (a β )σ(f ( β )) β =m = 1 n F a β (G(y)) j n F (G(y))η β j j (G(y))η j β =m j=1 x 1 j=1 x n ( = σ(p ) G(y), F ) x (G(y))η. K n (cotangent bundle) T K n T K n := {(x, ξ 1 dx ξ n dx n ) x K n, ξ 1,..., ξ n K}. x y, dx i dy 1,..., dy n., R K = C K = R. P R σ(p ) T K n., (x, ξ 1 dx ξ n dx n ) = (x, ξ 1,..., ξ n ). β n 3.2,., ( [HU], [Kaw] ). X, F X (presheaf), X U F(U) (F( ) = {0} ), U V U, V, ρ V U : F(U) F(V ) ( ), ρ UU, U V W, ρ W V ρ V U. f F(U) ρ V U (f) = f V. F (sheaf), {U λ } λ Λ X U := λ Λ U λ, (S1), (S2) ( ) : 46

53 (S1) f F(U) λ Λ f Uλ = 0, f = 0; (S2) λ Λ f λ F(U λ ), U λ U µ f λ Uλ U µ = f µ Uλ U µ, λ f Uλ = f λ f F(U). F, F(U), U V ρ V U, F (sheaf of rings). R X, X F ( ) R- (sheaf of R-modules) ( ) R- (R-module), F(U) ( ) R(U)-, U V f F(U) a R(U) ρ V U (af) = ρ V U (a)ρ V U (f). F(U) F U (section). f F(U), f F X := C n (holomorphic function) O = O X, U X, O(U) U. (.) O C n. X F p X, F p (stalk) U p (inductive limit) F p := lim F(U) = U p F(U)/. f F(U) g F(V ) (f g), p W U V f W = g W. p U ρ p,u : F(U) F p. f F(U) ρ p,u (F) = f p, f p (germ) C n, O 0 = C{x} X F (support) ( 0 {0} ). Supp(F) := {p X F p 0} F X, X F, p X lim F (U) = F p ( p ). F F (sheafification)., X, U, F U F (U). 47

54 (1) A,,, X U A (U ) A. U U = λ U λ, A(U) = λ A ( ). (S1),(S2). p X A p = lim A = A (U p ). (2) C n D U O(U) (U ). U U = λ U λ, D(U) = λ O(U λ ). D D 0 = C{x}. O D-. D O-. (3) m D(m) U {P O(U) ord(p ) m} (U C n ). D(m) D O-, D-. (4) C n (regular function) O alg, U O(U) C(x) (U ). U U = λ U λ, O alg (U) = λ(o(u λ ) C(x)). p C n { } f (O alg ) p = lim(o(u) C(x)) = g f, g C[x], g(0) 0 (U p ). (5) C n D alg U O alg (U) (U ). O alg D alg -. D alg O alg O alg (C n ) = C[x], D alg (C n ) = A n (Weyl ). R X, F, G X R-. G F (R- ) (subsheaf), U G(U) F(U) (R(U)-) (G F ).,, U F(U)/G(U) H F G (quotient sheaf) F/G. H R-. R X, F, G R-. ϕ : F G X R-, U R(U)- ϕ(u) : F(U) G(U), U V, ϕ(v ) ρ V U = ρ V U ϕ(u). ρ V U G. p U, R p - ϕ p : F p G p., ϕ(u) ϕ p ϕ., R- Ker ϕ Im ϕ, ϕ (kernel), (image) : (Ker ϕ)(u) := {f F(U) ϕ(u)(f) = 0}, (Im ϕ)(u) := {g G(U) p U, f p F p s.t. g p = ϕ p (f p )}. 48

55 Im ϕ U Im ϕ(u) X, Y, f : X Y. (1) F X, V F(f 1 (V )) (V Y ) Y. f F, F f (direct image). (2) G Y, X U lim G(V ) (U X ) ( f(u) Y V ) f 1 G, G f (inverse image). p X (f 1 G) p = G f(p). R X, F, G, H X R-, F ϕ G ψ H X R- (exact sequence), ϕ, ψ X R- Im ϕ = Ker ψ., p X F p ϕ p ψ p Gp Hp R p -. 0 F G H 0 R- F G R- H = G/F R X, F R-. (1) F X (locally finitely generated), X p p U r, U R- ϕ R r ϕ F 0 U R- ( Im ϕ = F). R r V r R(V ) r. (2) F ( R- ) (coherent sheaf), ( ) R-, F, r, U U R- ϕ : R r F Ker ϕ U. R ( ) R ( ) R-. 49

56 (2) e 1,..., e r r R(U) r g i := ϕ(u)( e i ) F(U), V U f := (f 1,..., f r ) R(V ) r ϕ(v )( f) r r = ϕ(v )( f i e i ) = f i (g i V ) F(V ). i=1 r (Ker ϕ)(v ) = {(f 1,..., f r ) R(V ) r f i (g i V ) = 0} i= R X, F X R-, F Supp(F) X. i=1 : p X. p Supp(F). p U U R- R r ϕ F 0. e 1,..., e r r f i := ϕ( e i ) F(U), F p = 0 p V U f i V = 0 (i = 1,..., r). q V, F q R q - f 1,..., f r q F q = 0. Supp(F) (Serre ) R X. 0 F G H 0 X R-. F, G, H R X. F R r R-, X F R-. : R s F R- ϕ R s R r R-, R r. F = H = R, G = R 2, R 2. R r., : (1) O C n ( ); (2) O alg C n (J.P. Serre); (3) D C n ( ). 50

57 (1) ([HU],[Hi],[Hö] ). 2 Weierstrass. (3) (1). [Kash1], [Bj1], [Bj2]. O- (coherent analytic sheaf), O alg - (coherent algebraic sheaf). 2. ( ) O alg C n R X, F, G X R- F G R- ϕ : F G X U R(U)- ϕ(u) : F(U) G(U) U, V U V ϕ(v ) ρ V U = ρ V U ϕ(u) ρ V U F ρ V U G X U F U G U R U - Hom R (F, G)(U) Hom R (F, G) F G R R X, F, G, H, I X R-. (1) F G H 0 X R-, Hom R (F, I) Hom R (G, I) Hom R (H, I) 0 X. (2) 0 F G H X R-, 0 Hom R (I, F) Hom R (I, G) Hom R (I, H) X... R, R X, R R. F X R-., F := R R F U R(U) R(U) F(U). F R R, R, X R- R-. F G H 0 R R F R R G R R H 0 51

58 R, R, F X R- F := R R F X R-. :, p, p U R- R r F 0. U R- R r F 0. F R, R, R R ( R- ) (flat) X R- 0 F G H 0 0 R R F R R G R R H 0 X R-. p X R p R p C n O O alg, C[x]. D D alg, A n. 7 ( ). 3.3 D- X C n. M X D (1), (2), p X p U D- 0 M ϕ D r ψ D s (3.1). e 1,..., e r r-, e 1,..., e s s- u j M(U) P ij D(U) u j := ϕ(u)( e j ) M(U) (j = 1,..., r), r ψ( e i ) = P ij e j = (P i1,..., P ir ) (i = 1,..., s) j=1. ϕ ψ = 0 u 1,..., u r M(U) r P ij u j = 0 (i = 1,..., s) (3.2) j=1 52

59 . F D- ( O, D, Schwartz distribution, (hyperfunction) ) Hom D (M, F) Hom D (D r, F) Hom D (D s, F), f Hom D (D r, F)(V ) (f( e 1 ),..., f( e r )) F(V ) r Hom D (D r, F) F r 0 Hom D (M, F) ϕ F r ψ F s (3.3). ψ, V U (f 1,..., f r ) F(V ) r r r ψ (V )((f 1,..., f r )) = P 1j f j,, P sj f j F(V ) s j=1. (3.3) Hom D (M, F)(V ) (Ker ψ )(V ) = r {(f 1,..., f r ) F(V ) r P ij f j = 0 (1 i s)} j=1., M D- u 1,..., u r M (3.2), f M F D-, f j := f(u j ) i = 1,..., s F r r P ij f j = f P ij u j = 0 (i = 1,..., s) j=1 j=1. f 1,..., f r F, j f(u j ) = f j D- f., D- M U, ( M ) u 1,..., u r (3.2), Hom D (M, F) F ( u 1,..., u r F ). M, (3.1),, (3.2) ( r ), Hom D (M, F) (3.1), M F D-., X X D-. (3.1) U. P ij D(X)(i = 1,..., s; j = 1,..., r). X D- N U X N (U) := { Q i (P i1,..., P ir ) Q 1,..., Q s D(U)} D(U) r i=1 53 j=1

60 M := D r /N. N = D(P 11,..., P 1r ) D(P s1,..., P sr ) D r N X D M X D-. X (3.1). e j M(X) u j, ϕ, ψ U X r ϕ(u)(a 1,..., A r ) = A j u j U M(U) j=1 ψ(u)(b 1,..., B s ) = B i (P i1,..., P ir ) D(U) r i=1 (A 1,..., A r D(U)), (B 1,..., B s D(U))., P ij D(X) (i = 1,..., r; j = 1,..., s) X D- M., M M : r P ij u j = 0 (i = 1,..., s) j=1. D- M (3.1) X, P ij M X D-. X D-, X (3.1). P ij A n = C[x] C n D- M. (A n ) r A n - N N := { Q i (P i1,..., P ir ) Q 1,..., Q s A n } i=1 M := (A n ) r /N M A n -, M M = D An M ( A n M C n )., ψ ((Q 1,..., Q s )) := s i=1 Q i (P i1,..., P ir ) 0 M (A n ) r ψ (A n ) s A n -, D An M D r ψ D s D-. D An M = D r /Im ψ = M.,, A n - M D D-., P ij C[ ] M ( ). 54

61 . M C n X D, p X U P ij D(U) M : r P ij u j = 0 (i = 1,..., s) j=1. v 1,..., v m M(U), p V U A ij D(V ) r v i V = A ij u j V (i = 1,..., m) j=1. v 1,..., v m M D-. M M := Dv Dv m M W {Q 1 (v 1 W ) + + Q m (v m W ) Q 1,..., Q m D(W )} (W V ). M V D-, M. M = M (V ) M M. M ( v 1,..., v m ) p 0 M ϕ D m D l. ϕ ϕ (Q 1,..., Q m ) = Q 1 v Q m v m D-. M,, D C n O D-. D- h : D O P D h(p ) = P 1 O ( P 1 1 O P ). P (a α O(U) ) (1.3) P 1 = a 0, P 1 = 0 Q 1,..., Q n D P = Q Q n n. P P 1 = 0, 0 O h D ψ D n C n D- ( ψ(q 1,..., Q n ) := Q Q n n ). O D- C n ( ) 1 u =... = n u = 0. U C n {f O(U) 1 f =... = n f = 0} = C O Hom D (O, O) C n C. 55

62 n = 1, λ C D- M λ M λ := D/D(x λ) = Du λ (u λ 1 D(C) M λ ). M λ C (x λ)u λ = 0. D- F : M λ+1 M λ F (Au λ+1 ) = Axu λ (A D). F well-defined, p C F D p - F : D p /D p (x λ 1) D p /D p (x λ). A D p (x λ 1), A = B(x λ 1) B D p., (1.1) 0 = F (Au λ+1 ) = AF (u λ+1 ) = Axu λ Ax = B(x λ 1)x = Bx(x λ) D p (x λ), F well-defined. (A Ax D D D- F D-.) λ 1 D- G : M λ M λ+1 G(Au λ ) = 1 λ + 1 A u λ+1 (A D). B D p A = B(x λ), A = B(x λ) = B (x λ 1) G D- well-defined. F (G(u λ )) = = = 1 λ + 1 F ( u λ+1) 1 λ + 1 xu λ = 1 λ + 1 (x + 1)u λ 1 λ + 1 (λ + 1)u λ = u λ G(F (u λ+1 )) = G(xu λ ) = 1 λ + 1 x u λ+1 1 = λ + 1 (λ + 1)u λ+1 = u λ+1 56

63 , F, G. λ 1 M λ M λ+1 D- C. v λ := F (u λ+1 ) M λ (C) M λ = Du λ = Dv λ, M λ v λ, M λ (x λ 1)v λ = 0. Hom D (M 0, O) 0 C, Hom D (M 1, O) 0 = 0 M 0 M 1 D-. 1. U C, a 1,..., a m O(U) P := m + a 1 m a m 1 + a m D(U), U D- M M := D/DP = Du (u 1 D ). P 1,..., P m D(U) m, U D- L P 1 := (, 1, 0,..., 0), P 2 := (0,, 1, 0,..., 0),... P m 1 := (0,..., 0,, 1), P m := (a m, a m 1,..., a 2, + a 1 ) L := D m /DP DP m = Dv Dv m (v i e i D m )., F (u) = v 1, G(Q 1 v Q m v m ) = (Q 1 + Q Q m m 1 )u (Q 1,..., Q m D) U D- F : M L, G : L M. M p C n U (3.1) D-. D p V U 0 M ϕ D r ψ D s χ D t D-. Q ij D(V ), A 1,..., A t D ( t ) t t χ(a 1,..., A t ) = A i Q i1,..., A i Q is = A i (Q i1,..., Q is ) i=1 i=1 57 i=1

64 . ψ χ = 0 Q ik P kj = 0 (i = 1,..., t; j = 1,..., r). F D- f j, g i F, r P ij f j = g i (i = 1,..., s) (3.4) j=1 r Q ki g i = Q ki P ij f j = 0 (k = 1,..., t) (3.5) i=1 j=1 i=1. (3.5) g 1,..., g s F (3.4) f 1,..., f r F ( ) O h D ψ D n χ i<j D (3.6) C n D-. D = {(P ij ) 1 i<j n ) P ij D}, i<j χ((p ij ) i<j ) = 1 i<j n P ij (0,..., (i) j,..., (j) i,..., 0) D n., { 1,..., n } C[ ] C[ξ], C h ψ C[ ] C[ ] n χ C[ ] i<j C[ ]-. h, ψ, χ h, ψ, χ D C[ ]. D C[ ], D C[ ], (3.6)., i f = g i (i = 1,..., n) D- F f,. j g i = i g j (1 i < j n) 58

65 3.4 M C n X D-. M 0 M O- X M = DM 0 ( U D(U)M 0 (U) = { m i=1 P i v i m N, P i D(U), v i M 0 (U)} )., M U X (3.1), M 0 := Ou Ou r M ( U O(U)u 1 U O(U)u r U ) U. M 0., M 0 k Z M k := D(k)M 0 ( k < 0 M k = 0 ). D(k) O- O, M k M O-., O- gr(m) gr(m) := (M k /M k 1 ) k=0. gr(m), O[ξ]- ( O[ξ] = O[ξ 1,..., ξ n ] U O(U)[ξ] ): P D(m) u M k P u D(m)M k = D(m)D(k)M 0 = D(m + k)m 0 = M m+k, [P u] M m+k /M m+k 1 P σ(p ) O[ξ]. gr(m) O[ξ], O[ξ]-. T X := {(x, ξ) = (x, ξ 1 dx ξ n dx n ) x X, ξ = (ξ 1,..., ξ n ) C n } X, O T X T X (x, ξ). π : T X X π(x, ξ) = x π 1 O[ξ] O T X, O T X π 1 O[ξ]. X O[ξ]- F µ(f) := O T X π 1 O[ξ] π 1 F. T X O T X T X µ(gr(m)) M (characteristic variety) Char(M). Char(M) M M X D-. {M k } k Z M (X ) good filtration, (1) M k M O- ; 59

66 (2) k Z M k M k+1 l Z M l = 0; (3) k, l Z D(l)M k M k+l ; (4) k Z M k = M, p X k Z(M k ) p = M p ; (5) k 0 Z, k k 0 M k = D(k k 0 )M k0. M k /M k 1 O-. gr(m) gr(m) := k Z M k /M k 1, gr(m) D M O[ξ] k 0 D(k)/D(k 1) O[ξ]-. O[ξ]- gr(m) M filtration {M k } graded module. M 0 M = DM 0 M O-, M k := D(k)M 0 {M k } good filtration., (5) k 1 k 0 k k 1 M k = D(k k 0 )M k0 = D(k k 1 )D(k 1 k 0 )M k0 = D(k k 1 )M k1 k 0 k 1 (5) {M k } {M k} X D- M good filtration, p X, p U l N, k Z U M k l M k M k+l. : (5), k k 0 M k = D(k k 0 )M k0 M k = D(k k 0 )M k 0 k 0 N. p X, p U u 1,..., u r M k0 (U) U M k0 = Ou Ou r. v 1,..., v s M k 0 (U) U M k 0 = Ov Ov s. M k 0 M = M k = D(k k 0 )M k0 = Du Du r k k 0 k k 0, p U, P ij D(U) r v i = P ij u j (i = 1,..., s) j=1 60

67 . P ij l M k 0 D(l)M k0 = M k0 +l. M k 1 = 0 k 1 k 0. k Z M k M l+k0 k 1 +k., k k 0 M k = D(k k 0 )M k 0 D(k k 0 )M l+k0 = M l+k M l+k0 k 1 +k, k 1 < k < k 0 M k M k 0 M l+k0 M l+k0 k 1 +k, k k 1 M k = 0. l + k 0 k 1 l. M k M k M X D-, {M k } {M k} M good filtration gr(m) := M k /M k 1, gr (M) := M k/m k 1 k Z k Z, Supp(µ(gr(M))) = Supp(µ(gr (M))) = Char(M). : l N, k Z M k l M k M k+l. k Z M k+l /M k+l 1 = k Z M k /M k 1, {M k }, k Z M k l M k M k., l. (1) l = 1 : k Z, O-. 0 M k/m k 1 M k /M k 1 M k /M k 0, 0 M k 1 /M k 1 M k/m k 1 M k/m k 1 0 L := M k/m k 1, N := M k /M k k Z k Z 61

68 , M k 1 M k 1 M k 1 M k L, N O[ξ]-. O[ξ]-. O T X π 1 O[ξ], O T X-. 0 L gr(m) N 0, 0 N gr (M) L 0 0 µ(l) µ(gr(m)) µ(n ) 0, 0 µ(n ) µ(gr (M)) µ(l) 0 Supp(µ(gr(M))) = Supp(µ(L)) Supp(µ(N )) = Supp(µ(gr (M))). (2) l 2 : M k := M k 1 + M k {M k} M good filtration. k M k 1 M k M k, M k l+1 M k M k. gr (M) := k Z M k/m k 1,. Supp(µ(gr(M))) = Supp(µ(gr (M))) = Supp(µ(gr (M))) O C n D-. M 0 := O M k := D(k)O = O {M k } O good filtration, filtration graded module gr(o) = O = O[ξ]/(O[ξ]ξ O[ξ]ξ n ), µ(gr(o)) = O T X/(O T Xξ O T Xξ n ) Char(O) = Supp(µ(O)) = {(x, ξ) T X ξ 1 =... = ξ n = 0}.., [Kash1] M 0 D- M O-, M 0 O M D- {M k } (1) (4) gr(m) := k Z M k /M k 1, gr(m) O[ξ]- {M k } (5). 62

69 L M ϕ N 0 D- Char(M) = Char(L) Char(N ). : L M D-. {M k } M good filtration L k := M k L, N k := ϕ(m k ), M k = D(k k 0 )M k0 N k = ϕ(d(k k 0 )M k0 ) = D(k k 0 )N k0, {N k } N good filtration. k 0 L k M k ϕ N k 0 (4.1) O M k N k O-, L k O-,. (4.1) filtration graded module, O[ξ]- 0 gr(l) gr(m) gr(n ) 0. {M k }, {N k } good filtration, {L k } good filtration. O T X- 0 µ(gr(l)) µ(gr(m)) µ(gr(n )) ,. M X D-, p X U D- 0 M ϕ D r. M k := ϕ(d(k) r ) {M k } M good filtration. N := Ker ϕ, N k := Ker ϕ D(k) r, 0 M ϕ D r N 0 D- N U D-, {N k } N good filtration. filtration graded module O T X- 0 µ(gr(m)) (O T X) r µ(gr(n )) 0. µ(gr(n )) (O T X) r, Char(M) T U = {p = (x, ξ) T U µ(gr(n )) p (O T U) r p }. r = 1 µ(gr(n )) p (O T U) p f gr(n ) p, f(x, ξ) = 0,. P N p, σ(p )(x, ξ) = 0 63

70 M u M(X) M = Du D-, D I I(U) := {P D(U) P u = 0},. Char(M) = {(x, ξ) T X P I x σ(p )(x, ξ) = 0} M X D-, Char(M) T X. (x, ξ) Char(M), c C \ {0} (x, cξ) Char(M), Char(M) (x, ξ). : M r. r = 1. M = Du u M I := {P D P u = 0}, I k := I D(k), {I k } I good filtration, graded ideal gr(i), P 1,..., P s I, gr(i) O[ξ] σ(p 1 ),..., σ(p s ) Char(M) = {(x, ξ) σ(p 1 )(x, ξ) =... = σ(p s )(x, ξ) = 0}, Char(M) T X. r > 1, M U u 1,..., u r, N := Du Du r 1 M, N D- M D-. L := M/N, 0 N M L 0 D-. L = D[u r ] ([u r ] u r M/N ), L N, Char(M) = Char(L) Char(N ). 3.5 Cauchy X C n, Y X ( ). p Y X U, ϕ 1,..., ϕ d O(U) Y U = {x U ϕ 1 (x) =... = ϕ d (x) = 0}. ι : Y X. X F, F Y := ι 1 F. F Y Y p F p Y. 64