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1 ,2,4

3 iii 1 Hilbert Hilbert

5 (Hilbert) = /() 1 (a 1, b 1 ) 1.1 (a 2, b 2 ) (a 3, b 3 ) md 2 x/dt 2 = F (a 1 a 2 ) 2 + (b 1 b 2 ) 2, (a 1 a 2 )(a 1 a 3 ) + (b 1 b 2 )(b 1 b 3 ) (1.1) xy ax 2 + 2bxy + cy 2 + 2dx + 2ey + f = 0 (1.2) (a + c) 2 : ac b 2 (1.3) ɛ (a + c) 2 ac b 2 = 2 + (1 ɛ2 ) ɛ 2 0 < 1 1 > 1 (a+c) 2 ac b 2 4 > D := det a b b c d e d e f D 2 : (ac b 2 ) π 4 D 2 /(ac b 2 ) a+c, ac b 2 D

6 cos θ sin θ r g = sin θ cos θ s a b d b c e d e f t g a b d b c e d e f g (1.4) D ( a10 + a 11 x + a 12 y (x, y) a 00 + a 01 x + a 02 y, a ) 20 + a 21 x + a 22 y a 00 + a 01 x + a 02 y det(a ij ) = D 4.3 a + c, ac b 2 D 2 {1, 2,..., n} n! S n x 1,..., x n f(x 1,..., x n ) f(σ(x 1 ),..., σ(x n )) = f(x 1,..., x n ) σ S n σ(x i ) x σ(i) x 3 + y 3 + z 3 xyz 3 x, y, z 1. 3 x, y, z 4 *1 4 a, b, c, d a(x 4 +y 4 +z 4 )+b(x 3 y+xy 3 +x 3 z +xz 3 +y 3 z +yz 3 ) +c(x 2 y 2 + x 2 z 2 + y 2 z 2 ) + d(x 2 yz + xy 2 z + xyz 2 ) n (t + x i ) = (t + x 1 )(t + x 2 ) (t + x n ) i=1 t s 1 = x 1 + x x n s 2 = i<j x ix j s 3 = i<j<k x ix j x k s n = x 1 x 2 x n s r x 1, x 2,..., x n r r 2 ( ). x 1, x 2,..., x n s 1, s 2,..., s n f = x 3 y + x 3 z + xy 3 + xz 3 + y 3 z + yz 3 f xxxy = x 3 y s a 1s b 2s c 3 x a (xy) b (xyz) c = x a+b+c y b+c z c x 3 y s 2 1s 2 f s 2 1s 2 = x 2 yz xy 2 z xyz 2 2(xy + xz + yz) 2 xxyy 2 2s 2 2 f s 2 1s 2 + 2s 2 2 = x 2 yz xy 2 z xyz 2 f = s 2 1s 2 2s 2 2 s 1 s 3 *1 x p y q z r p + q + r

7 4 3 3 f(x 1,..., x n ) 1 i < j n 5. f(x 1,..., x n ) f(σ(x 1 ),..., σ(x n )) = f(x 1,..., x n ) (1.6) f(x 1,..., x i,..., x j,..., x n ) = f(x 1,..., i ˇx j,..., jˇx i,..., x n ) (1.5) σ A n x 2 1 x 2 2 x 1, x 2 (x 1, x 2,..., x n ) := (x i x j ) 1 i<j n = (x 1 x 2 )(x 1 x 3 )(x 1 x 4 ) (x 1 x n ) (x 2 x 3 )(x 2 x 4 ) (x 2 x n ) (x 3 x 4 ) (x 3 x n ) (x n 1 x n ) x 1, x 2,..., x n 3. (x) 1.5 x i = x j f(x 1,..., x n ) f(x) x i x j (x) f(x)/ (x) 4. x 3 (y z) + y 3 (z x) + z 3 (x y) x, y, z (x y)(x z)(y z)(x + y + z) (x) 2 4 σ S n (σ(x 1 ),..., σ(x n )) = ± (x 1, x 2,..., x n ) 6. f = f(x 1,..., x n ) x 1 x 2 f 1 f 1 (f + f 1 )/2 (f f 1 )/2 3 f(x), g(x) f(x) + (x)g(x) 2 7. s 1, s 2,..., s n (x) S S n 8. f(x 1,..., x n ) 1.6 S σ S S S n A n 9. S (12)(34), (14)(23) *2 S f(x 1, x 2, x 3, x 4 ) = f(x 2, x 1, x 4, x 3 ) = f(x 4, x 3, x 2, x 1 ) f s 1, s 3 x 1 x 2 + x 3 x 4, x 1 x 4 + x 2 x 3, x 1 x 3 + x 2 x 4 + σ A n *2 (13)(24) Klein 4

8 S n n GL(n) A = (a ij ) 1 i,j n x i x i A := n a ji x j, j=1 f(x 1,..., x n ) 1 i n f(x 1 A,..., x n A) = f(x 1,..., x n ) A S GL(n) S ( ) f(x, y) A = 0 1 f(x, y) = f(x, y) y A x y 2 ( ) f(x, y) A = 0 1 f(x, y) = f( x, y) A x 2, xy y S 1 1 1, S f(x, y, z) x 2, y 2, z 2 xyz f(x 1, x 2, x 3, x 4 ) = f(x 2, x 1, x 4, x 3 ) = f(x 4, x 3, x 2, x 1 ) y 1 = x 1 + x 2 + x 3 + x 4 y 2 = x 1 + x 2 x 3 x 4 y 3 = x 1 x 2 + x 3 x 4 y 4 = x 1 x 2 x 3 + x 4 (12)(34), (14)(23) y 1 y 2, y 3, y 4 12 y 1 y 2 2, y 2 3, y 2 4 y 2y 3 y Q R C *3 k k f(x 1,..., x n ) k[x 1,..., x n ] *4 k n S GL(n, k) S k[x 1,..., x n ] S S S k[x 1,..., x n ] S k[x 1,..., x n ] 14. S f(x) f 1 (x),..., f N (x) k[x 1,..., x n ] S N k f(x) = F (f 1 (x),..., f N (x)) k[x 1,..., x n ] S k f 1 (x),..., f N (x) k R k[x 1,..., x n ] *3 Z/pZ *4 Z = {0, ±1, ±2,...}

9 S k[x 1,..., x n ] S R k[x 1,..., x n ] 15. y = 0 2 a + yg(x, y) a k R R k[x, y] x n y, n 0 n S G(S) S G = G(S) G GL(n, k) G S G(S) G(S) G GL(n, k) G k[x 1,..., x n ] G * 5 m R x m y n 16. k[x, y] n 3m x m y n R S k[x 1,..., x n ] S A B AB A 1 S GL(n, k) S G(S) G(S) GL(n, k) G(S) GL(n, k) 1. (x 1 + x 2 x 3 x 4 )(x 1 x 2 + x 3 x 4 )(x 1 x 2 x 3 + x 4 ) x 1, x 2, x 3, x (1234) k[x 1, x 2, x 3, x 4 ] (1234) 4. l (2 + 3) l = n l + m l 3 (ml, n l ) 10 R x m l y n l l 0 *5 G Noether

11 7 2 ax 2 +2bx+c = 0 ( a x + b ) 2 ac b2 + = 0 (2.1) a a ax 3 +3bx 2 +3cx+d = 0 x y = x+b/a ay 3 + 3(ac b2 ) y + a2 d 3abc + 2b 3 a a 2 = 0 (2.2) (1) (2) ac b 2 a 2 d 3abc + 2b 3 n ( ) ( ) n n F n (x) = ax n +nbx n 1 + cx n 2 + dx n x = b/a a n 1 a n 1 F n ( b/a) n = 2, 3 1 E.T. Men of Mathematics (Sylvester, ) (Cayley, ) Boole, ) (Lagrange, ) (Gauss, ) 2 ac b 2, a 2 d 3abc + 2b 3, F n (x) = 0 x q F n (x + q) = 0 ax 2 + 2bx + c = 0 a(x + q) 2 + 2b(x + q) + c = 0 ax 2 + 2(aq + b)x + (aq 2 + 2bq + c) = 0 a(aq 2 + 2bq + c) (aq + b) 2 = ac b 2 ac b 2 4 a, ac b 2 k[a, b, c] H(2) := 1 q q q q k GL(3, k) (2.3)

12 8 2 k C n F n (x) = 0 a, af 2 ( b/a) = ac b 2,, a n 1 F n ( b/a) (2.4) F n (x) F n (x + q) 1. F n (x) = 0 f(a, b, c,...) f(a, b, c, ) = f(a, aq + b, aq 2 + 2bq + c, ) q k F n (x) = 0 * 1 (n + 1) 1 q... q n nq n 1 H(n) := q k (2.4) k[a, b, c,...] H(n) a ac b 2 k[a, b, c] H(2) n 3 k[a, b, c,...] H(n) a 2b c a 2b c b 2c d = 4(ac b2 ) 3 + (a 2 d 3abc + 2b 3 ) 2 a 2 b 2c d (2.4) * 2 (covariant) *1 semi-invariant (relative invariant) *2 k(a, b, c,...) H(n) 3 x + q ax 2 + 2bx + c = 0 x (px + q)/(rx + s) := y a(px + q) 2 + 2b(px + q)(rx + s) + c(rx + s) 2 = 0 Ax 2 + 2Bx + C = 0 A, B, C a, b, c A = ap 2 + 2bpr + cr 2 B = apq + b(ps + qr) + crs C = aq 2 + 2bqs + cs 2 1/4 b 2 ac B 2 AC = (ps qr) 2 (b 2 ac) (ps qr) 2 p, q, r, s b 2 ac 0 x 2 y

13 3 9 4 n F n (x) = 0 α 1,..., α n (α 1,..., α n ) := (α i α j ) (2.5) 1 i<j n (α) 2 b/a, c/a, d/a,... a 2n 2 a, b, c,... D(a, b, c,...) a 2n 2 (α) 2 2 (). x 1, x 2,..., x n x i e s 1, s 2,..., s n e a 3b 3c d a 2b c a 3b 3c d 3 3 a 2b c b 2c d, 4 4 a 3b 3c d b 3c 3d e b 2c d b 3c 3d e b 3c 3d e 4 a, b, c, d, e ps qr x (px + q)/(rx + s) ps qr = 1 3. F n (x) = 0 f(a, b, c,...) ( ) px + q F n (x) (rx + s) n F n rx + s (2.6) F n (x) = a, b, c f(a, b, c) k[a, b, c] p, q, r, s k ps qr = 1 f(ap 2 +2bpr+cr 2, apq+b(ps+qr)+crs, aq 2 +2bqs+cs 2 ) f(a, b, c) 3 G(2) := p2 pq q 2 2pr ps + qr 2qs ps qr = 1 r 2 rs s GL(3, k) n G(n) GL(n + 1, k) D(a, b, c,...) = a 2n 2 (α) 2 k[a, b, c] G(2) k[a, b, c, d] G(3) k[a, b, c, d, e] G(4) (P. Gordan, ) 4 (). n f(a, b, c,...) a, b, c,... e (1 U n+1 )(1 U n+2 ) (1 U n+e ) (1 U 2 ) (1 U e ) U U ne/2 ne/2 5. k[a, b, c,...] G(n)

14 F n (x) = a n i=1 (x α i) D(a, b, c,...) = a 2n 2 1 i<j n (α i α j ) 2 (2.7) F n (x) 1 (2.6) n n a (x α i ) a {px + q α i (rx + s)} i=1 i=1 a n i=1 ( rα i + p) (sα i q)/(rα i p) =: α i 1 i n a 2n 2 i ( rα i + p) 2n 2 i<j (α i α j) 2 α i α j = (ps qr)(α i α j ) (rα i p)(rα j p) (2.8) F n (x) = i<j n (α i α j ) mij, m ij Z 0 (2.9) α i e α 1,..., α n ne/2 m ij e = 2n Φ(α 1,..., α n ) Φ(α) b/a, c/a, d/a,... 2 f(a, b, c,...) = a e Φ(α) (2.10) f f *3 a e i ( rα i + p) e i<j(α i α j) mij 2.8 a e i<j (α i α j ) mij F 4 (x) = 0 A = (α 1 α 2 )(α 3 α 4 ) B = (α 1 α 3 )(α 4 α 2 ) C = (α 1 α 4 )(α 2 α 3 ) = A B (2.11) S 4 S 3 A 2, B 2, C 2 A B, B C, C A A 2 + B 2 + C 2 (A B)(A C)(B C) α 1, α 2, α 3, α 4 e = 2, 3 a 2 (A 2 + B 2 + C 2 ) = ae 4bd + 3c 2 =: g 2 a 3 (A B)(A C)(B C) = a b c b c d c d e =: g 3 D(a) = a 6 A 2 B 2 C 2 g g 2 3 α i i (2.9) i j m ij n e A, B, C *3

15 F n (x) = 0 2 n ( ) n F n (x, y) = ax n + nbx n 1 y + cx n 2 y n = 6 15(= 6! (2!) 3 3! ) e=1 15 (α 1 α 2 )(α 3 α 4 )(α 5 α 6 ) 2.11 A + B + C = 0 1 *4 2.2 (Kempe) 6. (2.9) Φ(α) (2.10) g 2, g 3 2 *4 n = ,, 14, x, y F n (x, y) F n (px + qy, rx + sy), ps qr = 1 a, b, c,... m n F n (x 1,..., x m ) F n ( a 1i x i,..., a mi x i ) i i m n (a ij ) 1 i,j m 1890 m n 14 S n S k[x 1,..., x n ] S L. Maurer

16 F n (x) = 0 a b c b c d c d e catalecticant

17 13 3 ( 3) f(x) f ev (x) = {f(x) + f( x)}/2 x x, k C 1 n f(x 1,..., x n ) f((x 1,..., x n )A) n A n G GL(n, C) A G G G A, B G AB 1 G G G GL(n, C) G C[x 1,..., x n ] G n F n (x) = 0 ( ) ( ) n n F n (x) = ax n +nbx n 1 + cx n 2 + dx n (n + 1) f(a, b, c, d,...) ( ) px + q F n (x) (rx + s) n F n, (3.1) rx + s p, q, r, s C, ps qr = 1 (3.2) a, b, c, d,... G(n) n = 2 p 2 pq q 2 G(2) = 2pr ps + qr 2qs ps qr = 1 r 2 rs s 2 ( ) p q (2) r s SL(2, C) (1) ρ : SL(2, C) GL(n + 1, C) (3.3) G(n) 2. 5 C[a, b, c, d,...] G(n) 2 G = {A 1, A 2,..., A m } A G

18 14 3 A 1 A, A 2 A,..., A m A A 1, A 2,..., A m G n f(x) G f av (x) := {f(xa 1 ) + f(xa 2 ) + + f(xa m )}/m f(xa) = f(x) A G 1. f av (x) G 2. f(x) G f av (x) = f(x) 3. (f + h) av = f av + h av 4. G f G (fg) av = fg av av C[x 1,..., x n ] f(x) (x) f f av C[x 1,..., x n ] G C[x 1,..., x n ] G 3. G m = n! av f(x 1,..., x n ) f av (x) = 1 f(σ(x 1 ),..., σ(x n )) n! σ S f(σ(x σ: 1),..., σ(x n )) 2 n! 3 {U i } i=1 m {U i } m i=1 U i m j=1 U j 4. R n S n 1 n t (α 1,..., α n ) C n 5. a 1, a 2, a 3,... C n m a i a 1,,..., a m n 0 n = 0 n 1 a 1, a 2, a 3,... n α 1n, α 2n, α 3n, a 1, a 2, a 3,... C n 1 OK l α ln C n 1 b1 := a l+1 α l+1,n a l, α b 2 := a l+2 α l+2,n a l,... ln α ln b i b 1,,..., b m m a i a 1,,..., a l,,..., a m+l C n a 1, a 2, a 3,... n α 11 α 21 α 31 α 12 α 22 α α 1n α 2n α 3n l r(l) 0 r(1) r(2) r(3) n m r(l) Z = {0, ±1, ±2, ±3,...}

19 n 1, n 2, n 3,... m n i n 1,,..., n m l n 1, n 2,..., n l gcd(l) n 1 = gcd(1) gcd(2) gcd(3) 1 m gcd(l) n n n 1, n 2,..., n m n i n 7. n f 1 (x), f 2 (x), f 3 (x),... m f i (x) f 1 (x),..., f m (x) m i=1 f i(x)g i (x) 4 1 n x k1 1 xkn n k 1+ +k n f(x 1,..., x n ) d 8. f(x 1,..., x n ) d d f f d + f d f 1 + f 0 f i i f f G f i G d d d G d G d 1 2 f f av G d d = 1, 2, 3,... f 1, f 2, f 3,... 7 *1 f i f 1,..., f m m G C[x 1,..., x n ] G f 1,..., f m G h f 1,..., f m h 0 h f 1, f 2, f 3,... h f 1,..., f m m i=1 h if i h f 1,..., f m h 1,..., h m h = m i=1 h if i 2 h = h av = m f i f i h av i i=1 hav i h f 1,..., f m h 5 1 S 1 = {e iθ θ R mod 2π} 4 SL(2, C) {( ) } e iθ 0 K := 0 e iθ θ R (3.3) ρ(k) GL(n + 1, C) ρ(k) (n + 1) (isobaric) *1 d = 0

20 16 3 f = f(a, b, c, d,...) K f av := 1 2π 2π 0 ( ) e iθ 0 f((a, b, c, d,...)ρ 0 e iθ )dθ C[a, b, c, d,...] ρ(k) SL(2, C) ( ) p q q p p 2 + q 2 3 {(p, q) p 2 + q 2 = 1} C 2 = R 4 SL(2, C) 2 SU(2) SU(2) f av := 1 ( ) p q 2π 2 f((a, b, c, d,...)ρ )ds S q p 3 *2 SU(2) = S 3 S 3 ds f av SU(2) 9. ρ(su(2)) GL(n + 1, C) C[a, b, c, d,...] ρ(su(2)) *3 (Weyl) (unitary trick) G(n) 1 q... q n nq n 1 H(n) = q C (n + 1) n H(n) *2 B 2n R 2n π n /n! S 2n 1 2n *3 D, ( ) N+ := n exp(qn + ) N + D = a b + 2b c + 3c d + f(a, b, c, d,...) Df = 0 H(n) U + := {( ) } 1 q q C 0 1 (3.3) ρ {( ) 1 0 U := r 1 } r C f = 0 := nb a + (n 1)c b + (n 2)d c +, U + U SL(2, C) 10. f(a, b, c, d,...) Df = f = 0 Aronhold 2 2 SU(2) 5 K S 1 { ( ) } {( )} K 0 θ cos θ sin θ := exp θ R =, θ 0 sin θ cos θ { ( ) } {( )} K 0 iθ cos θ i sin θ := exp θ R =. iθ 0 i sin θ cos θ ρ(su(2)) f = f(a, b, c, d,...) ρ(k ) ρ(k ) (D )f = 0 i(d + )f = 0 10 f 9

21 I C[x 1,..., x n ] 1. h(x), h (x) I h(x) + h (x) I. 2. f(x) C[x 1,..., x n ] h(x) I f(x)h(x) I. 12. f =(f 1 (x), f 2 (x), f 3 (x),...) f(x) = i f i(x)g i (x) I f f I I = I f f 13. I C[x 1,..., x n ] 12 f I f I f h 1 (x),..., h l (x) I f h i (x) j f j(x)g ij (x) f 1 (x), f 2 (x),..., f m (x) I f f i (x) f 1 (x),..., f m (x) 7 13 n n (n + 1) C[x 1,..., x n, y] 4 y(= x n+1 ) y d f(x, y) f d (x)y d + f d 1 (x)y d f 0 (x) f d (x) 0 I y d y d a d C[x 1,..., x n ] a d d a 0 a 1 a 2 a = d 0 a d C[x 1,..., x n ] I a d a a f 1 (x, y),..., f m (x, y) I d max y i a i i f (i) (i) 1 (x, y), f 2 (x, y),... I f(x, y) I f 1 (x, y),..., f m (x, y) d max I f 1 (x, y),..., f m (x, y) 0 i d max 1 f (i) 1 (i) (x, y), f 2 (x, y), y 1 = s 1 y 2 2, y 2 3, y 2 4 y 2y 3 y 4 y 2 y 3 y 4 s 3 1 4s 1 s 2 + 8s 3 y 2 2, y 2 3, y 2 4 s 1 x 1 x 2 + x 3 x 4, x 1 x 4 + x 2 x 3, x 1 x 3 + x 2 x 4 (1234) y 0 = x 1 + x 2 + x 3 + x 4, y 1 = x 1 + 1x 2 x 3 1x 4, y 2 = x 1 x 2 + x 3 x 4, y 3 = x 1 1x 2 x 3 + 1x 4 y i ( 1) i 7 y 0, y 1 y 3, y 2 2, y 2 1y 2, y 2 3y 2, y 4 1, y 4 3 M l = x m l y n l l = 0, 1, 2, 3,... y, xy 2, x 4 y 7, x 15 y 26,... l 1 n n l m l m x m y n k[x, y] R l n l m l 3 {R l } l 1 l R l = R n l 1 m l m l 1 n l = 1 R 1 y = M 0

22 18 3 xy 2 = M 1 R l R l 1 M l R M l, l 0,

23 m f(x 1,..., x m ) f(x 1 A,..., x m A) = f(x 1,..., x m ) m A *1 S S f(x 1,..., x m ) S k[x 1,..., x m ] S * 2 14 S k[x 1,..., x m ] S ZariskiRees 1958 k[x 1,..., x m ] S *1 x i A, 1 i m, (x 1,..., x m)a i *2 k Q R C S G GL(n) G A (unipotent) (4.1) S 4.1 S k[x 1,..., x m ] S

24 ax 2 + 2bxy + cy 2 + 2dx + 2ey + f = 0 (4.2) g = g r,s,θ = a b d b c e d e f t g cos θ sin θ r sin θ cos θ s a b d b c e d e f g (4.3) G GL(6) k[a, b, c, d, e, f] G 1. G u = {g r,s,0 } k[a, b, c, d, e, f] Gu 2. k[a, b, c, d, e, f] Gu G 0 = {g 0,0,θ } k[a, b, c, d, e, f] G G u (i) 14 k[a, b, c, d, e, f] G k[a, b, c, d, e, f] Gu a, b, c a b d D = b c e d e f a, b, c, D G 0 a + c, ac b 2 D G GL(m) G *3 G (unipotent radical) G u G u x 1,..., x m G u G *3 G G u k[x 1,..., x m ] Gu G/G u G/G u * 4 1. k[x 1,..., x m ] Gu k[x 1,..., x m ] G k[x 1,..., x m ] Gu 2 3 n n A (1), A (2),..., A (n) A (1) A (2) A (n) A (1) A (2) A (n) A (i) 2n ( ) 1 a1 0 1 ( ) 1 a2 0 1 ( ) 1 an 0 1 (4.4) A 2n (a 1, a 2,..., a n ) A 2n (a 1,..., a n ) A 2n (a 1,..., a n) A 2n (a 1 + a 1,..., a n + a n) 2n x 1, y 1,..., x n, y n A 2n (a 1, a 2,..., a n ) f(x 1, a 1 x 1 + y 1,..., x n, a n x n + y n ) = f(x 1, y 1,..., x n, y n ) (4.5) *4 (reductive algebraic group)

25 3 21 (4.4) S k[x 1, y 1,..., x n, y n ] S 2 ( 1958 ) (4.4) S GL(32, C) C[x 1, y 1,..., x 16, y 16 ] S S A i := A 18 (1 i, 2 i,..., 9 i ) i = 0, 1, 3 k[x 1, y 1,..., x 9, y 9 ] {A0,A1,A3} ( ). A k[x 1,..., x m ] A. (Weitzenböck) s A 2n (a 1, a 2,..., a n ) A 2n (1, 1,..., 1) x 1,..., x n A 2n (1, 1,..., 1) 2 n ( ) x1 x 2... x n y 1 y 2... y n 2 p ij = x i y i x j y j A 2n(1, 1,..., 1) 5. A 2n (1, 1,..., 1) k[x 1, y 1,..., x n, y n ] A(1,...,1) x 1,..., x n p ij 1 i < j n 3 4 A, B k[x 1,..., x m ] A,B 2 n Casteravat-Tevelev 2005 (2n + 1) A = B = (n + n + 1) 2 (n n) 2 6 ( 2005 ). A, B k[x 0,..., x n 1, y 0,..., y n ] {A,B} 1. x 0,..., x n 1,

26 (n + 1) x 0 x 1... x n 1 x 0... x n 1 x n 1 y 0 y 1... y n 1 y n 3 5 (n + 2) 7 (n + 3) 7, 2n + 1 ( ) ( ) ( ) ( ) ( ) n n + 1 n + 2 2n 2n n 1 2n f = f(x, y) p = (a, b) f(a, b) f p p m (m 1) p m (m + 1) m f p mult p f n p 1,..., p n µ(f; p 1,..., p n ) := n mult pi f/ deg f f µ(p 1,..., p n ) := sup µ(f; p 1,..., p n ) f n = 1 µ = n 9 µ(p 1,..., p n ) = n n p 1,..., p n d (d + 1)(d + 2)/2 p m m(m + 1)/2 (d + 1)(d + 2)/2 n m i (m i + 1)/2 (4.6) i=1 d p 1,..., p n m 1,..., m n (a; b 1,..., b n ) l d = la, m i = lb i l l l 2 (a 2 n 1 b2 i )/2 n 1 b i/a > n a 2 n 1 b2 i < 0 7. µ(p 1,..., p n ) n b 1 = = b n (=: b) n 1 b i/a < n a 2 n 1 b2 i > 0 l la p 1,..., p n lb 2 n > 9 n p 1,..., p n f µ(f; p 1,..., p n ) < n 8. n 9 p 1, p 2 f = f 1,2 n = 2, 3, 4 µ(f; p 1,..., p n ) 2 n n = 5, 6 n = 7, 8, 9 (4.6) (d; m 1,..., m n ) = (2; 1 5 ), (3; 2, 1 6 ), (6; 3, 2 7 ), (3; 1 9 ) 5 p 1,..., p 5 2 f = f 1,...,5 µ(f; p 1,..., p 5 ) = 5/2 n = 5, 6 µ(p 1,..., p n ) 5 2 > n n = 7, 8 µ(p 1,..., p 7 ) 8 3 > 7 µ(p 1,..., p 8 ) 17 6 > 8

27 6 23 f 1,...,9 µ(f; p 1,..., p 9 ) 9/3 = n 5 (4.4) (n 3) f(x 1, y 1,..., x n, y n ) x 1,..., x n x-deg f y 1,..., y n y-deg f (n 3) A 2n (1, a 1, b 1, 1, 0,..., 0), A 2n (1, a 2, b 2, 0, 1,..., 0), z 1 = y 1 /x 1..., A 2n (1, a n 3, b n 3, 0, 0,..., 1) n y i /x i, i=4 z 2 = y 3 /x 3 z 1 = y 2 /x 2 n b i 3 y i /x i i=4 n a i 3 y i /x i, i=4 (n 3) n 1 x i z 1, z 2, z 3 d f(x, y) p 1 = (a 1, b 1 ),..., p n = (a n, b n ) m 1,..., m n f( z2 z 1, z3 n 1 xmi i z 1 )z a 1 7 a n 1 n x- deg f/y-deg f = a f f f(z 2 /z 1, z 3 /z 1 )z1 a / n 1 xmi i x-deg f/y-deg f > n 1 n 10. k[x, y] n 3m x m y n 11. y = 0 2 a + yg(x, y) a k R n 4.1 R x m y n f(x, y) x 0,..., x n 1 y 0,..., y n 12 ( ). x 0,..., x n 1 y 0,..., y n f(x, y) l 4 l+1 2l + 3 j=1 sin 2 2jπ 2l+3 (2 cos 2jπ 2l+3 )2n 1 l = 1 m

28 F n (x) F n (x + q) b, c, d, e aq +b, aq 2 +2bq +c,, aq 4 +4bq 3 +6cq 2 +4dq +e q 1 0 a b c b c d 1 q q q q 2 2q 1 c d e (catalecticant) F n (x) = 0

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy