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Disc(X1,..., Xn) Z[X1,..., Xn] f = Xn + an-1Xn-1 + + a0 R[X], R 1,..., n Disc(1(1,..., n),..., n(1,..., n))) = f f = anXn + an-1Xn-1 + + a0, an = 0.

1,..., n Disc(f) = a2n-2W (1,..., n)2.

n an-1 aDisc(f) = a2n-2 Disc(-,..., (-1)n ).

n an an Disc(f) an,..., an f = anXn + an-1Xn-1 + + a0, g = bmXm + bm-1Xm-1 + + b R an, bm an an-1 an-2... a0 0 0... 0 an an-1... a1 a0 0....................................................

0 0 0... an an-1........ a R(f, g) = bm bm-1 bm-2... b1 b0 0... 0 Mat(m + n, R).

0 bm bm-1... b2 b1 b0....................... bm.................... b n f m f R(f, g) Xm-1f Xm-2f Xn+m- Xn+m- f R(f, g) = Xn-1g X Xn-2g g f, g R(f, g) = det(R(f, g)).

u, v R[X] R(f, g) = fu + gv.

R(f, g) = 0 R(f, g) = Uf + V G 1 =, U, V R[X].

R(f, g) U V u =, v =.

R(f, g) R(f, g) R(f, g) = (f, g) = (f, g) = 1 R(f, g) = 0 R(f, g) = (n+m-1,..., 0), R(f, g) = Xm-1f Xm-2f f = (n+m-1Xm-1 + + n)f + (n-1Xn-1 + + 0)g = 0.

Xn-1g Xn-2g g (f, g) = f|(n-1Xn-1 + + 0), g|(n+m-1Xm-1 + + n).

= 0 deg f = n deg g = m R(f, g) = 0 u, v u u R(f, g) = fu + gv 1 = f + g (f, g) = R(f,g) R(f,g) f, g C[X] R(f, g) = f, g R f, g R[X] R(f, g) = R f(X1,..., Xn) R[X1,..., Xn].

f(X1, X2, X3,..., Xn) = f(X1,..., Xn) = (X1 - X2)h(X1,..., Xn), h(X1,..., Xn) R[X1,..., Xn].

R[X1,..., Xn] = R[X1, X1 - X2, X3,..., Xn].

f(X1,..., Xn) = u(X1, X3,..., Xn) + (X1 - X2)h(X1,..., Xn).

K R = K[X1,..., Xn, Y1,..., Ym].

R[X] f = anXn + an-1Xn-1 + + a0, ak = (-1)kan(X1,..., Xn), g = bmXm + bm-1Xm-1 + + b0, bk = (-1)kbm(Y1,..., Ym), an, bn K R(f, g) = ambn (Xi - Yj).

n m i,j an = bm = Xi Yj R(f, g) R(f, g) = A (Xi - Yj), A R.

i,j (Xi - Yj) = g(Xi).

i,j i (Xi - Yj) Xi m i,j j(X1,..., Xn) Xi R(f, g) Xi m A Xi Xi, Yj A K X1 = = Xn = 0, Y1 = = Ym = 1.

m (Xi - Yj) = (-1)mn, i(0,..., 0) = 0, j(1,..., 1) =.

j i,j 1 0 0.............................. 0 1 0....................................................................................

0 0 0... 1 0........... m R(f, g)(0,..., 0, 1,..., 1) = -............. (-1)m 0... = (-1)mn.

0 1 0 (-1)m.............................................. (-1)m A = n(n-1) an Disc(f) = (-1) R(f, f ).

f = anXn + an-1Xn-1 + + a0, f = nanXn-1 + + a1.

x1,..., xn f n R(f, g) = an-1 f (xi).

n i= f (xi).

n n n f = an (X - xi), f = (X - xj).

i=1 i=1 j=i n f (xi) = an (xi - xj), j=i R(f, f ) = a2n-1 n (xi - xj) = n i=1 j=i n(n-1) n(n-1) 2 a2n-1(-1) (xi - xj)2 = an(-1) Disc(f).

n j

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