WWW.DISSERS.RU


...
    !

Pages:     | 1 || 3 |

i= R[X] R R[X] R[X] (a0, 0,... ) R[X] R a (a, 0,... ) R[X] a R (a, 0,... ) R[X] X = (0, 1, 0,... ) n Xn = (0,..., 0, 1, 0,... ) n a a = a0 + a1X + + anXn, ai R.

a R[X] an = 0. a deg a = n a0 a an a R[X] R R R[X] f, g r[X] \ 0 deg(fg) = deg f + deg g f, g, f + g = 0 deg(f + g) max(deg f, deg g) R R[X] R f, g R[X] g = q, r R[X] f = qg + r r = 0 deg r < deg g f = 0 deg f < deg g q = 0, r = g deg f = n an, bm an f, g h = f - Xn-mg n h = q g + r bm an an f = h + Xn-mg = (q + Xn-m)g + r.

bm bm f = qg + r = q g + r, deg r, deg r < deg g, g(q - q ) = r - r q - q = 0 r - r = 0.

deg g > deg(r - r ) = deg(g(q - q )) = deg g + deg(q - q ) deg g.

q r f g g f r = 0 g|f f1,..., fm d d|fi, i = 1,..., m;

d R[X], d |fi, i = 1,..., m, d |d f1,..., fm (f1,..., fm) (f1,..., fm) (f1,..., fm) (f1, (f2,..., fm)).

f, g g = f = q1g + r1, deg r1 < deg g;

g = q2r1 + r2, deg r2 < deg r1;

r1 = q3r2 + r3, deg r3 < deg r2;

............................................

rk = qk+2rk+1 + rk+2, deg rk+2 < deg rk+1;

rk+1 = qk+3rk+2.

k (f, g) = rk+ rk+2|rk+ rk+2|g, rk+2|f d |f, d |g d |r d |g d |rk+ d = (f1,..., fm) u1,..., um d = u1f1 + + umfm.

m = 2 f1 = f, f2 = g f1,..., fm (f1,..., fm) = f1,..., fm u1,..., um 1 = u1f1 + + umfm.

p deg p p X2 + 1 R[X] p R[X] f R[X] (p, f) = p|f d = (p, f) f = f d deg d = 0 deg d = deg p d d = deg d = deg p p = dc, c d = pc-1.

f = pc-1f p R[X] p|(fg) f, g R[X] p|f p|g (p, f) = 1 1 = fu + pv g = 1g = fgu + pvg. p|g f = p1 pn = q1 qm p1,..., pn, q1,..., qm n = m Sn pi = ciqi ci deg f deg f = 1 f f f = gh deg g, deg g < deg f g, h deg f deg f = p1|qi i i = 1 q q1 = c1p1 cp1(p2 pn - (c1q2)q3 qm) = 0.

p2 pn - (c1q2)q3 qm = 0, p2 pn = (c1q2)q3 qm.

n - 1 = m - 1 pi = ciqi, i 2, ci R f R[X] 1 n f = pl pl, c R, li N 0, 1 n p1,..., pn li pi f char R = 0 f R[X] \ 0 k p f p f k - f = pkg (p, g) = f = kpk-1p g + pkg = pk-1(kp g + pg ).

pk-1|f pk|f p|(kp g + pg ) p|kp g p|kp p|g kp = 0 deg(kp ) < deg p char R = 0 f R[X] \ f 1-n-d = (f, f ) = pl pl, = p1 pn.

1 n d R c R a a(c) = a0 + a1c + + ancn. a a(c) = f, g R[X] (f + g)(c) = f(c) + g(c), (fg)(c) = f(c)g(c).

c R f R[X] (X - c)|f f X - c f = (X - c)q + r r R r = f(c) c R X - c f f = b0 + b1(X - c) + + bn(X - c)n.

bi f X - c f = a0 + a1X + + anXn = g(X - c) + r, r R, g = s0 + s1X + + sn-1Xn-1.

a0 + a1X + + anXn = (X - c)(s0 + s1X + + sn-1Xn-1) + r.

X an = sn-an-1 = sn-2 - csn-........................

a1 = s0 - csa0 = r - cs0.

sn-1 = an sn-2 = an-1 + csn-........................

s0 = a1 + csr = a0 + cs0.

n r = f(c) an an-1... a1 ac a0 = sn-1 sn-2... s1 r g X - c an an-1... a1 ac an = sn-1 sn-2... s1 r = bc an = tn-2 tn-3... t0 = b, c an = u1 u0 = bn-c an = bn bi c f X - c f k (X - c)k|f (X - c)k+1 f c f k f = bk(X - c)k + bk+1(X - c)k+1 + + bn(X - c)n, bk = 0.

c1,..., cm f k1,..., km 1 m f = (X - c1)k (X - cm)k g.

R c R 2 c R a0, a - 1,..., an b0, b - 1,..., bn R n (X - a0) (X - ai-1)(X - ai+1) (X - an) F = bi.

(ai - a0) (ai - ai-1)(ai - ai+1) (ai - an) i= deg F n F (aj) = bj j = 0,..., n n (aj - a0) (aj - ai-1)(aj - ai+1) (aj - an) F (aj) = bi = bj.

(ai - a0) (ai - ai-1)(ai - ai+1) (ai - an) i= F n F (aj) = bj j = 0,..., n F1, FG = F1 - F2 = 0 deg G n a0, a - 1,..., an R f, g R f(c) = g(c) c R f = g R q Xq - X R R R q-1 c R c q- cq-1 = 1 cq = c C R f C[X] f f F = C f f C R f C[X] z |f(z)| f f = a0 + a1X + + anXn, an = 0.

an-1 a0 |an-1| |a0| |f(z)| = |z|n|an + + + | |z|n |an| - - -.

|z| |z|n |z| |z|n |z| |an-1| |a0| |an| - - -.

|z| |z|n M |f(z)| |z|nM f f(z0) = z0 z |f(z)| < |f(z0)|.

f(z) z f(z0) = f(z) = 1 + cpzp + cp+1zp+1 + + cnzn, cp, cn = 0.

f(z0) p z1 C z1 = - z = z0 + tz1, t (0, 1) R cp f(z) = 1 - tp + tp+1hn-p-1(t), f(z0) hn-p-1(t) C[t] n - p - 1 C hn-p-1(t) t < C(n - p) f(z) 1 - tp + |hn-p-1(t)| 1 - tp + C(n - p)tp+1 = 1 - tp(1 - C(n - p)t) < 1.

f(z0) f C[X] f > M = infzC |f(z)| > 0.

zk f(zk) M zi k zk C zk z0 f(zk) f(z0) = M C f C[X] 1 m f = a(X - c1)k (X - cm)k, a C, c1,..., cm f k1,..., km f R[X] c C f c f f 0 = f(c) = a0 + a1c + + ancn = a0 + a1c + + ancn = a0 + a1c + + ancn = f(c).

f R[X] c C f k c f k R 2.

f R[X] 1 m 1 s f = a(X - c1)k (X - cm)k (X2 + p1X + q1)l (X2 + psX + qs)l, a R, i c1,..., cm f k1,..., km (X2 + piX + qi)l R[X] f C[X] an-1 = = ak+1 = 0, an, ak = 0 0 k < n B = max(|ak|, |ak-1|,..., |a0|).

z C f B n-k |z| < 1 +.

|an| B n-k |z| 1 +.

|an| |f(z)| > |f(z)| = |anzn + akzk + + a1z + a0| |anzn| - |akzk + + a1z + a0| |z|k+1 - k k |an||z|n - |aj||z|j |an||z|n - B |z|j = |an||z|n - B = j=0 j=|z| - |an| B B |an| B |z|n(|z| - 1) - |z|k+1 + > |z|n(|z| - 1) - |z|k+1 = |z| - 1 |an| |an| |z| - 1 |an| |an||z|k+1 B |z|n-k-1(|z| - 1) -.

|z| - 1 |an| B |z|n-k-1(|z| - 1) >.

|an| B n-k |z| - 1 = > 0.

|an| B |z|n-k-1(|z| - 1) - (1 + )n-k-1 - n-k = [(1 + )n-k-1 - n-k-1] > 0, |an| 1 + >.

C R f R[X] f R[X] f (f, f ) f (f, f ) f R[X] f0 = f, f1 = f i 2 fi fi-2 fi-fi-2 = gi-1fi-1 - fi.

u < v f a W (a) f0(a), f1(a),..., fn(a), n = deg f.

[u, v] W (u) - W (v) fi(c) = 0 fi+1(c) = 0 i > 0 fi-1(c) = -fi+1(c).

fi(c) = fi+1(c) = 0 f(c) = f (c) = 0 c f fi(c) = 0 i > 0 > fi-1(x), fi+1(x) x (c -, c + ) fj j (u, v) f W (x) c fj(c) = c fj(c) W (x) c fj(c) = 0, j 1 c fj-1(x), fj+1(x) fj-1(x), fj(x), fj+1(x) j c W (x) f(c) = 0. f1(c) = 0 f1(c) > U c f(x) f(x) < 0, f(y) > 0, x, y U, x < c < y.

fj(c) = 0 fj(x) U fj(c) = 0, j > W (a) W (x) - W (y) = 1 x, y f1(c) < 0 U c f(x) f(x) > 0, f(y) < 0, x, y U, x < c < y.

fj(c) = 0 fj(x) U fj(c) = 0, j > W (a) W (x) - W (y) = 1 x, y Z Q Z[X] f, g Z[X] fg p fg Zp f, g f, g Zp[X] Zp[X] fg = Zp[X] n f Q[X] \ 0 f = f, n, m Z (n, m) = 1 f Z[X] m f Z[X] f = gh g, h Q[X] f = uv u, v Z[X] u = 1g, v = 1 h r c g = g, r, s Z (r, s) = 1 h = h s d rc c, d Z (c, d) = 1 f = g h sd (sd)f = (rc)g h, rs sd rs = cd. f = g h f Z[X] Z[X] Q[X] f Z[X] p p an p|an-1,..., p|a1, p|a p2 a f Q[X] d = f p d f d-1f f f = gh f Z[X] p Zp[X] a0Xn = gh Zp[X] p g = bXr, h = cXn-r, b, c Z 0 < r < n g, h p f = gh p p n 1 Xn + p Z[X] p Xp - p(X) = Xp-1 + xp-2 + + X + 1 = X - Q T = X + p p k T + pXp-1 + + T + + pT (T + 1)p - 1 k p(X) = = = T T p p-1 k-T + pXp-2 + + T + + p.

k K a a, b K b = b a c ad = bc.

b d ab b =.

ac c F F a c ad + bc a c ac + =, =.

b d bd b d bd F a : K F (a) = (a + b) = (a) + (b) (ab) = (a)(b) P F K = Z F = Q K = R[X] R F = R(X) f R R(X) g f deg f < deg g. R(X) p R[X] pk deg f < deg p f R(X) g = uv u, v R[X] (u, v) = g f h t = + g u v ua + vb = 1 a, b R[X] f = uaf + vbf f uaf vbf af bf = + = +.

g uv uv v u f pk p deg f deg f < deg p deg f deg p f p f = qp + r, r = 0 deg r < deg p.

f qp + r q r = = +.

pk pk pk-1 pk C(X) c R(X) (X - a)k c aX + b,, (X - a)k (X2 + pA + q)k X2 + pX + q R[X] R R[[X]] a = (a0, a1,... ) R R[[X]] a b = (b0, b1,... ), c = ab = (c0, c1,... ), k k ck = aibk-i.

i= R[[X]] R R[[X]] R[[X]] (a0, 0,... ) R[[X]] R R[[X]] R[X] a (a, 0,... ) R[[X]] a R (a, 0,... ) R[X] X = (0, 1, 0,... ) n n Xn = (0,..., 0,, 0,... ) a f = a0 + a1X + , ai R.

a R[X] a0 =... = an-1 = 0, an = 0.

o(f) n an f R R[[X]] o(f + g) o(f), o(g) f a R g = b0 + b1X + = f-1, bi R.

1 = gf 1 = a0b0 1 = fg 1 = b0a f a0 R g = f-b0 = a-1. j 1 fg = ajb0 + aj-1b1 + + a0bj = 0.

bj = -a-1 [ajb0 + aj-1b1 + + a1bj-1].

bj, j f g g g h gh = 1 f = f(gh) = (fg)h = h R f R[[X]] f = Xnu n = o(f) 0 u R[[X]] R[[X]] R((X)) R((X)) Xnu n Z u R[[X]] a = Xmu, b = Xmv n, m 0 u, v a R[[X]] = Xm-nuv-b Xsu = Xlv R((X)) s l Xs-l = uv-1.

s - l = 0. u = v f R((X)) \ 0 f f = anXn + an+1Xn+1 + , n Z, ai R, an = 0.

f R((X)) f = nanXn-1 + (n + 1)an+1Xn + R((X)).

f, g R((X)) (f + g) = f + g (fg) = f g + fg f, g R((X)) (g-1f) = g-2(f g - fg ) (gn) = ngn-1g n Z XR[[X]] g R[[X]] f R[[X]] g XR[[X]] g f f(g) = a0 + a1g + + angn + o(gn) n Xm Xm a0 + a1g + + am-1gm- f R[[X]] g XR[[X]] f(g) = f (g)g.

Xm f(g) Xm (a0 + a1g + + amgm) Xm f(g) a1g + + mamgm-1g = (a1 + 2a2g + + mamgm-1)g.

R f, g R[[X]] f = g f = g + c, c R f R[[X]] j f(j)(0) aj =.

j! f XR[[X]] f fn (-1)i-exp(f) = 1 + + + + , ln(1 + f) = fi.

1! n! i jf o(f), o(g) 1 exp(f + g) = exp f exp g (ln(1 + f)) =.

1 + f f fn g gn exp f exp g = (1 + + + + )(1 + + + + ) = 1! n! 1! n! fi fj 1 t! = figj = (f + g)t = exp(f + g).

i,j0 t0 i+j=t ti! j! t! i!j! t! (ln(1 + f)) = (-1)j-1fi-1f = (1 + f)-1f.

j o(f) ln(exp f) = f, exp(ln(1 + f)) = f.

f = X (exp X - 1) exp X (ln(exp X)) = (ln(1 + (exp X - 1))) = = = 1 = X.

1 + (exp X - 1) exp X ln(exp X) = X + c, c R X = ln(exp 0) = 0 = c.

exp(ln(1 + X)) 1 + X exp(ln(1 + X)) [exp(ln(1 + X))] (1 + X) - exp(ln(1 + X)) = = 1 + X (1 + X)exp(ln(1+X)) (1 + X) - exp(ln(1 + X)) 1+X = (1 + X)exp(ln(1 + X)) R.

1 + X X = exp(ln(1 + X)) = exp(ln 1)) = exp 0 = 1.

1 + X XR[[X]] 1 + XR[[X]] R[[X]] 1 + XR[[X]] exp : XR[[X]] 1 + XR[[X]] ln : 1 + XR[[X]] XR[[X]] ln((1 + f)(1 + g)) = ln(1 + f) + ln(1 + g) f, g XR[[X]] R R[X,..., Xn] = (R[X,..., Xn-1])[Xn].

R[X,..., Xn] i1 in ai,...,inX1 Xn, ai,...,in R, 1 i1,..., in N0 (N0)n (i1,..., in) N i1 in aiXi, i = (i1,..., in) (N 0)n, Xi = X1 Xn.

R R[X,..., Xn] (N 0)n m = (m1,..., mn), r = (r1..., rn) (N 0)n.

m > r 1 j < n m1 = r1,..., mj-1 = rj-1, mj > rj.

m > m, m > m m > m m m m m, m m m = m m = (m1,..., mn), m = (m 1..., m n), m = (m 1..., m n).

m1 = m 1,..., mj-1 = m j-1, mj > m j;

m 1 = m 1,..., m j -1 = m j -1, m j > m j.

s = min(j, j ) m1 = m 1,..., ms-1 = m s-1, ms > m s.

m > m M1 M2... (N 0)n n n = n - 1 Mi = (mi1, mi2,..., min) m11 m21... k mk,1 = mk+1,1 =... (N 0)n-(mk,2,..., mk,n) (mk+1,2,..., mk+1,n)...

t Mt = Mt+1 =....

m, m, m (N 0)n m > m m + m > m + m m, m, m m1 + m 1 = m 1 + m 1,..., mj-1 + m j-1 = m j-1 + m j-1, mj + m j > m j + m j.

m, m, m, m (N 0)n m m, m m m + m > m + m f = aiXi = 0.

i(N0)n amXm f m > j j (N 0)n aj = R f d aiXi i = (i1,..., in) i1 + + in = d f fd d d f = f0 + f1 + .

f 1 = 1(X1,..., Xn) = X1 + + Xn;

2 = 2(X1,..., Xn) = X1X2 + + Xn-1Xn;

.....................................................

n = n(X1,..., Xn) = X1 Xn.

k k Xi Xi 1 i1 < 1 k < ik n f f = f0 + f1 + f0, f1,...

k X1 Xk m1 mn aX1 , Xn m1 m2 mn.

m1 < m2 X1 Xmmn aX1 X2m1 Xn f m1-m2 mn-mn-1 mn h = a1 n-1 n h m1-m2 m1 mn 2-mn-mn-n aX1 (X1X2)m (X1 Xn-1)m (X1 Xn)m = aX1 Xn f f - h f 3 3 3 f = X1 + X2 + X3 X1 (3, 0, 0) (m1, m2, m3) (N 0) (m1, m2, m3) < (3, 0, 0) m1 m2 m m1 + m2 + m + 3 = 3.

(2, 1, 0), (1, 1, 1).

(3, 0, 0), (2, 1, 0), (1, 1, 1) f = 1 + a12 + b3, a, b X1 = X2 = 1, X3 = 2 = f = 8 + 2a a = -3 X1 = X2 = X3 = 3 = f = 27 - 27 + b b = 3 3 3 X1 + X2 + X3 = 1 - 312 + 33.

R[X, X1,..., Xn] f = a(X - X1) (X - Xn), a R.

f = aXn + a(-1)1Xn-1 + + a(-1)kkXn-k + + a(-1)nn.

R f = anXn + + a0 R[X] 1,..., n k = 1,..., n an-k = (-1)kk.

an X1 = 1,..., Xn = n, a = an k k k pk = X1 + + Xn Z[X1,..., Xn].

Z[X1,..., Xn] k = 0 k > n 0 = 1. k pk - pk-11 + pk-22 + + (-1)k-1p1k-1 + (-1)kk = 0.

Z[X1,..., Xn] Q[X, X1,..., Xn] n n F (X) = kXk = (1 + XkX).

k=0 k=n ln(F (X)) = ln(1 + XkX).

k= X F (X) n Xk n = = Xk(1 - XkX + XkX2 + ) = k=1 k=F (X) 1 + XkX k p1 - Xp2 + X2p3 + + (-1)kT pk+1 + F (X) n kkXk-k==.

n F (X) kXk k=n n k kkXk-1 = ( kXk)(p1 - Xp2 + X2p3 + + (-1)kT pk+1 + ) k=1 k= k = 1,..., n kk = (-1)kpk + (-1)k-11 + + p1k-1.

pk R p1,..., pn k p1,..., pn 1 = p 1p1 -22 = p 1p2 -2p1 32 = p......................................................

pk-1 -2pk-2 3pk-2... (-1)k-1kk = pk k 1 0 0... 0 pp1 -2 0... 0 p.............................

p1 1 0... 0 pk-1 pk-2...... p1 pk p2 p1 2... 0 k = =.

..........................

1 0 0... 0 0 k! pk pk-1...... p2 pp1 -2 0... 0....................................

pk-1 pk-2...... p1 (-1)k-1k pk 1 1 0... 0 22 1 1... 0 pk =.......................................

(k - 1)k-1 k-2...... 1 kk k-1...... 2 - 2 p2 = 1 - 22, p3 = 1 - 312 + 33.

Z[X1,..., Xn] W (X1,..., Xn) = (Xj - Xi).

1i

2 n-1 Xn Xn... Xn n-1 n-n-X1 X2... Xn n p1 p2... pn-p1 p2 p3... pn.

.............................

pn-1 pn pn+1... p2n- n = 2 p2 2 Disc(x1, x2) = = 2p2 - p2 = 2(1 - 22) - 1 = 1 - 42.

p1 pW (X1,..., Xn)2 = Disc(1,..., n).

Pages:     | 1 || 3 |



2011 www.dissers.ru -

, .
, , , , 1-2 .