- 22= Zt,Zc,= 50f), ndn f u 12 100 r,
chilng chua tdt cd cdc di€m cti aS ihung
khong chil'a didm (0, 0, 0).
(-
.s oro,n-ror* T?!I#8S
(ar Ldi gidị Tathdy 3n m4t phing v : i, !: i vd
z : i ehua t6t ca c6c di6m cira S vd khdng chil'a di6m (0, 0, 0). Nhu v6y s6 mflt phlng
cAn tim kh6ng vuot qu5 3n.
D6 chring to s6 mdt phing cAn tim dfng bing 3n,ta chftngminh b6 dA sau:
gO 46. Xdt da thtrc k bidn P(x1, x2, ..., x). Nilu P ftiil fiAu Qi cdc diAm cira tdp hqP
S: {(ar,a2,..., a*):a1 e {0, 1,..., n\, ar* az
+ ... + ak> 0\
vd khong triil fiAu @i di€m (0, 0, , 0) thi P cd bqc kh6ng nhd hon kn.
Ch*ng minh. Ta chfíng minh ki5t qu6 b6 dC
bing luy nap theo n. oE, ttt6.y k6t luan cta b6 d€ Jring-vdir : ọ GiA su ttit lu4n uo aA anng cho fr -l,tachftng minh ktlt lufn cta bO dA cfing dring cho [.
^A
Thq'c v4y, n6u da th*c k bi6n P(xr. x1, ."'
xr-r, x) thoa m6n diAu kiQn cria b6 di: (trong
d6 x ld biiin thf'ft), thi ta thuc hi6n phdp chia P(x1, x2, ..., xk-;, x) cho da thric x(x-l)...(x--n)
dC dugc thuong ld Q(x1, x2, ..., xr-r, x) vd da
thfc du R(x1, x2, ..., xk-r, x). Vi6t lai R(x1, x2, ..., xk-1, x) dang chinh tic theo lfiy thira cria x
ta c6
R(x1, x2, ..., x1,-1, x)
: Rr(xr, xz, ..., xt-r{'+...+ R6(x1, xz, ..., xr-t) (*)
Ta s€ chimg minh R,(x1, x2, ..., xt-r) ld cla th0c
ft-l bii5n th6a m6n di6u kiQn cria bO d6.
a) T(x): R(0, 0, ..., 0, x) ld da th[rc cira x vcn
b6c kh6ng vugt qu6 r vh tri6t ti6u t4i c6cdi,5m x :1,2,..., n.Do f(0): R(0,0, ...,0;0) di,5m x :1,2,..., n.Do f(0): R(0,0, ...,0;0) + 0, cho n6n I(0) ld da th[rc bQc n crtạx, suy
ra hQ s6 cira bAc cao nh6t x' trong khai triiSn
(*) ld R"(0, 0,...,0) + 0.
b) V6i mQt bQ (at, az, ... , a1,1) thoa a; e
{0,1,..., n\, ar+az*...* a* P0taco R(a1' a2,
..., ar-r, x) tri6t ti6u tai n+l di6m x :0, 1,2,
..., n.Yi bdc cria R(a1, a2,..., at*t, x) khdng
vugt qu6 n, cho n6n R(a1, az, ..., a6,.x) lit da
thirc d6ng nh6t 0, do d6 tlt cir hQ s5 cfia n6