điều kiện đủ để đa thức là tổng bình phương và ứng dụng

❱■➏◆ ❍⑨◆ ▲❹▼ ❑❍❖❆ ❍➴❈ ❱⑨ ❈➷◆● ◆●❍➏ ❱■➏❚ ◆❆▼

❱■➏◆ ❚❖⑩◆ ❍➴❈

✖✖✖✖✖✖✖♦✵♦✖✖✖✖✖✖✕

✣■➋❯ ❑■➏◆ ✣Õ ✣➎ ✣❆ ❚❍Ù❈ ▲⑨ ❚✃◆● ❇➐◆❍ P❍×❒◆●
❱⑨ Ù◆● ❉Ö◆●
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ ❚➼❝❤
▼➣ sè✿ ✻✵ ✹✻ ✵✶ ✵✷

✣é ❚❤à ❚❤❛♥❤ ◆❣❛
❈❛♦ ❤å❝ ❑✷✵

❍å❝ ✈✐➯♥ t❤ü❝ ❤✐➺♥✿
▲î♣✿

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿

❚❙✳ ❍ç ▼✐♥❤ ❚♦➔♥

❍⑨ ◆❐■ ✲ ✷✵✶✹


▼ö❝ ❧ö❝
▲í✐ ♥â✐ ✤➛✉

✷

✶ ✣❛ t❤ù❝ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✈➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣

✺

✶✳✶ ▼ð ✤➛✉ ✈➲ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✷ ▼❛ tr➟♥ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✈➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝→❝ ✤❛
t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✶✳✸ ✣❛ t❤ù❝ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✈➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✶✸

✷ ✣✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✤❛ t❤ù❝ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ✈➔ ù♥❣
❞ö♥❣
✷✵

✷✳✶ ✣✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✤❛ t❤ù❝ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✷ ×î❝ ❧÷ñ♥❣ ❝➟♥ ❞÷î✐ ❝õ❛ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

❑➳t ❧✉➟♥

✸✹

❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖

✸✻

✶


▲í✐ ♥â✐ ✤➛✉
❈❤♦ f ∈ R[X] := R[X1, ..., Xn] ❧➔ ✤❛ t❤ù❝ n ❜✐➳♥ ❜➟❝ m✳ ❍✐➸♥ ♥❤✐➯♥✱
♠ët ✤❛ t❤ù❝ f ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ t❤➻ ♥â ❧➔ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✈➻ ❜➻♥❤
♣❤÷ì♥❣ tr♦♥❣ R ❧➔ ❦❤æ♥❣ ➙♠✳ ❱➜♥ ✤➲ ✤➦t r❛ ❧➔✿ ❑❤✐ ♥➔♦✱ ♠ët ✤❛ t❤ù❝

♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ f ✤÷ñ❝ ❜✐➸✉ ❞✐➵♥ ❞÷î✐ ❞↕♥❣ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣❄
• ◆➳✉ n = 1 ✈➔ f ❧➔ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ t❤➻ f ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣
✭①❡♠ ▼➺♥❤ ✤➲ ✶✳✸✮✳
• ❙②❧✈❡st❡r ✭✶✽✺✵✮ ✤➣ ❝❤➾ r❛ r➡♥❣✿ ◆➳✉ f ❧➔ ✤❛ t❤ù❝ ♥û❛ ①→❝ ✤à♥❤
❞÷ì♥❣ ❜➟❝ 2 t❤➻ f ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣✳
◆➠♠ ✶✽✽✽✱ ❍✐❧❜❡rt ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❤❛✐ ✤à♥❤ ❧þ q✉❛♥ trå♥❣ ❧➔✿
• ◆➳✉ f ❧➔ ✤❛ t❤ù❝ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ 2 ❜✐➳♥ ❜➟❝ 4 t❤➻ f ❧➔ tê♥❣
❜➻♥❤ ♣❤÷ì♥❣✳
• ❈→❝ tr÷í♥❣ ❤ñ♣ ❝á♥ ❧↕✐✱ ❧✉æ♥ tç♥ t↕✐ ✤❛ t❤ù❝ f ❧➔ ♥û❛ ①→❝ ✤à♥❤
❞÷ì♥❣ ♥❤÷♥❣ ❦❤æ♥❣ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣✳
❍✐❧❜❡rt ✤➣ ❦❤æ♥❣ ✤÷❛ r❛ ✈➼ ❞ö ❝ö t❤➸ ❝❤♦ ✤❛ t❤ù❝ ❧➔ ♥û❛ ①→❝ ✤à♥❤
❞÷ì♥❣ ♠➔ ❦❤æ♥❣ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣✳ ❱➼ ❞ö ♥ê✐ t✐➳♥❣ ✤➛✉ t✐➯♥ ✤÷ñ❝ ✤÷❛
r❛ ❝❤♦ ✈➜♥ ✤➲ ♥➔② ❧➔ ✤❛ t❤ù❝ ▼♦t③❦✐♥ ✭✶✾✻✵✮
s(X, Y ) = 1 − 3X 2 Y 2 + X 4 Y 2 + X 2 Y 4 .

◆➠♠ ✶✽✾✸✱ ❍✐❧❜❡rt ✤➣ ❝❤ù♥❣ ♠✐♥❤ ♠ët ✤❛ t❤ù❝ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣
❤❛✐ ❜✐➳♥ ❜➜t ❦ý ❧✉æ♥ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝ ❞÷î✐ ❞↕♥❣ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛
✷


ỳ t ổ ổ ự ữủ trữớ ủ
tờ qt õ tr t t tự tr t ừ
rt ữủ ữ r t ữủ tr ớ
rt
ở ừ tổ q s s số ừ
ỡ tự õ t ợ số õ t ỡ ừ ởt
tự ữ r ừ tự õ tờ ữỡ
õ ự ử ởt số ữợ ừ tự
ữ s

f := inf {f (a)|a Rn } ,

.
t fsos f t f õ t t trỹ
t f t t ữợ ừ õ ú t s r r t
t t 2d ừ tự f ởt tr ừ t tt
tự n 2d tờ ữỡ t s õ sỹ ồ ỹ
ũ ủ k r s tự f (kX) r tọ ừ
ữủ ữ r tr q q ú
t ữủ ữợ fsos õ rL rF K ỵ
ỵ ỹ ỵ t ữủ ởt ữợ ỳ
ừ fsos õ rdmt ỵ
ỗ t ử t t
ữỡ ữ s
ữỡ tự ỷ ữỡ tờ ữỡ tr
tờ q t tự ỷ ữỡ t
tờ ữỡ
fsos := sup r r R, f r




ữỡ ừ tự tờ ữỡ ự ử
ữỡ ú t tr ừ tự õ t
ữợ tờ ữỡ ự ử õ
t tố ữ tự
ữủ t t ồ
ồ ổ t ữợ sỹ ữợ ừ ỗ
t ỡ t t t ữợ
ú ù t tr sốt q tr ồ t ự t t

ừ
r q tr ồ t ớ ừ
ổ tr ồ ồ ổ
t t tr ỗ t tự ử ử ổ
ồ t ự ừ t tọ ỏ
ỡ s s tợ ổ
t ỡ sỹ t tứ sỹ ở
ợ tứ ố ổ tr trữớ
tr ợ ồ K20 t ồ ú ù
t tr sốt tớ ồ t ự ồ
ở t
ồ

ộ




❈❤÷ì♥❣ ✶
✣❛ t❤ù❝ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✈➔
tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣
✶✳✶ ▼ð ✤➛✉ ✈➲ ✤❛ t❤ù❝
✣❛ t❤ù❝ f
❞÷î✐ ❞↕♥❣

∈ R[X]

❜➟❝ d ✭❦➼ ❤✐➺✉ ❧➔ deg(f ) = d✮ ❧✉æ♥ ✤÷ñ❝ ♣❤➙♥ t➼❝❤
f = f0 + f1 + ... + fd ,

tr♦♥❣ ✤â ♠é✐ fi ∈ R [X] ❧➔ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ i, i = 1, ..., n✳
▼ët ✤❛ t❤ù❝ f t❤ä❛ ♠➣♥ f (tX) = tdf (X) ✤÷ñ❝ ❣å✐ ❧➔ ✤❛ t❤ù❝ t❤✉➛♥
♥❤➜t ❜➟❝ d✳
k
✣❛ t❤ù❝ f ✤÷ñ❝ ❣å✐ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ♥➳✉ f (X) = fi(X)2✱ ✈î✐
i=1
fi (X) ∈ R[X]✳
❙❛✉ ✤➙② s➩ ❧➔ ♠ët sè ❦➳t q✉↔ ✈➲ ✤❛ t❤ù❝ ♠➔ ❧✉➟♥ ✈➠♥ s➩ sû ❞ö♥❣ tî✐✳
▼➺♥❤ ✤➲ ✶✳✶✳ ❬✹✱ ▼➺♥❤ ✤➲ ✶✳✶✳✶❪ ◆➳✉ f ∈ R[X]✱ f = 0 t❤➻ tç♥ t↕✐ ♠ët
✤✐➸♠ x ∈ Rn s❛♦ ❝❤♦ f (x) = 0.

❈❤ù♥❣ ♠✐♥❤✳

▼➺♥❤ ✤➲ ✤ó♥❣ ✈î✐ n = 1✱ ✈➻ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ❦❤→❝ 0 ❝â ❜➟❝ ❤ú✉ ❤↕♥ ❝❤➾
✺


❝â ❤ú✉ ❤↕♥ ♥❣❤✐➺♠✳ ●✐↔ sû✱ ♠➺♥❤ ✤➲ ✤ó♥❣ ✈î✐ n − 1 ❜✐➳♥✳
❱➻ f = 0 t❛ ♣❤➙♥ t➼❝❤ f ❞÷î✐ ❞↕♥❣
f = g0 + g1 Xn + ... + gk Xnk ,

tr♦♥❣ ✤â g0, ..., gk ∈ R [X1, ..., Xn−1]✱ gk = 0✳
❚❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣✱ tç♥ t↕✐ ♠ët ✤✐➸♠ (x1, ..., xn−1) ∈ Rn−1 s❛♦ ❝❤♦
gk (x1 , ..., xn−1 ) = 0✳ ❉♦ ✤â
k

gi (x1 , ..., xn−1 )Xni

f (x1 , ..., xn−1 , Xn ) =
i=0

❧➔ ✤❛ t❤ù❝ ❦❤→❝ 0 ♠ët ❜✐➳♥ Xn✳ ❙✉② r❛✱ tç♥ t↕✐ xn ∈ R s❛♦ ❝❤♦ f (x1, ..., xn) =
0✳ ✷
❍➺ q✉↔ ✶✳✶✳ ❬✹✱ ❍➺ q✉↔ ✶✳✶✳✸❪ ●✐↔ sû f = f12 + ... + fk2✱ tr♦♥❣ ✤â
f1 , ..., fk ∈ R[X]✱ f1 = 0✳ ❑❤✐ ✤â
✭✐✮ f = 0.
✭✐✐✮ deg (f ) = 2 i=1,...k
max {deg (fi )}✳

❈❤ù♥❣ ♠✐♥❤✳

✭✐✮ ●✐↔ sû f1 = 0✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶✱ tç♥ t↕✐ x ∈ Rn s❛♦ ❝❤♦ f1(x) = 0✳
❑❤✐ ✤â
f (x) = f12 (x) + ... + fk2 (x) > 0.

❉♦ ✤â✱ f = 0✳
✭✐✐✮ P❤➙♥ t➼❝❤ ♠é✐ fi t❤➔♥❤ tê♥❣ ❝→❝ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t✱ tù❝ ❧➔
fi = fi0 + fi1 + ... + fid ,

✈î✐ fij ❧➔ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ j ✈➔ d := 2 i=1,...k
max {deg (fi )}✳
❑❤✐ ✤â✱ deg(f ) ≤ 2d✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû deg(f ) > 2d✱ t❛ ❧↕✐ ❝â deg(f12 +
... + fk2 ) ≤ 2d✱ ♠➔ f = f12 + ... + fk2 ✭✈æ ❧➼✮✳
✻


t tỷ õ 2d tr tờ f12 +...+fk2 f1d2 +...+fkd2
2
d := 2 max {deg (fi )} tỗ t i s fid = 0 õ f1d
+...+fkd =

i=1,...k
0 deg(f ) = 2d
ỵ ỵ f (X) = X n b1X n1 ... bn tr
õ tt bi ổ t t ởt bi = 0 tự f õ ởt
ữỡ t p tr tt ố ừ ỏ ổ ữủt
q p

ự
t

F (X) =

f (X)
b1
bn
=
+
...
+
1.
Xn
X
Xn

X = 0 t õ f (X) = 0 F (X) = 0 X t tứ 0 +
số F (X) t tứ + 1 x > 0 số F trt
t t t ởt p
ự x0 ởt ừ f t q = |x0| p t
sỷ q > p t õ f (q) > 0 t x0 ừ f
xn0 = b1 xn1

+ ... + bn q n b1 q n1 + ... + bn f (q) 0.
0


C(f ) ữỡ t ừ f
ữợ C(X n) := 0.
n
q ởt tự t ý q (t) = i=0 biti ợ bn = 0
ừ q õ ợ tr tr tt ố ừ
n1

C

n

t
i=0



bi i
t .
bn


✶✳✷ ▼❛ tr➟♥ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✈➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣
❝→❝ ✤❛ t❤ù❝
▼ët ♠❛ tr➟♥ A ✤è✐ ①ù♥❣ t❤ü❝✱ ✈✉æ♥❣ ❝➜♣ n ✤÷ñ❝ ❣å✐ ❧➔ ♥û❛ ①→❝ ✤à♥❤
❞÷ì♥❣ ♥➳✉ xT Ax ≥ 0✱ ∀x ∈ Rn✳
▼➺♥❤ ✤➲ ✶✳✷✳ ❬✹✱ ▼➺♥❤ ✤➲ ✵✳✷✳✶❪ ❈❤♦ A ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣✱ t❤ü❝ ❝➜♣

n × n✳ ❈→❝ ✤✐➲✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✭✐✮ xT Ax ≥ 0✱ ∀x ∈ Rn✳
✭✐✐✮ ▼å✐ ❣✐→ trà r✐➯♥❣ ❝õ❛ A ✤➲✉ ❦❤æ♥❣ ➙♠✳
✭✐✐✐✮ A = U T U ✱ ✈î✐ ❯ ❧➔ ♠ët ♠❛ tr➟♥ ❝➜♣ n × n ♥➔♦ ✤â✳
✭✐✈✮ A ❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ ➙♠ ❝õ❛ ❝→❝ ♠❛ tr➟♥ ❞↕♥❣ xxT ✱
✈î✐ x ∈ Rn✳

❈❤ù♥❣ ♠✐♥❤✳

✭✐✮ ⇒ ✭✐✐✮ ❉♦ A ✤è✐ ①ù♥❣ ♥➯♥ ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ A ❧➔ t❤ü❝✳ ▲➜② d
❧➔ ♠ët ❣✐→ trà r✐➯♥❣ ❝õ❛ A✱ t÷ì♥❣ ù♥❣ ✈î✐ ✈❡❝tì r✐➯♥❣ x✳ ❑❤✐ ✤â Ax = dx
♥➯♥
xT Ax = xT dx = dxT x = d x 2 .

❱➻ xT Ax ≥ 0 ✈➔ x = 0 ♥➯♥ d ≥ 0✳
✭✐✐✮ ⇒ ✭✐✐✐✮ ❱➻ A ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ♥➯♥ A = C −1DC ✱ ✈î✐ C ❧➔ ♠❛
tr➟♥ trü❝ ❣✐❛♦ (C T = C −1)✱ D ❧➔ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦ ✭❝→❝ ♣❤➛♥ tû ✤÷í♥❣
❝❤➨♦ ❝õ❛ D ❧➔ ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ A✮✳
●✐↔ sû


d1 . . .



D=


0

✳✳ ✳ ✳ ✳ ✳✳

0 · · · dn



 = diag (d1 , ..., dn ) .


✽


di 0 i = 1, ..., n D =
1

A = C DC = C

T


D



T

D

T

D

õ



T
DC = ( DC) DC = U T U,



tr õ U = DC
A = v1v1T + ... + vnvnT tr õ v1, ..., vn tỡ
ừ U
sỷ A = r1v1v1T + ... + rnv1v1T ợ ri 0
õ
n
n
xT Ax =


ri xT vi viT x =
i=1

ri viT x

2

0.

i=1

ởt tự õ tờ ữỡ ổ t õ t ỹ tr
s
ỵ ờ tự p(X) R[X] 2d õ
s tữỡ ữỡ
p(X) tờ ữỡ
ỗ t tr A ỷ ữỡ s p(X) = zT Az ợ
z = (zi ) zi X : || d

ự

sỷ p(X) tờ ữỡ p(X) = p2j (X)
j=1
ợ deg(pj (X)) d t z1 = 1 z2 = X1 zn+1 = Xn zn+2 = X12
zn+3 = X1 X2 z2n+1 = X1 Xn z2n+2 = X2 X3 zC(n+d,d) = Xnd ữ
ỵ r số ỡ tự X1d+1...Xnd ợ d1 + ... + dn d di 0
m

n

C(n + d, d) =



(n + d)!
.
d!n!

❚❛ ❝â✿


pj (X) =

z1 z2 ... zC(n+d,d)

♥➯♥

m

z T pj

p (X) =

2









pj1
pj2

✳✳

pjC(n+d,d)


m




 = z T pj






m

z T pj pTj z = z T 

=

j=1



j=1

pj pTj  z,
j=1

tr♦♥❣ ✤â




T
pj pj = 






pj1
pj2

✳✳

pjC(n+d,d)









pj1 pj2 ... pjC(n+d,d)

= Aj ,

✈î✐ Aj (j = 1, ..., m) ❧➔ ❝→❝ ♠❛ tr➟♥ ❝➜♣ C (n + d, d) × C (n + d, d)✱ ❝ö t❤➸
❧➔





Aj = 




p2j1

pj1 pj2

pj2 pj1

p2j2

✳✳

✳✳

· · · pj1 pjC(n+d,d)
· · · pj2 pjC(n+d,d)
···

pjC(n+d,d) pj1 pjC(n+d,d) pj2 · · ·

✳✳

p2jC(n+d,d)




.




≥ 0 ♥➯♥
Aj ❧➔ ♠ët ♠❛ tr➟♥ ♥û❛
❉♦ p (X) = z Aj z =
j=1
j=1
j=1
①→❝ ✤à♥❤ ❞÷ì♥❣ ✈➔ ❝â ❝→❝ ♣❤➛♥ tû ❝❤➼♥❤ ❧➔ tê♥❣ ❝õ❛ t➼❝❤ ❧➛♥ ❧÷ñt ❝→❝ ❤➺
T

m

m

p2j (X)

✶✵

m


số ừ tự pj (X) ợ j = 1, ..., m ữ s
m





m


Aj =

j=1




j=1
m

m

m

p2j1

pj1 pj2
j=1
m

pj2 pj1
j=1

m



p2j2



j=1

m

ããã

pj1 pjC(n+d,d)
j=1
m

ããã

pj2 pjC(n+d,d)

ããã

j=1

j=1

j=1

m

pjC(n+d,d) pj2 ã ã ã

pjC(n+d,d) pj1



j=1

p2jC(n+d,d)







.

sỷ A tr ỷ ữỡ p(X) = zT Az, z =
X
tr A õ t số ỡ tự õ ọ ỡ d tự
||d
C (n + d, d)
A tr ỷ ữỡ A tr ố ự
tr r ừ õ số tỹ tỗ t ởt tr ữớ
ỗ t tr r
õ A = SS 1 ợ S tr trỹ tr ữớ
ỗ tt tr r ừ A
ú ỵ S trỹ S 1 = S T = diag(1, 2, ...)





= diag( i) t õ A = SS 1 = S T S T = S (S )T

u1, u2, ..., uC(n+d,d) tỡ ừ (S )T tỡ

uT1 , uT2 , ..., uTC(n+d,d) tỡ ởt ừ (S ) t



A= S


S

T

=

uT1 uT2 ã ã ã uTC(n+d,d)









u1
u2



uC(n+d,d)

= uT1 u1 + ... + uTC(n+d,d) uC(n+d,d) .

❙✉② r❛
p (X) = z T Az = z T uT1 u1 z + ... + z T uTC(n+d,d) uC(n+d,d) z
(uj z)T (uj z),

=
j

✈î✐


uj z =

uTj1 uTj2 · · · uTjC(n+d,d)











z1
z2

✳✳

zC(n+d,d)




=




ujk zk .
k

❉♦ ✤â
2

2
T

p (X) =

(uj z) (uj z) =
j

ujk zk
j

ujk X β

=
j

k

, (|β| ≤ d) .

k

▲÷✉ þ✿ ◆➳✉ tr♦♥❣ ✤❛ t❤ù❝ ❜❛♥ ✤➛✉ ❦❤æ♥❣ ❝❤ù❛ sè ❤↕♥❣ tü ❞♦ t❤➻
t❛ ❝â t❤➸ ❜ä z1 = 1 ✤✐ tr♦♥❣ ❦❤✐ ❧➟♣ ♠❛ tr➟♥ z✳
❱➼ ❞ö ✶✳✶✳ ✣❛ t❤ù❝ p(X, Y, Z) = X 2Z 2 + 2XY Z 2 + 2Y 2Z 2 − 2Y Z 3 + Z 4
❧➔ ♠ët tê♥❣
 ❜➻♥❤♣❤÷ì♥❣✳ ❚❤➟t ✈➟②✿
❳➨t

XZ


z = YZ

Z2



✱


t❛ ❝â


p (X, Y, Z) = z T Az =

XZ Y Z Z 2

✶✷





a b c
XZ






 b d e  Y Z .



2
c e f
Z



❑❤✐ ✤â



1 1 0




A =  1 2 −1 


0 −1 1

❧➔ ♠❛ tr➟♥ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣✳

❈❤✉➞♥ ❤â❛ ♠❛ tr➟♥ A t❛ ✤÷ñ❝ A = SΛS T ✱ tr♦♥❣ ✤â



S=



−1
√
3
1

√
3
1
√
3

√1
2

√1
6
2
0 √6
−1
√1
√
2 6




,


0 0 0




Λ =  0 1 0  = diag (0, 1, 3) .



0 0 3

❱➻ S trü❝ ❣✐❛♦ ♥➯♥ S −1 = S T ♥➯♥
A = SΛS T
√
√
= Sdiag(0, 1, 3)diag(0, 1, 3)S T
√
√
= Sdiag(0, 1, 3)[Sdiag(0, 1, 3)]T .

❱➟②


p (X, Y, Z) =

XZ Y Z Z

2

0


0

0

√1

2

0
√1
2

√1
2
2
√
2
−1
√
2







0
√1
2
√1
2

0

0

0

√1
2
−1
√
2

√2
2



XZ


 Y Z

Z2







1
1
= (XZ + Z 2 )2 + (XZ + 2Y Z − Z 2 )2 .

2
2

✶✳✸ ✣❛ t❤ù❝ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✈➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣
❑➼ ❤✐➺✉ f ≥ 0 tr➯♥ Rn ❧➔ ✤❛ t❤ù❝ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣✳
▼➺♥❤ ✤➲ ✶✳✸✳ ❬✹✱ ▼➺♥❤ ✤➲ ✶✳✷✳✶❪ ●✐↔ sû f = 0 ❧➔ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ X
✶✸


✈➔
(X − ai )ki

f =d
i

((X − bj )2 + c2j )lj
j

❧➔ ♣❤➨♣ ♣❤➙♥ t➼❝❤ t❤➔♥❤ ♥❤➙♥ tû tr♦♥❣ R[X]✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ s❛✉ ❧➔ t÷ì♥❣
✤÷ì♥❣✿
✭✐✮ f ≥ 0 tr➯♥ Rn✳
✭✐✐✮ d > 0 ✈➔ ♠é✐ ki ❧➔ ❝❤➤♥✳
✭✐✐✐✮ f = g2 + h2✱ ✈î✐ g, h ∈ R[X]✳

❈❤ù♥❣ ♠✐♥❤✳

✭✐✮ ⇒ ✭✐✐✮ ❍✐➸♥ ♥❤✐➯♥✳
✭✐✐✮ ⇒ ✭✐✐✐✮ ⑩♣ ❞ö♥❣ ❝æ♥❣ t❤ù❝✿
(a2 + b2 )(c2 + d2 ) = (ac − bd)2 + (ad − bc)2 .

❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
✭✐✐✐✮ ⇒ ✭✐✮ ❍✐➸♥ ♥❤✐➯♥✳ ✷
❈❤♦ ♠ët ✤❛ t❤ù❝ ❜➜t ❦➻
f (X1 , ..., Xn ) =
f

cX1d1 ...Xndn ∈ R [X] , deg(f ) ≤ d.

❝â t❤➸ ✤÷❛ ✈➲ ❞↕♥❣ ♠ët ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t✿
f (X0 , X1 , ..., Xn ) = X0d f
=

Xn
X1
, ...,
X0
X0
X1
c
X0

X0d

d1

Xn
...
X0

dn

n

d−

di

X1d1 ...Xndn

=

cX

=

cX0d0 X1d1 ...Xndn .

i=1

❧➔ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ d✱ (n + 1) ❜✐➳♥ X0, X1, ..., Xn✱ tr♦♥❣ ✤â
n
d0 := d −
di ✳ ✣❛ t❤ù❝ f ❣å✐ ❧➔ t❤✉➛♥ ♥❤➜t ❤â❛ ❝õ❛ ✤❛ t❤ù❝ f ✳
f

i=1

✶✹

◆❣÷ñ❝ ❧↕✐✱ t❛ ❝ô♥❣ ❝â t❤➸ ✤÷❛ ♠ët ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ✈➲ ❞↕♥❣ ❦❤æ♥❣
t❤✉➛♥ ♥❤➜t✿ f (1, X1, ..., Xn) = f (X1, ..., Xn)✳
▼➺♥❤ ✤➲ ✶✳✹✳ ❬✹✱ ▼➺♥❤ ✤➲ ✶✳✷✳✹❪ ✣➦t Vd,n ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❝õ❛ t➜t
❝↔ ❝→❝ ✤❛ t❤ù❝ n ❜✐➳♥ ❜➟❝ ≤ d ①→❝ ✤à♥❤ tr♦♥❣ Rn✱ Fd,n ❧➔ ❦❤æ♥❣ ❣✐❛♥
✈❡❝tì ❝õ❛ t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ n ❜✐➳♥ ❜➟❝ d ①→❝ ✤à♥❤ tr♦♥❣ Rn✳ ⑩♥❤ ①↕
Vd,n → Fd,n+1 ❧➔ ♠ët ✤➥♥❣ ❝➜✉✳ ◆➳✉ d ❝❤➤♥ t❤➻
✭✐✮ f ≥ 0 tr➯♥ Rn ⇔ f ≥ 0 tr➯♥ Rn+1✳
✭✐✐✮ f ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝→❝ ✤❛ t❤ù❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f ❧➔ tê♥❣ ❜➻♥❤
♣❤÷ì♥❣ ❝→❝ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ d2 ✳

❈❤ù♥❣ ♠✐♥❤✳

●✐↔ t❤✐➳t d ❝❤➤♥✱ deg(f ) ≤ d✳
✭✐✮ ●✐↔ sû f ≥ 0 tr➯♥ Rn t❛ ❝â✿
◆➳✉ X0 = 0 t❤➻
f (X0 , X1 , ..., Xn ) = X0d f (

X1
Xn
, ...,
) ≥ 0.
X0
X0

◆➳✉ X0 = 0 t❤➻
f (0, X1 , ..., Xn ) = lim (ε, X1 , ..., Xn ) ≥ 0.
ε→0

●✐↔ sû f ≥ 0 tr➯♥ Rn+1✱ t❛ ❝â
f (X1 , ..., Xn ) = f (1, X1 , ..., Xn ) ≥ 0.

✭✐✐✮ ◆➳✉ f =

k
i=1

fi2

t❤➻ deg(fi) ≤ d2 ✈➔
k

f=

X1
Xn
, ...,
X0
X0

d
2

X 0 fi
i=1

✶✺

2

❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ d2 ✳
k
◆➳✉ f = gi2 t❤➻
i=1

k

gi2 (1, X1 , ..., Xn )

f = f (1, X1 , ..., Xn ) =
i=1

❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ ❝→❝ ✤❛ t❤ù❝✳ ✷
❈❤♦ f ∈ R[X] ❧➔ ✤❛ t❤ù❝ n ❜✐➳♥ ❜➟❝ d✳ ▼ët ✤❛ t❤ù❝ ♥û❛ ①→❝ ✤à♥❤
❞÷ì♥❣ f ✤÷ñ❝ ❜✐➸✉ ❞✐➵♥ ❞÷î✐ ❞↕♥❣ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ tr♦♥❣ ❝→❝ tr÷í♥❣
❤ñ♣✿ ✤❛ t❤ù❝ 1 ❜✐➳♥ ✭①❡♠ ▼➺♥❤ ✤➲ ✶✳✸✮✱ ✤❛ t❤ù❝ ❜➟❝ 2 ✤÷ñ❝ ✤÷❛ r❛
❜ð✐ ❙②❧✈❡st❡r ✭✶✽✺✵✮✱ ✤❛ t❤ù❝ 2 ❜✐➳♥ ❜➟❝ 4 ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ❍✐❧❜❡rt
✭✶✽✽✽✮✳ ❍✐❧❜❡rt ❝ô♥❣ ✤➣ ❝❤➾ r❛ r➡♥❣✿ ❝→❝ tr÷í♥❣ ❤ñ♣ ❝á♥ ❧↕✐ ❧✉æ♥ tç♥ t↕✐
✤❛ t❤ù❝ f ❧➔ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ♥❤÷♥❣ ❦❤æ♥❣ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣✳ ❚✉②
♥❤✐➯♥✱ æ♥❣ ✤➣ ❦❤æ♥❣ ✤÷❛ r❛ ✈➼ ❞ö ❝ö t❤➸ ❝❤♦ ✤❛ t❤ù❝ ❧➔ ♥û❛ ①→❝ ✤à♥❤
❞÷ì♥❣ ❦❤æ♥❣ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣✳ ❱➼ ❞ö ♥ê✐ t✐➳♥❣ ✤➛✉ t✐➯♥ ✤÷ñ❝ ✤÷❛ r❛
❝❤♦ ✈➜♥ ✤➲ ♥➔② ❧➔ ✤❛ t❤ù❝ ▼♦t③❦✐♥ ✭✶✾✻✵✮✳
❱➼ ❞ö ✶✳✷✳ ✣❛ t❤ù❝
s(X, Y ) = 1 − 3X 2 Y 2 + X 2 Y 4 + X 4 Y 2

❧➔ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ♥❤÷♥❣ ❦❤æ♥❣ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣✳
❚❤➟t ✈➟②✱ →♣ ❞ö♥❣ ✤à♥❤ ❧➼ ❈❛✉❝❤② t❛ ❝â
X 4 Y 2 + X 2 Y 4 + 1 ≥ 3X 2 Y 2 ,

❞♦ ✤â s(X, Y ) ≥ 0✳

▼➦t ❦❤→❝✱ ♥➳✉ s(X, Y ) ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝→❝ ✤❛ t❤ù❝ t❤➻ s(X, Y ) =
k
fi2 ✳ ⑩♣ ❞ö♥❣ ❍➺ q✉↔ ✶✳✶✱ t❛ t❤➜② ♠é✐ ✤❛ t❤ù❝ fi ❝â ❜➟❝ ❦❤æ♥❣ q✉→ 3✳
i=1
❉♦ ✤â ❝→❝ ✤ì♥ t❤ù❝ ❝â ♠➦t tr♦♥❣ fi ❧➔
1, X, Y, X 2 , Y 2 , XY, X 3 , Y 3 , X 2 Y, XY 2 .
✶✻


X, Y t tr fi t X 2 Y 2 s t tr s(X, Y )
ữỡ tỹ ợ X 2, Y 2 X 3, Y 3 fi õ s
fi = ai + bi XY + ci X 2 Y + di XY 2 .

ứ s(X, Y ) =

i=1

fi2

số t t số ừ ỡ tự

õ b2i = 3 ổ
i=1
i=1
s(X, Y ) ổ tờ ữỡ
tớ õ õ ử t tự ỷ
ữỡ ữ ổ tờ ữỡ ữủ ú ỵ ữ
ử ữủ ữ r
2

X Y

2



k

k

k

b2i

q(X, Y, Z) = 1 + X 2 Y 2 + Y 2 Z 2 + Z 2 X 2 4XY Z.

ử ữủ ữ r uă
r(X, Y ) = 200[(X 3 4X)2 +(Y 3 4Y )2 ]+(Y 2 X 2 )X(X+2)[X(X2)+2(Y 2 4)]

ử ữủ ữ r r rsts s
p(X, Y ) = 1 X 2 Y 2 + X 4 Y 2 + X 2 Y 4 .

ổ r r
p(X, Y ) =



1
26 + s
3X, 3Y

27

.

Pd,n t ủ ỗ tự ỷ ữỡ
t t n d d,n t ừ Pd,n ỗ
tự tờ ữỡ
ỵ ừ rt ữủ ự ởt ỡ
ỡ ữ s



✣à♥❤ ❧þ ✶✳✸✳ ❬✹✱ ✣à♥❤ ❧þ ✶✳✷✳✻❪ ❈❤♦ d ❝❤➤♥✱ Pd,n =
n≤2

d,n

❤♦➦❝ d = 2 ❤♦➦❝ ✭n = 3 ✈➔ d = 4✮✳

❦❤✐ ✈➔ ❝❤➾ ❦❤✐

❈❤ù♥❣ ♠✐♥❤✳

⑩♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✶✳✹✱ t❛ ❝â✿
❚❤✉➛♥ ♥❤➜t ❤â❛ ✤❛ t❤ù❝ ❝õ❛ ▼♦t③❦✐♥
X 6s

Y Z
,
X X

= X 6 + Y 4Z 2 + Y 2Z 4 − X 2Y 2Z 2

❧➔ ✤❛ t❤ù❝ t❤✉ë❝ P6,3 \ 6,3✳
❚❤✉➛♥ ♥❤➜t ❤â❛ ✤❛ t❤ù❝ ❝õ❛ ❈❤♦✐ ✲ ▲❛♠
W 4q

X Y Z
, ,
W W W

= W 4 + X 2 Y 2 + Y 2 Z 2 + X 2 Z 2 − 4XY ZW

❧➔ ✤❛ t❤ù❝ t❤✉ë❝ P4,4 \
◆➳✉ d ≥ 6 ✈➔ n ≥ 3 t❤➻
X1d s

◆➳✉ d ≥ 4 ✈➔ n ≥ 4 t❤➻
X1d q

4,4

✳

X2 X3
,
X1 X1

∈ Pd,n \

X 2 X3 X4
,
,
X 1 X1 X1

∈ Pd,n \

d,n

.

d,n

.

❍✐➸♥ ♥❤✐➯♥✳
Pd,2 = d,2 t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✸ ✈➔ ▼➺♥❤ ✤➲ ✶✳✹✳
P2,n = 2,n ✳ ❚❤➟t ✈➟②✱ ♠ët ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ 2 ❜➜t ❦➻ ✤➲✉ ❝â
t❤➸ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝ ❞÷î✐ ❞↕♥❣
Pd,1 =

d,1

n

f (X1 , ..., Xn ) =

aij Xi Xj ,
i,j=1

tr♦♥❣ ✤â A = (aij ) ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣✳ ◆➳✉ f ≥ 0 tr➯♥ Rn t❤➻ ♠❛ tr➟♥ A
❧➔ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣✳ ❱➻ ✈➟②✱ A ❝â t❤➸ ♣❤➙♥ t➼❝❤ ❞÷î✐ ❞↕♥❣ A = U T U ✳
❉♦ ✤â✱
f (X) = X T AX = X T U T U X = (U X)T U X = ||U X||2
✶✽


❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣✳
❈❤ù♥❣ ♠✐♥❤ P4,3 = 4,3 ❧➔ ♠ët ✈➜♥ ✤➲ ♣❤ù❝ t↕♣ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤
❜ð✐ ♥❤â♠ t→❝ ❣✐↔✿ ❏✳ ❇♦❝❤♥❛❦✱ ▼✳ ❈♦st❡✱ ▼✳ ✲ ❋✳ ❘♦② ✭✶✾✾✽✮✳ ✷
◆➠♠ ✶✽✾✸✱ ❍✐❧❜❡rt ✤➣ ❝❤ù♥❣ ♠✐♥❤ ♠ët ✤❛ t❤ù❝ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣
2 ❜✐➳♥ ❜➜t ❦ý ❧✉æ♥ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝ ❞÷î✐ ❞↕♥❣ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ ❝→❝
❤➔♠ ❤ú✉ t➾✳ ❚✉② ♥❤✐➯♥✱ æ♥❣ ❦❤æ♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ❝❤♦ tr÷í♥❣ ❤ñ♣ tê♥❣
q✉→t✳ ❱➻ ✈➟②✱ ♥â trð t❤➔♥❤ ❜➔✐ t♦→♥ t❤ù ✶✼ tr♦♥❣ ✷✸ ❜➔✐ t♦→♥ ❝õ❛ ❍✐❧❜❡rt
✤÷ñ❝ ✤÷❛ r❛ ♥➠♠ ✶✾✵✵✳
❇➔✐ t♦→♥✿ ❈❤♦ ✤❛ t❤ù❝ f ∈ R[X] ❜➜t ❦ý✳ ◆➳✉ f ≥ 0 tr➯♥ Rn t❤➻ ❝â ❦➨♦
t❤❡♦ f ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ ❝→❝ ❤➔♠ ❤ú✉ t➾ t❤✉ë❝ R[X] ❤❛② ❦❤æ♥❣❄
❇➔✐ t♦→♥ ♥➔② ✤➣ ✤÷ñ❝ tr↔ ❧í✐ ❜ð✐ ❊♠✐❧ ❆rt✐♥ ✭✶✾✷✼✮✳
▼➦❝ ❞ò✱ ✤❛ t❤ù❝ ▼♦t③❦✐♥ ❦❤æ♥❣ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝→❝ ✤❛ t❤ù❝
♥❤÷♥❣ ♥â ❧↕✐ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ ✹ ❤➔♠ ❤ú✉ t➾ ♥❤÷ s❛✉✿
X 2 Y 2 (X 2 + Y 2 + 1)(X 2 + Y 2 − 2)2 + (X 2 − Y 2 )2
s(X, Y ) =
(X 2 + Y 2 )2
2

X 2 Y (X 2 + Y 2 − 2)
XY 2 (X 2 + Y 2 − 2)
=
+
X2 + Y 2

X2 + Y 2
2
2
XY (X 2 + Y 2 − 2)
X2 − Y 2
+
+
.
X2 + Y 2
X2 + Y 2

❙ü ♣❤➙♥ t➼❝❤ ♥➔② ✤÷ñ❝ ✤÷❛ r❛ ❜ð✐ ▼✳❇r❡♠♥❡r✳

✶✾

2


❈❤÷ì♥❣ ✷
✣✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✤❛ t❤ù❝ ❧➔ tê♥❣
❜➻♥❤ ♣❤÷ì♥❣ ✈➔ ù♥❣ ❞ö♥❣
◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✤❛ t❤ù❝
❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ✭✈✐➳t t➢t ❧➔ ❙❖❙✮ ✈➔ tø ✤â s➩ ❝❤➾ r❛ ♠ët ✈➔✐ ❝➟♥
❞÷î✐ ❝õ❛ ✤❛ t❤ù❝✳

✷✳✶ ✣✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✤❛ t❤ù❝ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣
❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ tr➻♥❤ ❜➔② ❤❛✐ ✤à♥❤ ❧þ q✉❛♥ trå♥❣ ✤➸ ♠ët ✤❛ t❤ù❝
t❤✉➛♥ ♥❤➜t ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣✳ ❙❛✉ ✤â ❧➔ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ♠ët ✤❛ t❤ù❝
❜➜t ❦➻ ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣✳
▼ët ✤❛ t❤ù❝ ❜➜t ❦➻ f ∈ R[X] ❝â t❤➸ ✈✐➳t ❞÷î✐ ❞↕♥❣ f (X) = α∈N fαX α✱

tr♦♥❣ ✤â X α := X1α ...Xnα ✈➔ α ∈ Nn s❛♦ ❝❤♦ |α| := α1 + ... + αn✱ fα ∈ R
✈➔ fα = 0 ♥❣♦↕✐ trø ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ α✳
❑➼ ❤✐➺✉ Ω(f ) = {α ∈ Nn|fα = 0} \ {0, 2dε1, ..., 2dεn}✱ tr♦♥❣ ✤â 2d =
n

1

n

✷✵


deg(f )✱ εi = (δi1 , ..., δin )✱

✈î✐

 1, i = j
δij =
 0, i = j

❧➔ Ω✳ ◆❤÷ ✈➟②✱ ✤❛ t❤ù❝ f ❝â t❤➸ ✈✐➳t ❞÷î✐ ❞↕♥❣
n
α

f2d,i Xi2d .

fα X +

f = f0 +

i=1

α∈Ω(f )

❑➼ ❤✐➺✉
❦❤æ♥❣ ❧➔ ❙❖❙ tr♦♥❣ R[X]
α ∈ Ω(f )|fα < 0 ❤♦➦❝ αi ❧➔ sè ❧➫✱ ✈î✐ i ∈ {1, ..., n}

∆(f ) = α ∈ Ω(f )|fα X α
=

❧➔ ∆✳
✣❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ 2d
f˜ (X, Y ) = Y 2d f

X1
Xn
, ...,
Y
Y
n

= f0 Y

2d

α

fα X Y

+

2d−|α|

f2d,i Xi2d

+
i=1

α∈Ω

❧➔ t❤✉➛♥ ♥❤➜t ❤â❛ ❝õ❛ ✤❛ t❤ù❝ f ✳
❚❤æ♥❣ q✉❛ ✈✐➺❝ s♦ s→♥❤ ❝→❝ ❤➺ sè ❝õ❛ Xi2d(i = 1, ..., n) ✈î✐ ❤➺ sè ❝õ❛
✤ì♥ t❤ù❝ X α(α ∈ Ω) t❛ ✤÷❛ r❛ ♠ët ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t
❧➔ ❙❖❙✳
✣à♥❤ ❧þ ✷✳✶✳ ❬✶✱ ✣à♥❤ ❧þ ✷✳✸❪ ❈❤♦ ♠ët ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ 2d
n

βi Xi2d − µX α ,

E(X) =
i=1

tr♦♥❣ ✤â αi > 0✱ βi ≥ 0 ✭✐ ❂ ✶✱✳✳✳✱♥✮ ✈➔ µ ≥ 0 ♥➳✉ t➜t ❝↔ αi ❧➔ ❝❤➤♥✳ ❈→❝
✤✐➲✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✭✐✮ E ❧➔ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣✳
✷✶


n

αi
2d

✭✐✐✮ |µ| ≤ 2d
✳
i=1
✭✐✐✐✮ E ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ ❝→❝ ♥❤à t❤ù❝✳
✭✐✈✮ E ❧➔ ❙❖❙✳
▼✳ ●❤❛s❡♠✐ ✈➔ ▼✳ ▼❛rs❤❛❧❧ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ ♥➔② ❝❤♦ ✤❛ t❤ù❝ t❤✉➛♥
♥❤➜t ❜➟❝ 2d ❜➜t ❦➻ ♥❤÷ s❛✉✿
✣à♥❤ ❧þ ✷✳✷✳ ❬✷✱ ✣à♥❤ ❧þ ✷✳✶❪ ●✐↔ sû f ∈ R[X] ❧➔ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t
❜➟❝ 2d ✈➔
βi
αi

f2d,i ≥

|fα |
α∈∆

❑❤✐ ✤â f ❧➔ ❙❖❙✳

αi
,
2d

i = 1, ..., n.

❈❤ù♥❣ ♠✐♥❤✳

❚r÷í♥❣ ❤ñ♣ ✶✿ α ∈ ∆.
◆❤➟♥ t❤➜②

αi
|fα | 2d
αi

2d
αi =0

αi
2d

= 2d

|fα |
= |fα | ≥ |fα |
2d

✈➔ fα < 0 ♥➳✉ t➜t ❝↔ αi ❧➔ ❝❤➤♥✳ ⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✶ t❛ ❝â
|fα |
αi =0

αi 2d
Xi + f α X α
2d

❧➔ ❙❖❙✱ ✈î✐ ♠é✐ α ∈ ∆✳ ❉♦ ✤â
n

|fα |
i=1

αi 2d
Xi + f α X α
2d

❧➔ ❙❖❙✱ ✈î✐ ♠é✐ α ∈ ∆✳ ❙✉② r❛
n

|fα |
i=1

❧➔ ❙❖❙✳ ❚❛ ❧↕✐ ❝â
n

n

f2d,i Xi2d
i=1

−
i=1

α∈∆

✭✷✳✶✮

αi

Xi2d +
fα X α
2d
α∈∆

αi
|fα |
Xi2d =
2d
α∈∆
✷✷

n

f2d,i −
i=1

|fα |
α∈∆

αi
Xi2d
2d

✭✷✳✷✮


❧➔ ❙❖❙✱ ❞♦
f2d,i ≥

❈ë♥❣ ✭✷✳✶✮ ✈➔ ✭✷✳✷✮ t❛ ❝â✿

|fα |
α∈∆

αi
,
2d

i = 1, ..., n.

n

fα X α

f2d,i Xi2d +
i=1

α∈∆

❧➔ ❙❖❙✳
❚r÷í♥❣ ❤ñ♣ ✷✿ α ∈ Ω \ ∆.
❍✐➸♥ ♥❤✐➯♥ t❛ ❝â✱ fαX α ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ ✤ì♥ t❤ù❝ ✭t❤❡♦ ✤à♥❤ ♥❣❤➽❛
❝õ❛ ∆✮✳ ❉♦ ✤â
n
fα X α

f2d,i Xi2d +
i=1

α∈Ω\∆

❧➔ ❙❖❙✳
❱➟② f ❧➔ ❙❖❙✳ ✷
❱➼ ❞ö ✷✳✶✳ ❈❤♦ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ 4✿
f (X, Y, Z) = X 4 + Y 4 + 4Z 4 + 4XZ 3 .

⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✷ ❝❤♦ ❝→❝ ❤➺ sè ❝õ❛ f t❛ ❝â t❤➸ ❝❤➾ r❛ f ❧➔ ❙❖❙✳
❍➺ q✉↔ ✷✳✶✳ ❈❤♦ ✤❛ t❤ù❝ ❜➜t ❦➻ f ∈ R[X] ❝â ❜➟❝ ❧➔ 2d✱ ♥➳✉
f0 ≥

|fα |
α∈∆

✈➔
f2d,i ≥

t❤➻ f ❧➔ ❙❖❙✳

|fα |
α∈∆

2d − |α|
2d

αi
, i = 1, ..., n
2d

❈❤ù♥❣ ♠✐♥❤✳

✭✷✳✸✮
✭✷✳✹✮

❚❤✉➛♥ ♥❤➜t ❤â❛ ✤❛ t❤ù❝ f t❤➔♥❤ ✤❛ t❤ù❝ f ✳ ❙❛✉ ✤â →♣ ❞ö♥❣ ✣à♥❤ ❧þ
✷✸


✷✳✷✱ t❛ ❝â f ❧➔ ❙❖❙✳ ❚✐➳♣ t❤❡♦ sû ❞ö♥❣ ▼➺♥❤ ✤➲ ✶✳✹ t❛ ✤÷ñ❝ f ❧➔ ❙❖❙✳
❱➼ ❞ö ✷✳✷✳ ⑩♣ ❞ö♥❣ ❍➺ q✉↔ ✷✳✶ ❝❤♦ ✤❛ t❤ù❝ f (X, Y, Z) = X 4 + 3Y 4 +
2Z 4 + 3Y 2 Z 2 − 2X 2 Y + 1 t❛ ❝â f ❧➔ ❙❖❙✳
✣à♥❤ ❧þ ✷✳✸✳ ❬✷✱ ✣à♥❤ ❧þ ✷✳✸❪ ●✐↔ sû f ∈ R[X] ❧➔ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t
❜➟❝ 2d ✈➔
min f2d,i ≥

i=1,...,n

1
1
|fα |(αα ) 2d ,
2d α∈∆

tr♦♥❣ ✤â αα := α1α ...αnα .
1

n

❑❤✐ ✤â f ❧➔ ❙❖❙✳
◗✉② ÷î❝ 00 := 1✳

❈❤ù♥❣ ♠✐♥❤✳

✣➦t eα = 2d1 |fα|(αα) ≥ 0✳
❚r÷í♥❣ ❤ñ♣ ✶✿ α ∈ ∆✳
❱➻
1
2d

2d
αi =0

αi
2d

eα
αi

=

|fα | eα
1

|fα |(αα ) 2d
2d

= |fα | ≥ |fα |

✈➔ fα < 0 ♥➳✉ t➜t ❝↔ αi ❧➔ ❝❤➤♥ ♥➯♥

Xi2d + fα X α

eα
αi =0

❧➔ ❙❖❙✱ ✈î✐ ♠é✐ α ∈ ∆ ✭t❤❡♦ ✣à♥❤ ❧þ ✷✳✶✮✳ ❉♦ ✤â
n

Xi2d + fα X α

eα
i=1

❧➔ ❙❖❙✱ ✈î✐ ♠é✐ α ∈ ∆✳ ❙✉② r❛
n

α∈∆

i=1

✭✷✳✺✮

fα X α

Xi2d +

eα

α∈∆

❧➔ ❙❖❙✳ ❚❛ ❧↕✐ ❝â

n

n

f2d,i Xi2d
i=1

n

eα Xi2d

−
i=1 α∈∆

i=1

✷✹

eα Xi2d ,

f2d,i −

=

α∈∆

✭✷✳✻✮