Thác triển của ánh xạ chỉnh hình giữa các siêu mặt thực có số chiều khác nhau (LV thạc sĩ)

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

❍⑨ ❚❍➚ ▲➒◆❍

❚❍⑩❈ ❚❘■➎◆ ❈Õ❆ ⑩◆❍ ❳❸ ❈❍➓◆❍ ❍➐◆❍
●■Ú❆ ❈⑩❈ ❙■➊❯ ▼➄❚ ❚❍Ü❈
❈➶ ❙➮ ❈❍■➋❯ ❑❍⑩❈ ◆❍❆❯

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

❍⑨ ❚❍➚ ▲➒◆❍

❚❍⑩❈ ❚❘■➎◆ ❈Õ❆ ⑩◆❍ ❳❸ ❈❍➓◆❍ ❍➐◆❍
●■Ú❆ ❈⑩❈ ❙■➊❯ ▼➄❚ ❚❍Ü❈
❈➶ ❙➮ ❈❍■➋❯ ❑❍⑩❈ ◆❍❆❯

❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝
❚❙✳ ◆❣✉②➵♥ ❚❤à ❚✉②➳t ▼❛✐

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


▲í✐ ❝❛♠ ✤♦❛♥
❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣
t❤ü❝ ✈➔ ❦❤æ♥❣ trò♥❣ ❧➦♣ ✈î✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ❊♠ ❝ô♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣
♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷ñ❝ ❝↔♠ ì♥ ✈➔ ❝→❝
t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ✤÷ñ❝ ❝❤➾ rã ♥❣✉ç♥ ❣è❝✳

❚❤→✐ ♥❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✺
◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥

❍➔ ❚❤à ▲➽♥❤
❳→❝ ♥❤➟♥

❳→❝ ♥❤➟♥

❝õ❛ tr÷ð♥❣ ❦❤♦❛ ❝❤✉②➯♥ ♠æ♥

❝õ❛ ♥❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝

❚❙✳ ◆❣✉②➵♥ ❚❤à ❚✉②➳t ▼❛✐

✐


▼ö❝ ❧ö❝
▲í✐ ❝❛♠ ✤♦❛♥

✐

▼ö❝ ❧ö❝

✐✐

▼ð ✤➛✉

✶

✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✹

✶✳✶

✣❛ t↕♣ ♣❤ù❝ ❬✷❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷

❙✐➯✉ ♠➦t t❤ü❝ tr♦♥❣ Cn ❬✸❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✸

❍➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐✱ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ❬✶❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✸✳✶

❍➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✸✳✷

❍➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

▼✐➲♥ ❣✐↔ ❧ç✐✱ ❣✐↔ ❧ç✐ ❝❤➦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✹✳✶

▼✐➲♥ ❣✐↔ ❧ç✐ ❬✶✹❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✹✳✷

▼✐➲♥ ❣✐↔ ❧ç✐ ❝❤➦t ❬✽❪

✶✳✹

✶✳✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

❚➟♣ ❣✐↔✐ t➼❝❤ ♣❤ù❝ ❬✺❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✶✳✺✳✶

✶✶

❚➟♣ ❣✐↔✐ t➼❝❤ ♣❤ù❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✐✐


✶✳✺✳✷

❙è ✤è✐ ❝❤✐➲✉ ❝õ❛ t➟♣ ❣✐↔✐ t➼❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✶✳✺✳✸

❚➟♣ ❣✐↔✐ t➼❝❤ ❜➜t ❦❤↔ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

⑩♥❤ ①↕ r✐➯♥❣ ❬✺❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✶✳✻✳✶

⑩♥❤ ①↕ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✶✳✻✳✷

❷♥❤ ❝õ❛ t➟♣ ❣✐↔✐ t➼❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✶✳✼

❍➔♠ ①→❝ ✤à♥❤ ❝❤➼♥❤ t➢❝ ❬✺❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶

✶✳✽

✣❛ t↕♣ ✤↕✐ sè ①↕ ↔♥❤ ♣❤ù❝ ❬✺❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

✶✳✽✳✶

❑❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

✶✳✽✳✷

❚➟♣ ✤↕✐ sè✱ ✤❛ t↕♣ ✤↕✐ sè ①↕ ↔♥❤ ♣❤ù❝

✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

❙è ❝❤✐➲✉ ❣❡♥❡r✐❝ ❬✹❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✹

✶✳✻

✶✳✾

✷ ❚❤→❝ tr✐➸♥ ❝õ❛ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❣✐ú❛ ❝→❝ s✐➯✉ ♠➦t t❤ü❝
❝â sè ❝❤✐➲✉ ❦❤→❝ ♥❤❛✉
✷✻
✷✳✶

✷✳✷

▼ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❬✶✻❪

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✻

✷✳✶✳✶

✣❛ t↕♣ ❙❡❣r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✻

✷✳✶✳✷

❈ü❝ t✉②➳♥ ❝õ❛ ♠ët t➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

✷✳✶✳✸

◗✉ÿ t➼❝❤ r➩ ♥❤→♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✾

✷✳✶✳✹

✣÷í♥❣ ❝♦♥❣ ❈❘✱ q✉ÿ ✤↕♦ ❈❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✾

❚❤→❝ tr✐➸♥ ❝õ❛ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❣✐ú❛ ❝→❝ s✐➯✉ ♠➦t t❤ü❝

❝â sè ❝❤✐➲✉ ❦❤→❝ ♥❤❛✉ ❬✶✻❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✵

✷✳✷✳✶

❚❤→❝ tr✐➸♥ t❤❡♦ Qξ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✶

✷✳✷✳✷

❚❤→❝ tr✐➸♥ t❤❡♦ Qa ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✷

✐✐✐


✷✳✷✳✸

❚❤→❝ tr✐➸♥ t÷ì♥❣ ù♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✷✳✹

▼ët sè ✤à♥❤ ❧þ t❤→❝ tr✐➸♥ ❝õ❛ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤
❣✐ú❛ ❝→❝ s✐➯✉ ♠➦t t❤ü❝ ❝â sè ❝❤✐➲✉ ❦❤→❝ ♥❤❛✉

✳ ✳ ✳

✸✽

✹✶

❑➳t ❧✉➟♥

✹✽

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✾

✐✈



tr ởt tr ỳ t tr t ừ
t ự ỳ ụ ữ ổ r t ợ õ t
ồ q t tợ ữ
r t ữủ t q ự q trồ
t tr ữủ tr t
ữợ
ữợ tự t tr
ỏ ồ t tr rts
ữợ tự tr q t ọ tự t õ
ở s tr ữủ ồ t tr

ởt tr ỳ ữợ ự ừ t tr
t tr ừ ỳ s t
tr ừ ởt ừ ỳ s t tỹ t

út ữủ sỹ ú ỵ ừ t ồ ữ P
ts Pr ữớ ữợ tr trữớ
ủ s t ỗ s t õ ũ số
rữớ ủ s t ỗ s t số t ữớ




ữ õ ỡ P ữớ ữ r t q t trữớ
ủ ự r ởt

ừ tứ s
Cn S 2N 1 t tr

t t tỹ ỗ t M
ồ t ữớ tr M r
ự r ởt ừ tứ s t t
tỹ ỗ M Cn S 2N 1 t tr ồ
t ữớ tr M s s r tờ qt
t q tr ữ r ỵ M s t ỹ t t
tỹ tổ tr Cn M s t số tỹ ỗ t t
tr CN 1 < n N. sỷ f ởt ừ t
p M f (M ) M õ f t tr ữ ởt tứ
M M t t ý ữớ tr M ỡ ỳ tr trữớ
ủ dimM = dimM ỵ tr tờ qt t q ừ P
tr tr õ s t M ữủ sỷ ỳ
ỡ M ỹ t
ử ừ ự t tr ừ ởt
ừ tứ s t t tỹ s t
số tỹ õ số ợ ỡ ở ừ ữủ tr

tr ữỡ
ữỡ tr ỳ tự ỡ s t ự t t
ự t t ỡ ừ t t ự ổ
ự t số ự
ữỡ tr ởt tt ró r t tr ừ
ỳ s t tỹ t s t s t
số õ số ợ ỡ




t ữủ ổ ữủ sỹ ữợ
ú ù t t ừ t ồ ữ P
t tọ ỏ t ỡ s s ổ
ỷ ớ tr t ừ ố ợ ỳ ổ
t ỡ ỏ s ồ qỵ t
ổ ợ ồ rữớ ồ ữ P
ồ t t tr t ỳ tự qỵ
ụ ữ t t õ ồ
ỷ ớ ỡ t t tợ ỳ
ữớ ổ ở ộ trủ t ồ tr sốt
q tr ồ t tỹ
ũ ố rt ữ tr ổ t tr
ọ ỳ t sõt rt õ ữủ ỳ ỵ õ õ ừ
t ổ




❈❤÷ì♥❣ ✶

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶ ✣❛ t↕♣ ♣❤ù❝ ❬✷❪
❈❤♦ M ❧➔ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ❍❛✉s❞♦r❢❢✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈➦♣ (V, ϕ) ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ❜↔♥ ✤ç ✤à❛ ♣❤÷ì♥❣ ❝õ❛
M✱

tr♦♥❣ ✤â V ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ M ✈➔ ϕ : V
❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷ñ❝ t❤ä❛ ♠➣♥ ✿

→ Cn

❧➔ ♠ët →♥❤ ①↕✱ ♥➳✉

✐✮ ϕ(V ) ❧➔ t➟♣ ♠ð tr♦♥❣ Cn✱
✐✐✮ ϕ : V

→ ϕ(V )

❧➔ ♠ët ✤ç♥❣ ♣❤æ✐✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❍å A = {(Vi, ϕi)}i∈I ❝õ❛ M ✤÷ñ❝ ❣å✐ ❧➔ ♠ët t➟♣ ❜↔♥
✤ç ❣✐↔✐ t➼❝❤ ✭❛t❧❛s✮ ❝õ❛ M ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷ñ❝ t❤ä❛ ♠➣♥
✐✮ {Vi}i∈I ❧➔ ♠ët ♣❤õ ♠ð ❝õ❛ M ✱
✐✐✮ ❱î✐ ♠å✐

♠➔ Vi ∩ Vj = ∅✱ →♥❤ ①↕
ϕj (Vi ∩ Vj ) ❧➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤✳
Vi , Vj

ϕj ◦ ϕi −1 : ϕi (Vi ∩ Vj ) →

❳➨t ❤å ❝→❝ ❛t❧❛s tr➯♥ M ✳ ❍❛✐ ❛t❧❛s ❣å✐ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ♥➳✉ ❤ñ♣ ❝õ❛
❝❤ó♥❣ ❝ô♥❣ ❧➔ ♠ët ❛t❧❛s tr➯♥ M ✳ ❉➵ t❤➜② sü t÷ì♥❣ ✤÷ì♥❣ ❣✐ú❛ ❝→❝ ❛t❧❛s

✹


❧➟♣ t❤➔♥❤ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣✳ ▼é✐ ❧î♣ t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ q✉❛♥ ❤➺
t÷ì♥❣ ✤÷ì♥❣ tr➯♥ ❣å✐ ❧➔ ♠ët ❝➜✉ tró❝ ❦❤↔ ✈✐ ♣❤ù❝ tr➯♥ M ✳ M ❝ò♥❣ ✈î✐
❝➜✉ tró❝ ❦❤↔ ✈✐ ♣❤ù❝ tr➯♥ ♥â ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ✤❛ t↕♣ ♣❤ù❝ n ❝❤✐➲✉✳
❚❛ ❜✐➳t r➡♥❣✱ ♠ët ❧î♣ t÷ì♥❣ ✤÷ì♥❣ ❤♦➔♥ t♦➔♥ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ♠ët
✤↕✐ ❞✐➺♥ ❝õ❛ ♥â✳ ❉♦ ✤â ♠ët ❛t❧❛t ❦❤↔ ✈✐ ❤♦➔♥ t♦➔♥ ①→❝ ✤à♥❤ ♠ët ❝➜✉ tró❝
❦❤↔ ✈✐✳

❱➼ ❞ö ✶✳ ❈❤♦ D ⊂ Cn ❧➔ ♠ët ♠✐➲♥✳ ❑❤✐ ✤â✱ D ❧➔ ♠ët ✤❛ t↕♣ ♣❤ù❝ n
❝❤✐➲✉ ✈î✐ ❜↔♥ ✤ç ✤à❛ ♣❤÷ì♥❣ {(D, IdD )}.

❱➼ ❞ö ✷✳ ✣❛ t↕♣ ①↕ ↔♥❤ P n(C).

❳➨t q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ Cn+1 \ {0} ①→❝ ✤à♥❤ ❜ð✐ x ∼ y ↔ ∃λ = 0
✤➸ y = λx. ❚❛ ❣å✐ P n (C) = Cn+1 \ {0} ∼ ✈î✐ tæ♣æ t❤÷ì♥❣✳
✣➦t Vi = {x = (x0 , . . . , xn ) ∈ Cn+1 \ {0} : xi = 0} ✈î✐ i = 0, 1, . . . , n✳
❘ã r➔♥❣ {Vi }i=0,...,n ❧➔ ♠ët ♣❤õ ♠ð ❝õ❛ P n (C)✳
❳➨t ❝→❝ ✤ç♥❣ ♣❤æ✐ ϕi : Vi → Cn ❝❤♦ ❜ð✐

x0
xˆi
xn

ϕi (x) = ( i , . . . , i , . . . , i ).
x
x
x
ð ✤â✱ ❦➼ ❤✐➺✉ ˆ ❝â ♥❣❤➽❛ ❧➔ sè ❤↕♥❣ ❞÷î✐ ♠ô ✤â ✤÷ñ❝ ❜ä ✤✐✳ ❑❤✐ ✤â →♥❤
0
n−1
①↕ ♥❣÷ñ❝ ✤÷ñ❝ ❝❤♦ ❜ð✐✿ ϕ−1
) = [(y 0 , . . . , y i−1 , 1, y i , . . . , y n−1 )].
i (y , . . . , y

●✐↔ sû (Vi , ϕi ) ✈➔ (Vj , ϕj ) ❧➔ ❤❛✐ ❜↔♥ ✤ç ✤à❛ ♣❤÷ì♥❣ tr➯♥ P n (C) ✈➔ i < j
t❤➻ ϕj ◦ ϕ−1
i : ϕi (Vi ∩ Vj ) → ϕj (Vi ∩ Vj ) ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝

(y 0 , . . . , y n−1 ) → (

y i−1 1
yˆj
y n−1
y0
,
.
.
.
,
,
,
.
.

.
,
,
.
.
.
,
).
yj
yj yj
yj
yj

❘ã r➔♥❣ ϕj ◦ ϕ−1
i ❧➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤✳ ❱➻ ✈➟② ❤å {(Vi , ϕi )} ❧➔ t➟♣ ❜↔♥
✤ç ✤à❛ ♣❤÷ì♥❣ ①→❝ ✤à♥❤ ❝➜✉ tró❝ ❦❤↔ ✈✐ ♣❤ù❝ tr➯♥ P n (C)✱ ✈➻ ✈➟② P n (C)
❧➔ ✤❛ t↕♣ ❦❤↔ ✈✐ ♣❤ù❝ n ❝❤✐➲✉✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ●✐↔ sû M, N ❧➔ ❤❛✐ t↕♣ ❦❤↔ ✈✐ ♣❤ù❝✳ ⑩♥❤ ①↕ ❧✐➯♥
✺


tử f : M N ữủ ồ tr M ợ ồ ỗ
ữỡ (U, ) ừ M (V, ) ừ N s f (U ) V, t
f 1 : (U ) (V )

M Cn ữủ ồ t ự õ số
ợ ộ a M õ
f1 , ..., fnp tr U s
p

U

ừ a tr

Cn



M U = {z U : f1 (z) = ã ã ã = fnp (z) = 0}

rankaf = n p tr õ f = (f1, ..., fnp)

t ự M tr Cn ữủ

f = (f1, ..., fnp) tr ừ a M tỡ
v Cn tỡ t ú ợ M t a (dfk )a (v) = 0, k = 1, ..., n p
tự tt fk t a t ữợ v trt t tt
tỡ õ ữ tr ữủ ồ ổ t ú t a ỵ
Ta M
(dfk )a (z) t t ự Ta M ổ ự
tr Cn
ở ự t ừ t M t a M ữủ t
ự Tac M = (Ta M ) i(Ta M ) ừ ổ t ú ố ự ừ

Tac M ữủ ồ ừ M t a ỵ CRdima M t õ
số ữủ ồ t t

M t tỹ = (1, ..., d) trt

t tr õ d ố ừ M j tr tỹ ử

j
tở t t t tỡ ự (
, ..., j )(0), 1
z1
zn
j d, ử tở t t t t tỹ M ữủ ồ r



✶✳✷ ❙✐➯✉ ♠➦t t❤ü❝ tr♦♥❣ Cn ❬✸❪
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❈❤♦ M ❧➔ t➟♣ ❝♦♥ tr♦♥❣ Cn✳ M ✤÷ñ❝ ❣å✐ ❧➔ s✐➯✉ ♠➦t

t❤ü❝ ♥➳✉ ✈î✐ ♠å✐ z ∈ M tç♥ t↕✐ ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ z tr♦♥❣ Cn ✈➔ ❤➔♠
❦❤↔ ✈✐ ❧✐➯♥ tö❝ ρ : U → R ♥❤➟♥ ❣✐→ trà t❤ü❝ tr♦♥❣ U s❛♦ ❝❤♦
M ∩ U = {z ∈ U : ρ(z) = 0}

✈î✐ dρ(z) = 0, ∀z ∈ U ✳
❍➔♠ ρ ♥❤÷ tr➯♥ ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❤↕♥ ❝❤➳ ✤à❛ ♣❤÷ì♥❣✳ ◆➳✉ ρ ❧➔ ❤➔♠
❣✐↔✐ t➼❝❤ t❤ü❝ trì♥ t❤➻ M ✤÷ñ❝ ❣å✐ ❧➔ s✐➯✉ ♠➦t ❣✐↔✐ t➼❝❤ t❤ü❝ trì♥✳
❚❤ü❝ r❛✱ s✐➯✉ ♠➦t ❧➔ ❝→✐ t➯♥ ✤➦❝ ❜✐➺t ❝õ❛ ✤❛ t↕♣ ❝♦♥ ❝â sè ✤è✐ ❝❤✐➲✉ ✶✳
❱➻ ✈➟②✱ ♥➳✉ M ❧➔ ✤❛ t↕♣ tæ♣æ ❝â sè ✤è✐ ❝❤✐➲✉ ✶ t❤➻ M ❝ô♥❣ ✤÷ñ❝ ❣å✐ ❧➔
s✐➯✉ ♠➦t✳
❱➼ ❞ö ✶✿ ❙✐➯✉ ♠➦t ✤÷ñ❝ ❝❤♦ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤

Imzn = 0
❧➔ s✐➯✉ ♣❤➥♥❣ tr♦♥❣ Cn .
❱➼ ❞ö ✷✿ ▼➦t ❝➛✉ ✤ì♥ ✈à tr♦♥❣ Cn ✤÷ñ❝ ❝❤♦ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤
n

|zj |2 = 1
j=1

❧➔ ♠ët s✐➯✉ ♠➦t ❝♦♠♣❛❝t✳
❱➼ ❞ö ✸✿ ❙✐➯✉ ♠➦t tr♦♥❣ CN ✤÷ñ❝ ❝❤♦ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤
N −1

|zj |2

ImzN =
j=1

❣å✐ ❧➔ s✐➯✉ ♠➦t ▲❡✈②✳

✼


✶✳✸ ❍➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐✱ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ❬✶❪
✶✳✸✳✶ ❍➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ●✐↔ sû D ❧➔ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✳ ❍➔♠
u : D → [−∞, +∞)

✤÷ñ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ tr➯♥ D ♥➳✉ ✈î✐ ♠é✐ r ∈ R t➟♣
Dr = {z ∈ D : u(z) < r}

❧➔ ♠ð tr♦♥❣ D✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✷✳ ●✐↔ sû Ω ❧➔ t➟♣ ♠ð tr♦♥❣ C✳ ❍➔♠ u : Ω → [−∞, +∞)

❣å✐ ❧➔ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ Ω ♥➳✉ ♥â ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ ✈➔ t❤ä❛ ♠➣♥ ❜➜t
✤➥♥❣ t❤ù❝ ❞÷î✐ tr✉♥❣ ❜➻♥❤ tr➯♥ Ω✱ ♥❣❤➽❛ ❧➔ ✈î✐ ♠å✐ w ∈ Ω tç♥ t↕✐ ρ > 0
s❛♦ ❝❤♦ ✈î✐ ♠å✐ 0 ≤ r < ρ t❛ ❝â
2π

1
u(w) ≤ 2π

u(w + reit )dt.
0

❈❤ó þ r➡♥❣ ✈î✐ ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❤➻ ❤➔♠ ✤ç♥❣ ♥❤➜t −∞ tr➯♥ Ω ✤÷ñ❝
①❡♠ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ Ω. ❚➟♣ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ Ω ❦þ
❤✐➺✉ ❧➔ SH(Ω)✳ ▼➺♥❤ ✤➲ s❛✉ ❝❤♦ t❛ ✈➼ ❞ö ✤→♥❣ ❝❤ó þ ✈➲ ❤➔♠ ✤✐➲✉ ❤á❛
❞÷î✐✳

▼➺♥❤ ✤➲ ✶✳✸✳✸✳ ◆➳✉ f : Ω → C ❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ Ω t❤➻ log |f | ❧➔
❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ Ω✳

✶✳✸✳✷ ❍➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✹✳ ●✐↔ sû Ω ⊆ Cn ❧➔ t➟♣ ♠ð✱ u : Ω → [−∞, +∞) ❧➔

❤➔♠ ♥û❛ ❧✐➯♥ tö❝ tr➯♥✱ ❦❤æ♥❣ ✤ç♥❣ ♥❤➜t ❜➡♥❣ −∞ tr➯♥ ♠å✐ t❤➔♥❤ ♣❤➛♥
✽


tổ ừ . u ữủ ồ ỏ ữợ tr ợ ồ
a b Cn u(a + b) ỏ ữợ tr

ồ t tổ ừ t { C : a + b }.
ỵ P SH() t ỏ ữợ tr õ ởt
số t t ừ ỏ ữợ ữ s
u P SH() t eu P SH()
u P SH(), u 0 1 t u P SH().
u1 , u2 ổ tr t Cn log u1

log u2 P SH() t u1 u2 P SH() log (u1 + u2 ) P SH().

ỗ ỗ t
ỗ
t D Rn ữủ ồ ỗ ợ ồ x, x D t

{tx + (1 t)x } D ợ t (0, 1) ứ tr t õ
tữỡ ữỡ s

D Rn ữủ ồ ỗ
f (x) = lnd(x, D)

ỗ tr D, tr õ d(x, D) t tứ x
D
ứ ỗ t trú tỹ trú ự t
õ ỗ




✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✷✳ ▼✐➲♥ D ⊂ Cn ✤÷ñ❝ ❣å✐ ❧➔ ❣✐↔ ❧ç✐✱ ♥➳✉ ❤➔♠
ϕ(z) = −lnd(z, ∂D),

tr♦♥❣ ✤â d(z, ∂D) ❧➔ ❤➔♠ ❦❤♦↔♥❣ ❝→❝❤ ❒❝❧✐t tø ✤✐➸♠ z ✤➳♥ ❜✐➯♥ ∂D, ❧➔
❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ tr♦♥❣ D.
❱➼ ❞ö✿ ❚r➯♥ ♠➦t ♣❤➥♥❣ C ♠ët ♠✐➲♥ tò② þ ❧➔ ❣✐↔ ❧ç✐✳

✶✳✹✳✷ ▼✐➲♥ ❣✐↔ ❧ç✐ ❝❤➦t ❬✽❪

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✸✳ ❈❤♦ D ❧➔ ♠ët ♠✐➲♥ ❜à ❝❤➦♥ tr♦♥❣ Cn✳ D ✤÷ñ❝ ❣å✐ ❧➔
❣✐↔ ❧ç✐ ❝❤➦t ✈î✐ ❜✐➯♥ C 2 ♥➳✉ tç♥ t↕✐ ♠ët ❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ Φ ①→❝ ✤à♥❤
tr♦♥❣ ❧➙♥ ❝➟♥ U ❝õ❛ ❜✐➯♥ ∂D s❛♦ ❝❤♦✿
✐✮ D ∩ U = {z ∈ D : Φ(z) < 0}✱
✐✐✮ dΦ = 0 tr♦♥❣ U ✳
❉↕♥❣ ▲❡✈✐ ❝õ❛ Φ t↕✐ x0 ∈ ∂D ❧➔ ♠ët ❞↕♥❣ ❍❡r♠✐t❛♥ ❝❤♦ ♥❤÷ s❛✉✿

LΦ,x0 (ζ) =

∂ 2 Φ (x )ζ ζ , ζ = (ζ , ..., ζ ).
1
n
∂zi zj 0 i j

❚❛ ❝â ♠ët sè t➼♥❤ ❝❤➜t ✤→♥❣ ❝❤ó þ ❝õ❛ t➟♣ ❣✐↔ ❧ç✐ ❝❤➦t ♥❤÷ s❛✉✳
✰✮ ❈❤♦ Br ❧➔ ❤➻♥❤ ❝➛✉ ❊✉❝❧✐❞ ❜→♥ ❦➼♥❤ r t➙♠ O✳ ❑❤✐ ✤â ✈î✐ ♠å✐ z ∈ Br

logr − logd(z, Br ) ≤ CBr (0, z) = dBr (0, z) ≤ log2r − logd(z, ∂Br ).
✰✮ ❈❤♦ X ∈ Cn ❧➔ ♠✐➲♥ ❣✐↔ ❧ç✐ ❝❤➦t ❜à ❝❤➦♥ ✈î✐ ❜✐➯♥ C2 ✈➔ K ❧➔ ♠ët
t➟♣ ❝♦♠♣❛❝t ❝õ❛ X. ❑❤✐ ✤â tç♥ t↕✐ ❤➡♥❣ sè c1 ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ X ✈➔ K
s❛♦ ❝❤♦

dX (z0 , z) ≤ c1 − logd(z, ∂X), ∀z ∈ X, z0 ∈ K.

✶✵


✶✳✺ ❚➟♣ ❣✐↔✐ t➼❝❤ ♣❤ù❝ ❬✺❪
✶✳✺✳✶ ❚➟♣ ❣✐↔✐ t➼❝❤ ♣❤ù❝
❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛ t➟♣ ❣✐↔✐ t➼❝❤ ♣❤ù❝ ✈➔ ♠ët sè
t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ t➟♣ ❣✐↔✐ t➼❝❤ ♣❤ù❝✳ ❚r÷î❝ ❤➳t t❛ ❝â ✤à♥❤ ♥❣❤➽❛ t➟♣
❣✐↔✐ t➼❝❤ ♥❤÷ s❛✉✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✶✳ ❈❤♦ Ω ❧➔ ✤❛ t↕♣ ♣❤ù❝✳ ❚➟♣ A ⊂ Ω ✤÷ñ❝ ❣å✐ ❧➔ t➟♣

❝♦♥ ❣✐↔✐ t➼❝❤ ✭♣❤ù❝✮ ❝õ❛ Ω ♥➳✉ ♠é✐ ✤✐➸♠ a ∈ Ω ❝â ❧➙♥ ❝➟♥ U ❝õ❛ a ✈➔ ❝→❝
❤➔♠ f1, f2, ..., fN ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ ❧➙♥ ❝➟♥ U s❛♦ ❝❤♦
A ∩ U = {z ∈ U : f1 (z) = · · · = fN (z) = 0}

✭❝→❝ ❤➔♠ fj ❧➔ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ t➟♣ A✮✳ ❍❛② ♥â✐ ❝→❝❤
❦❤→❝✱ t➟♣ A ❧➔ t➟♣ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ❝→❝ ❤➔♠ ❝❤➻♥❤ ❤➻♥❤ fj .

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✷✳ ❚➟♣ A tr➯♥ ✤❛ t↕♣ ♣❤ù❝ Ω ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ❣✐↔✐ t➼❝❤

✭✤à❛ ♣❤÷ì♥❣✮ ♥➳✉ tr♦♥❣ ♠é✐ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ p ∈ A ❜➜t ❦➻✱ ♥â ❧➔ t➟♣
❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤✳
▼ët ♠✐➲♥ D ⊂ Cn ❜➜t ❦➻ ❧➔ ♠ët t➟♣ ❣✐↔✐ t➼❝❤ tr♦♥❣ Cn ♥❤÷♥❣ ♥â ❧➔ t➟♣
❝♦♥ ❣✐↔✐ t➼❝❤ tr♦♥❣ Cn ❝❤➾ ❦❤✐ D = Cn ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✸✳ ▼å✐ t➟♣ ❣✐↔✐ t➼❝❤ ✭✤à❛ ♣❤÷ì♥❣✮ tr➯♥ ✤❛ t↕♣ ♣❤ù❝ ❧➔
t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ ❝õ❛ ♠ët ❧➙♥ ❝➟♥ ✤➣ ❜✐➳t ❝õ❛ ♥â✳

◗✉② ÷î❝✿ ♥➳✉ A, A ❧➔ ❝→❝ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ tr♦♥❣ Ω ✈➔ A ⊂ A t❤➻ A
✤÷ñ❝ ❣å✐ ❧➔ ♠ët t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ ❝õ❛ A✳

❙❛✉ ✤➙② t❛ ❝â ♠ët sè ❤➺ q✉↔ ✤ì♥ ❣✐↔♥ ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ❣✐↔✐ t➼❝❤✳

✶✶


❍➺ q✉↔ ✶✳✺✳✹✳ ❍ñ♣ ❤ú✉ ❤↕♥ ❝õ❛ ❝→❝ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ ❝ô♥❣ ❧➔ ♠ët t➟♣
❝♦♥ ❣✐↔✐ t➼❝❤✳

❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ tr♦♥❣ ♠✐➲♥ U ⊂ Ω ❝→❝ t➟♣ Aj ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ❤å ❝→❝
N

j
❤➔♠ ❝❤➾♥❤ ❤➻♥❤{fjk }k=1
t÷ì♥❣ ù♥❣✱ ❦❤✐ ✤â (

m
1 Aj ) ∩ U

❝❤✉♥❣ ❝õ❛ t➜t ❝↔ ❝→❝ ❤➔♠ ❞↕♥❣ Πm
j=1 fjkj tr♦♥❣ ✤â 1

❧➔ t➟♣ ❦❤æ♥❣ ✤✐➸♠

kj

Nj ✳

❍➺ q✉↔ ✶✳✺✳✺✳ ❈❤♦ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ φ : X → Y ❝õ❛ ❝→❝ ✤❛ t↕♣ ♣❤ù❝✳
❑❤✐ ✤â t↕♦ ↔♥❤ ❝õ❛ t➟♣ ❣✐↔✐ t➼❝❤ A ⊂ Y ❧➔ ♠ët t➟♣ ❣✐↔✐ t➼❝❤ tr♦♥❣ X ✳

❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ φ(b) = a ∈ A✱ ✈➔ f1, ..., fN ❧➔ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣
❧➙♥ ❝➟♥ U ❝õ❛ a✱ ①→❝ ✤à♥❤ tr➯♥ A ∩ U. ❑❤✐ ✤â V = φ−1 (U ) ❧➔ ❧➙♥ ❝➟♥ ❝õ❛

b ✈➔ tr♦♥❣ V t➟♣ φ−1 (A) trò♥❣ ✈î✐ t➟♣ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ❝→❝ ❤➔♠
❝❤➾♥❤ ❤➻♥❤ f1 ◦ φ, ..., fN ◦ φ.
❷♥❤ ❝õ❛ ♠ët t➟♣ ❣✐↔✐ t➼❝❤ q✉❛ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❦❤æ♥❣ ♣❤↔✐ ❧ó❝ ♥➔♦
❝ô♥❣ ❧➔ ♠ët t➟♣ ❣✐↔✐ t➼❝❤✳ ❱➼ ❞ö✿ q✉❛ →♥❤ ①↕ ξ → (ξ 2 − ξ, ξ 3 − ξ)✱ ↔♥❤ ❝õ❛
✤➽❛ ✤ì♥ ✈à {|ξ| < 1} ⊂ C tr♦♥❣ C2 ❦❤æ♥❣ ❧➔ t➟♣ ❣✐↔✐ t➼❝❤ tr♦♥❣ ❜➜t ❦➻ ❧➙♥
❝➟♥ ♥➔♦ ❝õ❛ ✤✐➸♠ (0, 0)✳

❍➺ q✉↔ ✶✳✺✳✻✳ A1 ⊂ Ω1✱ A2 ⊂ Ω2 ❧➔ ❝→❝ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤✳ ❑❤✐ ✤â✱ t➼❝❤
trü❝ t✐➳♣ A1 × A2 ❧➔ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ tr♦♥❣ Ω1 × Ω2.

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû πj ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❝õ❛ Ω1 × Ω2 ❧➯♥ Ωj , j = 1, 2✱ ❦❤✐ ✤â
A1 × Ω2 = π1−1 (A1 ) ✈➔ Ω1 × A2 = π2−1 (A2 ) ❧➔ ❝→❝ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ ❝õ❛
Ω1 × Ω2 ✳ ❑❤✐ ✤â✱ t➟♣ A1 × A2 = (A1 × Ω2 ) ∩ (Ω1 × A2 ) ♥➯♥ A1 × A2 ❧➔ t➟♣
❝♦♥ ❣✐↔✐ t➼❝❤✳

✶✷


❚ø ✤à♥❤ ❧þ ❞✉② ♥❤➜t ✤è✐ ✈î✐ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤✱ t❛ ❝â ✤à♥❤ ❧þ ❞✉② ♥❤➜t
✤è✐ ✈î✐ ❝→❝ t➟♣ ❣✐↔✐ t➼❝❤ ♥❤÷ s❛✉✳

✣à♥❤ ❧þ ✶✳✺✳✼✳ ✭✣à♥❤ ❧þ ❞✉② ♥❤➜t✮
◆➳✉ ✤❛ t↕♣ ♣❤ù❝ Ω ❧✐➯♥ t❤æ♥❣ ✈➔ t➟♣ ❣✐↔✐ t➼❝❤ A ⊂ Ω ❝❤ù❛ ♠ët t➟♣ ❝♦♥
♠ð ❦❤→❝ ré♥❣ t❤➻ A = Ω✳
❈❤ù♥❣ ♠✐♥❤✳ ✣➦t A0 ❧➔ t➟♣ ❝→❝ ✤✐➸♠ tr♦♥❣ ❝õ❛ A✳ ❚❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â
A0 = ∅✱ ✈➔ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ A0 ❧➔ t➟♣ ♠ð✳ ●✐↔ sû a ❧➔ ✤✐➸♠ ❣✐î✐ ❤↕♥ ❝õ❛
A0 ✳ ❱➻ A ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ Ω ♥➯♥ a ∈ A✱ ❞♦ ✤â tr♦♥❣ ❧➙♥ ❝➟♥ ❧✐➯♥ t❤æ♥❣

U ❝õ❛ a t➟♣ A ∩ U ❧➔ t➟♣ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤
f1 , ..., fN ✳ ❱➻ fj ≡ 0 tr➯♥ t➟♣ ♠ð ❦❤→❝ ré♥❣ A0 ∩ U ✱ fj ≡ 0 tr♦♥❣ U ♥➯♥
a ∈ U ⊂ A0 ✳ ◆❤÷ ✈➟②✱ A0 ❝ô♥❣ ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ Ω✳ ❱➻ Ω ❧➔ ❧✐➯♥ t❤æ♥❣
♥➯♥ A0 = Ω✳

✶✳✺✳✷ ❙è ✤è✐ ❝❤✐➲✉ ❝õ❛ t➟♣ ❣✐↔✐ t➼❝❤

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✽✳ ❈❤♦ A ❧➔ t➟♣ ❣✐↔✐ t➼❝❤ tr➯♥ ✤❛ t↕♣ ♣❤ù❝ Ω✳ ✣✐➸♠ a ∈ A

✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝❤➼♥❤ q✉② ♥➳✉ ❝â ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ a tr♦♥❣ Ω s❛♦ ❝❤♦
A ∩ U ❧➔ ✤❛ t↕♣ ❝♦♥ ♣❤ù❝ ❝õ❛ Ω✳ ❚➟♣ ❝→❝ ✤✐➸♠ ❝❤➼♥❤ q✉② ✤÷ñ❝ ❦þ ❤✐➺✉
regA✳
❙è ❝❤✐➲✉ ❝õ❛ ✤❛ t↕♣ ❝♦♥ A ∩ U ❜➡♥❣ sè ❝❤✐➲✉ ❝õ❛ t➟♣ A t↕✐ ✤✐➸♠ ❝❤➼♥❤
q✉② a ✈➔ ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ dima A✳ ❈→❝ ✤✐➸♠ ❝æ ❧➟♣ ❝õ❛ A ❧➔ ❝❤➼♥❤ q✉② t❤❡♦
✤à♥❤ ♥❣❤➽❛ tr➯♥✱ sè ❝❤✐➲✉ ❝õ❛ A t↕✐ ❝→❝ ✤✐➸♠ ♥➔② ✤÷ñ❝ ❝❤♦ ❧➔ ❜➡♥❣ ✵✳

❈→❝ ✤✐➸♠ t❤✉ë❝ A\regA =: sngA ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ ❦➻ ❞à ❝õ❛ A✳ ❚❤❡♦

✶✸


✤à♥❤ ♥❣❤➽❛✱ t➟♣ regA ❧➔ t➟♣ ♠ð✱ ❞♦ ✤â t➟♣ A\regA ❧➔ t➟♣ ✤â♥❣✳
❑❤→✐ ♥✐➺♠ s❛✉ ❝❤♦ t❛ ✤à♥❤ ♥❣❤➽❛ sè ❝❤✐➲✉ ❝õ❛ t➟♣ ❣✐↔✐ t➼❝❤✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✾✳ ❈❤♦ A ❧➔ t➟♣ ❣✐↔✐ t➼❝❤ tr➯♥ ✤❛ t↕♣ ♣❤ù❝ Ω✳ ❙è ❝❤✐➲✉
❝õ❛ A t↕✐ ✤✐➸♠ a ∈ A ❜➜t ❦ý ❧➔

dima A := z→a
lim dimz A.
z∈regA

❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❛ ❝â sè ❝❤✐➲✉ ❝õ❛ t➟♣ A ❧➔

dimA = maxdimz A = max dimz A.
z∈A

z∈regA

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✶✵✳ ❙è ✤è✐ ❝❤✐➲✉ ❝õ❛ t➟♣ ❣✐↔✐ t➼❝❤ A ⊂ Ω ❧➔ dimΩ−dimA.
✣➦t A(p) = {z ∈ A : dimz A = p}. ❑❤✐ ✤â t❛ ❝â ✤à♥❤ ♥❣❤➽❛ s❛✉✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✶✶✳ ▼ët t➟♣ ❣✐↔✐ t➼❝❤ ✤÷ñ❝ ❣å✐ ❧➔ t❤✉➛♥ ♥❤➜t ✭tr♦♥❣ sè
❝❤✐➲✉✮ ♥➳✉ sè ❝❤✐➲✉ ❝õ❛ ♥â t↕✐ t➜t ❝↔ ❝→❝ ✤✐➸♠ trò♥❣ ♥❤❛✉✳ ❚➟♣ ❣✐↔✐ t➼❝❤
t❤✉➛♥ ♥❤➜t sè ❝❤✐➲✉ p ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ❣✐↔✐ t➼❝❤ p−❝❤✐➲✉ t❤✉➛♥ tó②✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✶✷✳ ❈❤♦ A ❧➔ t➟♣ ❣✐↔✐ t➼❝❤ p−❝❤✐➲✉✳ ◗✉ÿ t➼❝❤ ❦➻ ❞à ❝õ❛
♥â ❧➔ t➟♣

δ(A) := (sngA) ∪ {z ∈ A : dimz A < dimA} = (sngA) ∪

A(k)
k
✶✳✺✳✸ ❚➟♣ ❣✐↔✐ t➼❝❤ ❜➜t ❦❤↔ q✉②

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✶✸✳ ▼ët t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ A ❝õ❛ ✤❛ t↕♣ ♣❤ù❝ Ω ✤÷ñ❝ ❣å✐

❧➔ ❦❤↔ q✉② ✭tr♦♥❣ Ω✮ ♥➳✉ A = A1 ∪ A2✱ tr♦♥❣ ✤â A1, A2 ❝ô♥❣ ❧➔ ❝→❝ t➟♣
❝♦♥ ❣✐↔✐ t➼❝❤ tr♦♥❣ Ω ✈➔ ❦❤→❝ A✳ ◆➳✉ A ❦❤æ♥❣ t❤➸ ❜✐➸✉ ❞✐➵♥ ❞÷î✐ ❞↕♥❣
♥❤÷ tr➯♥ t❤➻ A ✤÷ñ❝ ❣å✐ ❧➔ ❜➜t ❦❤↔ q✉② ✭tr♦♥❣ Ω✮✳

✶✹


✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✶✹✳ ▼ët t➟♣ ❣✐↔✐ t➼❝❤ A ⊂ Ω ✤÷ñ❝ ❣å✐ ❧➔ ❜➜t ❦❤↔ q✉② ♥➳✉
♥â ❧➔ ❜➜t ❦❤↔ q✉② tr♦♥❣ ❧➙♥ ❝➟♥ ❝õ❛ ♥â ✳

▼➺♥❤ ✤➲ s❛✉ ❝❤♦ t❛ ✤✐➲✉ ❦✐➺♥ ✤➸ ♠ët t➟♣ ❣✐↔✐ t➼❝❤ ❧➔ ❜➜t ❦❤↔ q✉②✳

▼➺♥❤ ✤➲ ✶✳✺✳✶✺✳ ❚➟♣ ❣✐↔✐ t➼❝❤ A ❧➔ ❜➜t ❦❤↔ q✉② ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ t➟♣ regA

❧➔ ❧✐➯♥ t❤æ♥❣✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû A ❧➔ ✤â♥❣ tr♦♥❣ Ω✳ ✣➦t S

❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ ❧✐➯♥

t❤æ♥❣ ❝õ❛ regA✳ ❑❤✐ ✤â✱ A1 = S ✈➔ A2 = (regA)\S ❧➔ ❝→❝ t➟♣ ❝♦♥ ❣✐↔✐
t➼❝❤ tr♦♥❣ Ω ✈➔ A = A1 ∪ A2 ✈➻ regA trò ♠➟t tr♦♥❣ A✳ ◆➳✉ A ❧➔ ❜➜t ❦❤↔
q✉② t❤➻ ❤♦➦❝ A1 = A ❤♦➦❝ A2 = A✳ ❱➻ A2 ∩ S = ∅✱ A1 = A ✱ tù❝ ❧➔ S trò
♠➟t tr♦♥❣ A ✈➻ ✈➟② S trò ♠➟t tr♦♥❣ regA✳ ❱➻ S ❧➔ ✤â♥❣ tr♦♥❣ regA✱ s✉②
r❛ regA = S ✱ tù❝ ❧➔ regA ❧➔ ❧✐➯♥ t❤æ♥❣✳

◆❣÷ñ❝ ❧↕✐✱ ♥➳✉ regA = S ❧➔ ❧✐➯♥ t❤æ♥❣ ✈➔ A1 , A2 ❧➔ ❝→❝ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤
tr♦♥❣ Ω s❛♦ ❝❤♦ A = A1 ∪ A2 ✱ ❦❤✐ ✤â Sj = Aj ∩ S ❧➔ ❝→❝ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤
❝õ❛ ✤❛ t↕♣ ♣❤ù❝ ❧✐➯♥ t❤æ♥❣ S ❀ ♥❣♦➔✐ r❛ S = S1 ∪ S2 ✱ ✈➻ ✈➟② ♠ët tr♦♥❣ ❝→❝

Sj ✱ ❣✐↔ sû ❧➔ S1 ✱ ❧➔ trò ♠➟t ♠ët sè ♥ì✐ tr➯♥ S ✳ ❚❤❡♦ ✤à♥❤ ❧þ ❞✉② ♥❤➜t s✉②
r❛ S = S1 ✳ ◆❤÷♥❣ A = S = S1 ⊂ A1 ✱ ✈➻ ✈➟② A = A1 ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✶✻✳ ❚➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ ❜➜t ❦❤↔ q✉② A ❝õ❛ t➟♣ ❣✐↔✐ t➼❝❤

✤÷ñ❝ ❣å✐ ❧➔ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ A ♥➳✉ ♠å✐ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤
A ⊂ A s❛♦ ❝❤♦ A = A ✈➔ A ⊂ A ❧➔ ❦❤↔ q✉②✳

A

✶✺


✶✳✻ ⑩♥❤ ①↕ r✐➯♥❣ ❬✺❪
✶✳✻✳✶ ⑩♥❤ ①↕ r✐➯♥❣

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳✶✳ ❈❤♦ X, Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ ✈➔ ❝♦♠♣❛❝t ✤à❛
♣❤÷ì♥❣✳ ⑩♥❤ ①↕ ❧✐➯♥ tö❝ f : X → Y ✤÷ñ❝ ❣å✐ ❧➔ →♥❤ ①↕ r✐➯♥❣ ♥➳✉ t↕♦ ↔♥❤
❝õ❛ ♠å✐ t➟♣ ❝♦♠♣❛❝t K ⊂ Y ❧➔ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ X.
❚ø ✤à♥❤ ♥❣❤➽❛ t❛ ❝â ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ →♥❤ ①↕ r✐➯♥❣✿
✰✮ ❷♥❤ ❝õ❛ ♠ët t➟♣ ✤â♥❣ q✉❛ →♥❤ ①↕ r✐➯♥❣ ❧➔ t➟♣ ✤â♥❣✳

❢

❣

− Y →
− Z ❧➔ ❝→❝ →♥❤ ①↕ ❧✐➯♥ tö❝ s❛♦ ❝❤♦ h = g ◦ f ❧➔ r✐➯♥❣✳
✰✮ ❈❤♦ X →
❑❤✐ ✤â f ✈➔ g|f (X) ❝ô♥❣ ❧➔ ❝→❝ →♥❤ ①↕ r✐➯♥❣✳
✰✮ ❈❤♦ D ⊂ X ✈➔ G ⊂ Y ❧➔ ❝→❝ t➟♣ ❝♦♥ ✈î✐ G ❧➔ ❝♦♠♣❛❝t✳ ❑❤✐ ✤â✱ ❤↕♥
❝❤➳ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ (x, y) → x ❧➯♥ t➟♣ ❝♦♥ ✤â♥❣ A ⊂ D × G ❧➔ r✐➯♥❣ ♥➳✉
✈➔ ❝❤➾ ♥➳✉ A ❦❤æ♥❣ ❝â ✤✐➸♠ ❣✐î✐ ❤↕♥ tr➯♥ D × ∂G.
✰✮ ❈❤♦ f : X −→ Y ❧➔ →♥❤ ①↕ r✐➯♥❣ ❣✐ú❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢
❝♦♠♣❛❝t ✤à❛ ♣❤÷ì♥❣✳ L ❧➔ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ Y ✈➔ K ❧➔ t➟♣ ❝♦♥ ❝õ❛

f −1 (L) ✈ø❛ ♠ð ✈ø❛ ✤â♥❣✳ ❑❤✐ ✤â✱ ❝â ❝→❝ ❧➙♥ ❝➟♥ Uj ⊃ K ✈➔ Vj ⊃ L s❛♦
❝❤♦ t➜t ❝↔ ❝→❝ ❤↕♥ ❝❤➳ f : Uj −→ Vj ❧➔ ❝→❝ →♥❤ ①↕ r✐➯♥❣✳ ❈❤ó þ✿ →♥❤ ①↕ f
✤÷ñ❝ ❣å✐ ❧➔ ❤ú✉ ❤↕♥ ♥➳✉ t↕♦ ↔♥❤ ❝õ❛ ♠é✐ ✤✐➸♠ ❝❤ù❛ ❤ú✉ ❤↕♥ ❝→❝ ✤✐➸♠✳
✰✮ ◆➳✉ f : X → Y ❧➔ →♥❤ ①↕ ❤ú✉ ❤↕♥ r✐➯♥❣✱ ✈➔ a ∈ X ✱ t❤➻ ❝â ❝→❝ ❧➙♥ ❝➟♥

Uj ❝õ❛ a ✈➔ Vj ❝õ❛ f (a) s❛♦ ❝❤♦ ♠å✐ →♥❤ ①↕ f : Uj → Vj ❧➔ ❝→❝ →♥❤ ①↕ r✐➯♥❣✳

✶✻


✶✳✻✳✷ ❷♥❤ ❝õ❛ t➟♣ ❣✐↔✐ t➼❝❤
❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ ①➨t ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ↔♥❤ ❝õ❛ t➟♣ ❣✐↔✐ t➼❝❤ q✉❛
→♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❜➜t ❦➻✳
❈❤♦ X, Y ❧➔ ❝→❝ ✤❛ t↕♣ ♣❤ù❝✱ A ❧➔ ♠ët t➟♣ ❣✐↔✐ t➼❝❤ tr➯♥ ✤❛ t↕♣ ♣❤ù❝

X ✈➔ f : X −→ Y ❧➔ ♠ët →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤✳ ▼ët ✤➦❝ tr÷♥❣ ❤➻♥❤ ❤å❝
q✉❛♥ trå♥❣ ❝õ❛ →♥❤ ①↕ f ❧➔ sè ❝❤✐➲✉ ❝õ❛ t❤î ❝õ❛ ♥â✭ tù❝ ❧➔ ❝õ❛ ❝→❝ t↕♦
↔♥❤ ❝õ❛ ❝→❝ ✤✐➸♠✮✳ ❑➼ ❤✐➺✉ dimz f ❧➔ sè ✤è✐ ❝❤✐➲✉ ❝õ❛ t❤î✱ tù❝ ❧➔ dimz A −

dimz f −1 (f (z)) ✈➔ ✤➦t dimf := maxz∈A dimz f ✳ ❑❤✐ ✤â ❤➔♠ dimz f −1 (f (z))
❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ A✳ ❱➻ ✈➟②✱ ♥➳✉ A ❧➔ p− ❝❤✐➲✉ t❤✉➛♥ tó② t❤➻ ❤➔♠ dimz f
❧➔ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐ ✭tù❝ ❧➔ −dimz f ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥✮✱ ✤➦❝ ❜✐➺t✱ t➟♣
❝→❝ ❣✐→ trà ❝ü❝ ✤↕✐ ❝õ❛ ♥â ❧➔ ♠ð tr♦♥❣ A✳ ❉♦ ✤â dimf

dim(f |sngA )✳

▼➺♥❤ ✤➲ s❛✉ ❝❤♦ t❛ t➼♥❤ ❝❤➜t ❝õ❛ ↔♥❤ ❝õ❛ ♠ët t➟♣ ❣✐↔✐ t➼❝❤ q✉❛ →♥❤
①↕ ❝❤➾♥❤ ❤➻♥❤ ✭❦❤æ♥❣ ❝➛♥ t❤✐➳t ❧➔ →♥❤ ①↕ r✐➯♥❣✮✳

▼➺♥❤ ✤➲ ✶✳✻✳✷✳ ❈❤♦ Y ❧➔ ✤❛ t↕♣ ♣❤ù❝✱ A ❧➔ t➟♣ ❣✐↔✐ t➼❝❤ p ❝❤✐➲✉✳ f :

❧➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤✳ ❑❤✐ ✤â✱ f (A) ✤÷ñ❝ ❝❤ù❛ tr♦♥❣ ❤ñ♣ ✤➳♠
✤÷ñ❝ ❝õ❛ ❝→❝ t➟♣ ❣✐↔✐ t➼❝❤ tr➯♥ Y ❝â sè ❝❤✐➲✉ ❦❤æ♥❣ ✈÷ñt q✉→ dimf ✳ ❍ì♥
♥ú❛✱ ✈î✐ ❜➜t ❦➻ q ≥ p − dimf t➟♣ {w ∈ Y : dimf −1(w) ≥ q} ✤÷ñ❝ ❝❤ù❛
tr♦♥❣ ❤ñ♣ ✤➳♠ ✤÷ñ❝ ❝õ❛ ❝→❝ t➟♣ ❣✐↔✐ t➼❝❤ ❝â sè ❝❤✐➲✉ ❦❤æ♥❣ ✈÷ñt q✉→ p − q✳

A −→ Y

❈❤ù♥❣ ♠✐♥❤✳ ❱➻ ♠➺♥❤ ✤➲ tr➯♥ ♠❛♥❣ t➼♥❤ ✤à❛ ♣❤÷ì♥❣ ♥➯♥ t❛ ❝â t❤➸ ❣✐↔ sû
A ❧➔ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ ❝õ❛ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ 0 tr♦♥❣ Cn ✈➔ ❣✐↔ sû δ(A) ✤÷ñ❝
❝❤ù❛ tr♦♥❣ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ A ⊂ A ❝â sè ❝❤✐➲✉ ♥❤ä ❤ì♥ p✳ ❑❤✐ ✤â✱ A\A
❧➔ ✤❛ t↕♣ ♣❤ù❝ ✭❦❤æ♥❣ ♥❤➜t t❤✐➳t ❧✐➯♥ t❤æ♥❣✮ p− ❝❤✐➲✉✳ ✣➦t S ❧➔ t❤➔♥❤
♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ A\A ✈➔ ❦➼ ❤✐➺✉ rS ❧➔ ❤↕♥❣ ❧î♥ ♥❤➜t ❝õ❛ f t↕✐ ❝→❝
✤✐➸♠ ❝õ❛ S ✳ ❑❤✐ ✤â✱ CS = {z ∈ S : rankz f < rS } ❧➔ ♠ët t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤

✶✼


tr♦♥❣ S ❝â sè ❝❤✐➲✉ ♥❤ä ❤ì♥ p ✈➔ C = ∪CS ❧➔ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ tr♦♥❣ A\A
❝â sè ❝❤✐➲✉ ❝ô♥❣ ♥❤ä ❤ì♥ p ✳ ❚➟♣ Eq = {w ∈ Y : dimf −1 (w) ≥ q} ⊂ Y ❝â
t❤➸ ❜✐➸✉ ❞✐➵♥ t❤➔♥❤ ❤ñ♣ ❝õ❛ ❜❛ t➟♣ s❛✉✿

E : dim f −1 (w) ∩ A ≥ q;
E : dim f −1 (w) ∩ C ≥ q; E 0 = Eq \ (E ∪ E ).
❙è ✤è✐ ❝❤✐➲✉ ❝õ❛ →♥❤ ①↕ ✭ tù❝ ❧➔ dimA − dimf ✮ ❧➔ sè ❝❤✐➲✉ ♥❤ä ♥❤➜t
❝õ❛ t❤î✱ ✈➻ ✈➟② ♥➳✉ w ∈ E ❤♦➦❝ w ∈ E t❤➻ q ≥ dim A − dim f |A ❤♦➦❝
t÷ì♥❣ ù♥❣ q ≥ dim C − dim f |C ✳ ❇➡♥❣ q✉② ♥↕♣ ✤è✐ ✈î✐ sè ❝❤✐➲✉ ❝õ❛ t➟♣
❣✐↔✐ t➼❝❤✱ t❛ ❝â ✤÷ñ❝ t➟♣ E ❝❤ù❛ tr♦♥❣ ❤ñ♣ ✤➳♠ ✤÷ñ❝ ❝õ❛ ❝→❝ t➟♣ ❣✐↔✐ t➼❝❤
tr➯♥ Y ❝â sè ❝❤✐➲✉

dim A − q < p − q tr♦♥❣ ❦❤✐ E ✤÷ñ❝ ❝❤ù❛ tr♦♥❣

❤ñ♣ ✤➳♠ ✤÷ñ❝ ❝õ❛ ❝→❝ t➟♣ ❣✐↔✐ t➼❝❤ ❝â sè ❝❤✐➲✉

dim C − q < p − q ✳

❳➨t t➟♣ E 0 ❝á♥ ❧↕✐✳ ▲➜② w ∈ E 0 ✱ ❦❤✐ ✤â dim(f −1 (w) \ (A ∪ C)) ≥ q ✱
✈➻ ✈➟② ❝â ♠ët t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ S tr♦♥❣ A\A s❛♦ ❝❤♦ dim(f −1 (w) ∩

(S\C)) ≥ q ✳ ❱➻ rankz f = rS tr➯♥ S\C ✱ tø ✤à♥❤ ❧þ ✈➲ ❤↕♥❣ s✉② r❛ t➜t ❝↔
❝→❝ t❤î ❝õ❛ fS\C ❝â sè ❝❤✐➲✉ p − rS ✱ ❞♦ ✤â p − rS

q. ❚❤❡♦ ✤à♥❤ ❧þ ✈➲

❤↕♥❣✱ ↔♥❤ ❝õ❛ t➟♣ S\C ❝❤ù❛ tr♦♥❣ ❤ñ♣ ✤➳♠ ✤÷ñ❝ ❝õ❛ ❝→❝ ✤❛ t↕♣ ♣❤ù❝
tr♦♥❣ Y ❝â sè ❝❤✐➲✉ rS

p − q ✳ ❈✉è✐ ❝ò♥❣✱ ❝❤ó þ sè ❝→❝ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥

t❤æ♥❣ ❝õ❛ A\A ❧➔ ❤➛✉ ❤➳t ✤➳♠ ✤÷ñ❝ ✈➔ E 0 ⊂ ∪S f (S \ C).
❇➙② ❣✐í t❛ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ →♥❤ ①↕ r✐➯♥❣ ❘❡♠♠❡rt✱ ♠ët
tr♦♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ✤➸ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ t❤→❝ tr✐➸♥ tr♦♥❣ ❝❤÷ì♥❣
✷✳ ✣à♥❤ ❧þ →♥❤ ①↕ r✐➯♥❣ ❘❡♠♠❡rt ❝❤♦ t❛ t❤➜② t➼♥❤ ❣✐↔✐ t➼❝❤ ❝õ❛ ↔♥❤ ❝õ❛
t➟♣ ❣✐↔✐ t➼❝❤ q✉❛ →♥❤ ①↕ r✐➯♥❣✳

✣à♥❤ ❧þ ✶✳✻✳✸✳ ✭✣à♥❤ ❧þ →♥❤ ①↕ r✐➯♥❣ ❘❡♠♠❡rt✮ ❈❤♦ A ❧➔ t➟♣ ❣✐↔✐ t➼❝❤
tr➯♥ ✤❛ t↕♣ ♣❤ù❝ ✈➔ f

: A −→ Y

❧➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❧➯♥ ✤❛ t↕♣ ♣❤ù❝
✶✽


ỡ ỳ f r tr A õ f (A) t t
tr Y dimf (A) = dimf
ự Pt tr f (A) ữỡ t õ t
sỷ Y tr Cm f (A) t t ứ õ tợ t
trữớ ủ A t q ự ỵ q
ợ p = dimA.
ợ p = 0 ỵ tr
sỷ ỵ ú ợ p 1 t ự ỵ ú ợ

p tt q E1 = f (sgnA) t t tr Y õ
số dim f |sgnA dim f =: q t ừ t t ự
ỵ tr ữợ

ữợ sỷ dimE1 = q t A

= f 1 (E1 ) õ A t

t tr A ỗ t ở tợ ừ f tr A dim f |A =

dim E1 = q ừ dim f |A tỗ t a A s t
f 1 (f (a)) A õ số dima A q pq. pq số ỹ t
ừ tợ tr A t dimf s r dimA = p
A = A t ỵ t õ tr trữớ ủ dimE1 = q
ừ t A trũ ợ ừ sngA t q f (sngA) t
t tr Y õ số dim f |sgnA = q = dim f f (A) t

t dimf (A) = dimf.

ữợ ớ sỷ dimE1 < q ử t õ t
z regA rankz f = q t tr regA ỵ A ũ
ừ t {z regA : rankz f = q} õ A t t tr