NHỮNG KẾT QUẢ MỚI CỦA LUẬN ÁN: Chứng minh một định lý tương tự Định lý thứ hai của Ritt cho hàm phân hình và một định lý tương tự Định lý thứ nhất của Ritt cho hàm phân hình trên trường số phức. Thiết lập một số định lý về vấn đề xác định duy nhất hàm phân hình trên một trường không-Acsimet và đa thức vi phân dạng , ở đó P là đa thức kiểu Fermat-Waring. Thiết lập một số kết quả đối với vấn đề duy nhất của tích q-sai phân dạng , của đa thức vi phân và q-sai phân dạng với f là hàm phân hình trên một trường không-Acsimet. CÁC ỨNG DỤNG, KHẢ NĂNG ỨNG DỤNG TRONG THỰC TIỄN HOẶC NHỮNG VẤN ĐỀ CÒN BỎ NGỎ CẦN TIẾP TỤC NGHIÊN CỨU: Các ứng dụng, khả năng ứng dụng trong thực tiễn Sử dụng các kết quả trong luận án để nghiên cứu bài toán về vấn đề xác định và vấn đề duy nhất đối với hàm phân hình trên trường không-Acsimet. Những vấn đề còn bỏ ngỏ cần tiếp tục nghiên cứu Tìm các tương tự của hai định lý Ritt về Vấn đề xác định và Vấn đề duy nhất đối với hàm phân hình và đa thức vi phân, đa thức sai phân, đa thức $q$-sai phân trong trường hợp phức và p-adic. Mở rộng các ứng dụng của hai định lý Ritt vào bài toán xác định hàm và tập xác định duy nhất.
ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM PHẠM NGỌC HOA MỘT SỐ DẠNG CỦA ĐỊNH LÝ RITT VÀ ỨNG DỤNG VÀO VẤN ĐỀ DUY NHẤT LUẬN ÁN TIẾN SĨ TỐN HỌC THÁI NGUN - NĂM 2018 ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ▲í✐ ❝↔♠ ì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ▼ö❝ ❧ö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐✐ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ữỡ ỵ tt ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ❝õ❛ ❘✐tt ✤è✐ ✈ỵ✐ ❝→❝ ✤❛ t❤ù❝ ❦✐➸✉ ❋❡r♠❛t✲❲❛r✐♥❣ ❝õ❛ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ỵ tự tt ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỵ tự tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tự ❘✐tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✸✳ ỵ tự tt ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ♥❤✐➲✉ ❜✐➳♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỵ tự ❤❛✐ ❝õ❛ ❘✐tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ t➼❝❤ q✲s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✸✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t❤ù ❤❛✐ ❝õ❛ ❘✐tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ t➼❝❤ q✲s❛✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tự ❤❛✐ ❝õ❛ ❘✐tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ❑➳t ❧✉➟♥ ✈➔ ❦✐➳♥ ♥❣❤à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ❉❛♥❤ ♠ö❝ ❝æ♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t ỵ ỡ ỵ tt số t ❜✐➸✉ r➡♥❣ ♠å✐ sè ♥❣✉②➯♥ n ≥ ✤➲✉ ❜✐➸✉ t ữợ t số tố ❝â ❞↕♥❣ mk n = pm pk , ✈ỵ✐ k ≥ 1, ð ✤â ❝→❝ t❤ø❛ sè ♥❣✉②➯♥ tè p1 , , pk ✤æ✐ ♠ët ♣❤➙♥ ❜✐➺t ✈➔ ❝→❝ sè ♠ơ t÷ì♥❣ ù♥❣ m1 ≥ 1, , mk ≥ ✤÷đ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤ ❞✉② ♥❤➜t t❤❡♦ n tt ữớ t tữỡ tỹ ỵ ♥➔② ✤è✐ ✈ỵ✐ ❝→❝ ✤❛ t❤ù❝✳ ✣➸ ♠ỉ t↔ ❦➳t q✉↔ ❝õ❛ ❘✐tt✱ t❛ ❦➼ ❤✐➺✉ M(C) ✭t÷ì♥❣ ù♥❣✱ A(C)✮ ❧➔ t➟♣ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭t÷ì♥❣ ù♥❣✱ ♥❣✉②➯♥✮ tr➯♥ C ✈➔ ❦➼ ❤✐➺✉ L(C) ❧➔ t➟♣ ❝→❝ ✤❛ t❤ù❝ ❜➟❝ 1✳ ✣➦t E, F ❧➔ ❝→❝ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ M(C)✱ ❦❤✐ ✤â ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ F (z) ữủ ổ t ữủ tr Eì F ♥➳✉ ❜➜t ❦ý ❝→❝❤ ✈✐➳t t❤➔♥❤ ♥❤➙♥ tû F (z) = f ◦ g(z) ✈ỵ✐ f (z) ∈ E ✈➔ g(z) ∈ F ✤➲✉ ❦➨♦ t❤❡♦ ❤♦➦❝ f ❧➔ t✉②➳♥ t➼♥❤ ❤♦➦❝ g ❧➔ t✉②➳♥ t➼♥❤✳ ◆➠♠ ✶✾✷✷✱ ❘✐tt ự ỵ s ỵ ỵ tự t tt F t ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ C[z] \ L(C) ◆➳✉ ♠ët ✤❛ t❤ù❝ F (z) ❝â ❤❛✐ ❝→❝❤ ♣❤➙♥ t➼❝❤ ❦❤→❝ ♥❤❛✉ t tự ổ t ữủ tr Fì F ✿ F = ϕ1 ◦ ϕ2 ◦ · · · ϕr = ψ1 ◦ ψ2 ◦ · · · ψs , t❤➻ r = s, ✈➔ ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ψ ❧➔ ❜➡♥❣ ✈ỵ✐ ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ϕ ♥➳✉ ❦❤æ♥❣ t➼♥❤ ✤➳♥ t❤ù tü ①✉➜t ❤✐➺♥ ❝õ❛ ❝❤ó♥❣✳ ❈ơ♥❣ tr♦♥❣ ❬✹✻❪✱ ❘✐tt ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ỵ s ỵ ỵ tự ❘✐tt✮✳ ●✐↔ sû r➡♥❣ a, b, c, d ∈ C[x]\ C t❤ä❛ ♠➣♥ a◦b = c◦d ✈➔ gcd(deg(a); deg(c)) = gcd(deg(b); deg(d)) = õ tỗ t t✉②➳♥ t➼♥❤ lj ∈ C[x] s❛♦ ❝❤♦ (l1 ◦ a ◦ l2 , l2−1 ◦ b ◦ l3 , l1 ◦ c ◦ l2 , l4−1 ◦ d ◦ l3 ) ❝â ♠ët tr♦♥❣ ❝→❝ ❞↕♥❣ (Fn , Fm , Fm , Fn ) ❤♦➦❝ ✷ (xn , xs h(xn ), xs h(x)n , xn ), ð ✤â m, n > ❧➔ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✱ s > ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ✈ỵ✐ n, ✈➔ h ∈ C[x]\xC[x], lj−1 ❧➔ ❤➔♠ ♥❣÷đ❝ ❝õ❛ lj ✱ Fn , Fm ❧➔ ❝→❝ ✤❛ t❤ù❝ ❈❤❡❜②❝❤❡✈✳ Ð ✤➙②✱ ♣❤➨♣ ♣❤➙♥ t➼❝❤ F (z) = f ◦ g(z) ❝❤➼♥❤ ❧➔ ♣❤➨♣ ❤ñ♣ t❤➔♥❤ F (z) = f (g(z))✳ ❉♦ ✤â✱ t❛ t❤➜② r ỵ tự tt ổ t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ a(b) = c(d)✱ ð ✤â a, b, c, d ❧➔ ❝→❝ ✤❛ t❤ù❝ ✈➔ ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ❧➔ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✳ ❘ã r➔♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤❛ t❤ù❝ ✤÷đ❝ ❘✐tt ♥❣❤✐➯♥ ❝ù✉ ❧➔ tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ P (f ) = Q(g), ð ✤â P, Q ❧➔ ❝→❝ ✤❛ t❤ù❝ ✈➔ f, g ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ P (f ) = Q(g) ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ♥❤✐➲✉ t→❝ ❣✐↔ ♥❤÷ ❚↕ ❚❤à ❍♦➔✐ ❆♥✲◆❣✉②➵♥ ❚❤à ◆❣å❝ ❉✐➺♣ ❬✸❪✱ ❍✳❋✉❥✐♠♦t♦ ❬✶✾❪✱ ❍➔ ❍✉② ❑❤♦→✐✲❈✳❈✳❨❛♥❣ ❬✸✺❪✱ ❋✳P❛❦♦✈✐❝❤ ❬✹✹❪✱ ỵ r ữỡ tr q✉❛♥ ♠➟t t❤✐➳t ✤➳♥ ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ố ợ ởt ự ỵ tt ♣❤➙♥ ❜è ❣✐→ trà✳ ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❧➛♥ ✤➛✉ t✐➯♥ ❜ð✐ ❘✳◆❡✈❛♥❧✐♥♥❛✳ ◆➠♠ ự ữủ r ợ ♣❤➙♥ ❤➻♥❤ f ✈➔ g tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✱ ♥➳✉ ❝❤ó♥❣ ❝â ❝❤✉♥❣ ♥❤❛✉ ↔♥❤ ♥❣÷đ❝ ✭❦❤ỉ♥❣ t➼♥❤ ❜ë✐✮ ❝õ❛ ✺ ✤✐➸♠ ♣❤➙♥ ❜✐➺t t❤➻ f = g ✭✣à♥❤ ỵ ú õ ♥❣÷đ❝ ✭❝â t➼♥❤ ❜ë✐✮ ❝õ❛ ✹ af + b ✭a, b, c, d ❧➔ ❝→❝ sè ♣❤ù❝ ♥➔♦ ✤â s❛♦ cf + d ad bc = ỵ ỗ tứ ỵ ỵ t ữủ ự tử ợ ữợ ự ②➳✉ ✈➔ ✤➣ ❝â r➜t ♥❤✐➲✉ ❦➳t q✉↔ s➙✉ s➢❝ ❝õ❛ ●✳❉❡t❤❧♦❢❢✱ ✣é ✣ù❝ ❚❤→✐✱ ▼✳ ❙❤✐r♦s❛❦✐✱ ❍✳❳✳❨✐✱ P✳❈✳❍✉✲❈✳❈✳❨❛♥❣✱ ❍➔ ❍✉② ❑❤♦→✐✱ ❍➔ ❍✉② ❑❤♦→✐✲❱ô ❍♦➔✐ ❆♥✱ ❍➔ ❍✉② ❑❤♦→✐✲❱ô ❍♦➔✐ ❆♥✲▲➯ ◗✉❛♥❣ ◆✐♥❤✱ ❚↕ ❚❤à ❍♦➔✐ ❆♥✱ ❚↕ ❚❤à ❍♦➔✐ ❆♥✲❍➔ ❚r➛♥ P❤÷ì♥❣✱ ▲✳▲❛❤✐r✐✱ ❚r➛♥ ❱➠♥ ❚➜♥✱ ❙➽ ✣ù❝ ◗✉❛♥❣✱ ❆✳❊s❝❛ss✉t✱ ❍✳❋✉❥✐♠♦t♦✱✳✳✳ ✤✐➸♠ ♣❤➙♥ ❜✐➺t t❤➻ g = ❚✐➳♣ t❤❡♦✱ sü ♥❣❤✐➯♥ ❝ù✉ ✤÷đ❝ ♠ð rë♥❣ s❛♥❣ ♠ët ♥❤→♥❤ ỵ tt t õ ①➨t t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ữớ t ữợ ữợ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❧➔ ❍❛②♠❛♥✳ ◆➠♠ ✶✾✻✼✱ ❍❛②♠❛♥ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❦➳t q✉↔ ♥ê✐ t✐➳♥❣ r➡♥❣ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ f tr➯♥ tr÷í♥❣ sè ♣❤ù❝ C ❦❤ỉ♥❣ ♥❤➟♥ ❣✐→ trà ✵ ✈➔ ✤↕♦ ❤➔♠ ❜➟❝ k ❝õ❛ f ✱ ợ k số ữỡ ổ tr ✶ t❤➻ f ❧➔ ❤➔♠ ❤➡♥❣✳ ❍❛②♠❛♥ ❝ơ♥❣ ✤÷❛ r❛ ❣✐↔ t❤✉②➳t s❛✉✳ ●✐↔n t❤✉②➳t ❍❛②♠❛♥✳ ❬✷✶❪ ◆➳✉ ♠ët ❤➔♠ ♥❣✉②➯♥ f t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (z)f (z) = ợ n số ữỡ ợ z ∈ C t❤➻ f ❧➔ ✸ ❤➔♠ ❤➡♥❣✳ ●✐↔ tt ữủ tr ợ n > ữủ tr ợ n 1✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q t ởt ữợ ự ữủ ❧➔ sü ❧ü❛ ❝❤å♥ ❝õ❛ ❍❛②♠❛♥✳ ❈æ♥❣ tr➻♥❤ q✉❛♥ trå♥❣ tú ữợ ự tở ❤❛✐ æ♥❣ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤ì♥ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ♥â ❝â ❞↕♥❣ f n f ✳ ❍❛✐ æ♥❣ ✤➣ ự ữủ r ợ f g ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣✱ n ❧➔ sè ♥❣✉②➯♥✱ n ≥ 11 ♥➳✉ f n f ✈➔ g n g ❝ò♥❣ ♥❤➟♥ ❣✐→ trà ♣❤ù❝ a t➼♥❤ ❝↔ ❜ë✐ t❤➻ ❤♦➦❝ f, g s❛✐ ❦❤→❝ ♥❤❛✉ ♠ët ❝➠♥ ❜➟❝ n + ❝õ❛ ✤ì♥ ✈à✱ ❤♦➦❝ f, g ✤÷đ❝ t➼♥❤ t❤❡♦ ❝→❝ ❝ỉ♥❣ t❤ù❝ ❝õ❛ ❤➔♠ ♠ơ ✈ỵ✐ ❝→❝ ❤➺ sè t❤ä❛ ♠➣♥ ♠ët ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤â✳ ❚ø ✤â✱ ❝→❝ ❦➳t q✉↔ t✐➳♣ t❤❡♦ ✤➣ ♥❤➟♥ ✤÷đ❝ ❞ü❛ tr➯♥ ①❡♠ ①➨t ❝→❝ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❞↕♥❣ (f n )(k) , [f n (f − 1)](k) ✭❇❤♦♦s♥✉r♠❛t❤ ✲ ❉②❛✈❛♥❛❧ ❬✶✵❪✱ ❋❛♥❣ ❬✶✽❪✮ ✈➔ ❝â ❞↕♥❣ [f n (af m + b)](k) , [f n (f − 1)m ](k) ✭①❡♠ ❩❤❛♥❣ ✈➔ ▲✐♥✱ ❬✺✹❪✮✱ ✈➔ ❝â ❞↕♥❣ (f )( ) P (f ),✭ ①❡♠ ❑✳ ❇♦✉ss❛❢✲ ❆✳ ❊s❝❛ss✉t✲ ❏✳ ❖❥❡❞❛❬✶✶❪✮✳ ◆➠♠ ✶✾✾✼✱ t❤❛② ✈➻ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤↕♦ ❤➔♠ ❜➟❝ n✱ ■✳ ▲❛❤✐r✐ ❬✸✻❪ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t ❤ì♥ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❦❤ỉ♥❣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ♥❤➟♥ ❣✐→ trà t ữợ ự ✷✵✵✷ ❈✳ ❨✳ ❋❛♥❣ ✈➔ ▼✳ ▲✳ ❋❛♥❣ ❬✶✼❪ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣✱ ♥➳✉ n ≥ 13, ✈➔ ✤è✐ ✈ỵ✐ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ f ✈➔ g, ♠➔ f (n) (f − 1)2 f ✈➔ g (n) (g − 1)2 g ♥❤➟♥ ❣✐→ trà ✶ t➼♥❤ ❝↔ ❜ë✐✱ t❤➻ f = g ❱➔♦ ❝✉è✐ ♥❤ú♥❣ ♥➠♠ ❝õ❛ t❤➟♣ ❦✛ ♥➔②✱ ✈➜♥ ✤➲ ♥❤➟♥ ❣✐→ trà ❝ơ♥❣ ✤÷đ❝ ①❡♠ ①➨t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥ ❝õ❛ ❝→❝ ❤➔♠ ♥❣✉②➯♥ ✈➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ▲❛✐♥❡ ✈➔ ❨❛♥❣ ❬✸✼❪ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ ♣❤➙♥ ❜è ❣✐→ trà ❝õ❛ t➼❝❤ s❛✐ ♣❤➙♥ ✤è✐ ✈ỵ✐ ❝→❝ ❤➔♠ ♥❣✉②➯♥✳ ❳✳ ❈✳✲◗✐✱ ▲✳✲❩✳ ❨❛♥❣ ✈➔ ❑✳ ▲✐✉ ❬✹✺❪ ①❡♠ ①➨t ❝→❝ t➼❝❤ s❛✐ ♣❤➙♥ ✈➔ ✈✐ ♣❤➙♥ ❝â ❞↕♥❣ f (z)(n) f (z + c), ✈➔ ✤➣ ❝❤➾ r❛ ✤✐➲✉ ❦✐➺♥ ✤➸ f = tg ✱ ✈ỵ✐ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♥❣✉②➯♥ s✐➯✉ ✈✐➺t ❝â ❜➟❝ ❤ú✉ ❤↕♥✳ ◆➠♠ ✷✵✵✼✱ t t tứ ỵ tự tt P õ ỵ tữ t ữủ t t ố ợ tự t ữủ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❤❛✐ ✤❛ t❤ù❝ f1 , f2 ✈➔ ❤❛✐ t➟♣ ❝♦♠♣❛❝t K1 , K2 t❤ä❛ ♠➣♥ f1−1 (K1 ) = f2−1 (K2 ) ❑➳t q✉↔ ❝õ❛ ❋✳P❛❝❦♦✈✐❝❤ ✤÷đ❝ ✣✐♥❤ ❚✐➳♥ ❈÷í♥❣ ♠ð rë♥❣ tr♦♥❣ ❬✶✸❪✱ ❬✶✹❪✳ ❚ø ✣à♥❤ ỵ tt tự t q P õ tr ú tổ õ t t ỵ ❘✐tt t❤ù ❤❛✐ ❝â t❤➸ ✤÷đ❝ ①❡♠ ❧➔ ❦➳t q✉↔ ✤➛✉ t✐➯♥ ✈➲ ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠ tø ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ P (f ) = Q(g)✱ tø ✤â s✐♥❤ r❛ ❝→❝ ❦➳t q✉↔ ❝❤♦ ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ✤❛ t❤ù❝ t❤ỉ♥❣ q✉❛ ✤✐➲✉ ❦✐➺♥ ↔♥❤ ♥❣÷đ❝ ❝õ❛ t➟♣ ❤đ♣ ✤✐➸♠✳ ❚ø ♥❤➟♥ ①➨t ♥➔② ✈➔ ❝→❝ ❦➳t q✉↔ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✭①❡♠ ❬✸❪✱ ❬✸✺❪✱ ✹ ❬✹✹❪✮ ♥➯✉ tr➯♥✱ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ ✤÷đ❝ ✤➦t r❛ tü ♥❤✐➯♥ ♥❤÷ s t sỹ tữỡ tỹ ỵ tt ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥✳ ❳❡♠ ①➨t ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠✱ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ tự s tự q s ữợ õ ỵ tt ứ õ ú tổ t ởt số ỵ ❘✐tt ✈➔ ù♥❣ ❞ö♥❣ ✈➔♦ ✈➜♥ ✤➲ ❞✉② ♥❤➜t✧ ✤➸ qt ự tr ỗ t❤í✐ ❣â♣ ♣❤➛♥ ❧➔♠ ♣❤♦♥❣ ♣❤ó t❤➯♠ ❝→❝ ❦➳t q✉↔ ự ỵ tt ❱➜♥ ✤➲ ✷✳ ✷✳ ▼ö❝ t✐➯✉ ❝õ❛ ❧✉➟♥ →♥ ✷✳✶✳ t ởt số ỵ tữỡ tỹ ỵ tt ố ợ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝ ✈➔ p✲❛❞✐❝✳ ✷✳✷✳ ❚✐➳♣ ❝➟♥ ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠✱ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ủ ự p ữợ õ ỵ tt ố tữủ ự ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ tr trữớ ủ ự p ữợ õ ỵ tt t ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ự p ữợ õ ỵ ❘✐tt✳ ✹✳ P❤÷ì♥❣ ♣❤→♣ ✈➔ ❝ỉ♥❣ ❝ư ♥❣❤✐➯♥ ❝ù✉ ❙û ỵ tữỡ tỹ ú ũ ợ r ỵ t❤✉②➳t ♣❤➙♥ ❜è ❣✐→ trà ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ♥➔② t÷ì♥❣ tü ♥❤÷ ♣❤÷ì♥❣ tr tr ỵ tt tự ỷ ỵ t ❤➔♠✱ ❜➔✐ t♦→♥ ❞✉② ♥❤➜t ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ◆❤í ✤â ✈➔ ❝→❝ ❦➳t q✉↔ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ♥â✐ tr➯♥ ✤➸ ✤÷❛ r❛ ❝→❝ ❦➳t q✉↔ ✈➲ ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠ ✈➔ ❱➜♥ ✤➲ ❞✉② ♥❤➜t✳ ✺✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ❝õ❛ ❧✉➟♥ →♥ ▲✉➟♥ →♥ ✤➣ ✤÷❛ r❛ ♠ët ❝→❝❤ t✐➳♣ ❝➟♥ ♠ỵ✐ ✤è✐ ✈ỵ✐ ❱➜♥ ✤➲ ①→❝ ✤à♥❤✱ ❱➜♥ ✺ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❤➔♠✱ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✳ ✣â ❧➔✱ t ữợ õ ỵ tt õ tt ữủ ❦➳t q✉↔ ♠ỵ✐ ❣â♣ ♣❤➛♥ ♠ð rë♥❣ t❤➯♠ ❝→❝ ù♥❣ ỵ tt trú t q ỗ õ ữỡ ❝ò♥❣ ✈ỵ✐ ♣❤➛♥ ♠ð ✤➛✉✱ ♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔ t➔✐ t ữỡ ợ tỹ ỵ tt t ố ợ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤✧✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❈❤÷ì♥❣ ✶ ✤÷đ❝ ✈✐➳t ❞ü❛ tr➯♥ ❝→❝ ❜➔✐ ❜→♦ ❬✺❪✱ ❬✼❪✱ ❬✷✾❪✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ♥➔② ỗ ữợ s ữợ t t q tữỡ tỹ ỵ tt ố ợ ữợ t t ữỡ tr ũ t q ữợ ữ t t tr ỵ tự t ❝õ❛ ❘✐tt ✤➣ ❝❤ù♥❣ tä r➡♥❣✿ ❜➜t ❦ý ❤❛✐ sü t ởt tự trữợ t ✤❛ t❤ù❝ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ s➩ ❝❤ù❛ ❝ò♥❣ ♠ët sè ✤❛ t❤ù❝ ♥❤÷ ♥❤❛✉ ✈➔ ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ tr♦♥❣ ♠é✐ ❝→❝❤ ♣❤➙♥ t➼❝❤ ❧➔ ♥❤÷ ♥❤❛✉ ♥➳✉ ❦❤ỉ♥❣ t➼♥❤ ✤➳♥ t❤ù tü ❝õ❛ ❝❤ó♥❣ tr♦♥❣ ❝→❝❤ ♣❤➙♥ t➼❝❤✳ ❚ø ✤â✱ ♠ư❝ t✐➯✉ t❤ù ♥❤➜t ❝õ❛ ❈❤÷ì♥❣ ✶ t t q tữỡ tỹ ỵ tự ♥❤➜t ❝õ❛ ❘✐tt ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤✳ ❚✉② ♥❤✐➯♥✱ t❛ t r ự ỵ tt tr♦♥❣ ❬✹✻❪ ❞÷í♥❣ ♥❤÷ ❦❤ỉ♥❣ t÷ì♥❣ tü ✤÷đ❝ ❝❤♦ ❤➔♠ ỵ ộ tt ũ ✤➳♥ ✤✐➲✉ ❦✐➺♥ ✧❤ú✉ ❤↕♥✧ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ ✤❛ t❤ù❝ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ỉ♥❣✳ ❑❤➢❝ ♣❤ư❝ ❦❤â ❦❤➠♥ ♥➔②✱ trữợ t ú tổ tt ỵ ỵ ởt ỵ tt tự ố ợ ữỡ tr P (f1 , f2 ) = Q(g1 , g2 ), ð ✤â P, Q ❧➔ ❝→❝ ✤❛ t❤ù❝ ❤❛✐ ❜✐➳♥ ❦✐➸✉ ❨✐ ✈➔ f1 , f2 , g1 , g2 ❧➔ ❝→❝ ❤➔♠ ♥❣✉②➯♥✳ ú ỵ r t q ữủ t ✈➔ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❬✷❪ ✈➔ ❬✸✷❪✱ t✉② ♥❤✐➯♥ ð ú tổ t q ữợ õ ỵ tt tự ữ r ởt ự ỵ ✈➔ ❝→❝ ❤➺ q✉↔ ❝❤ó♥❣ tỉ✐ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ✣à♥❤ ỵ ởt t q tữỡ tỹ ỵ tt tự t ố ợ r ❈❤÷ì♥❣ ✶ ❝á♥ tr➻♥❤ ❜➔② ❝→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ✣à♥❤ ỵ õ ỵ ỵ ỵ t t q ♠ỵ✐ ✈➲ Bi − U RSM ❝❤♦ ❝→❝ ❤➔♠ ♣❤➙♥ ỵ r t ố ợ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❞↕♥❣ (P (f ))(k) ✱ ð ✤â P ❧➔ ✤❛ t❤ù❝ ✈➔ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤✱ ❧➔ ♠ët ❜➔✐ t♦→♥ ❦❤â✳ ❑❤â ❦❤➠♥ ð ✤➙② ❧➔ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t ❤✐➺♥ ❝❤÷❛ ❝â ♠ët ♠è✐ ❧✐➯♥ ❤➺ tèt ❣✐ú❛ ❤➔♠ ✻ ✤➳♠✱ ❤➔♠ ✤➦❝ trữ f ợ trữ ❝õ❛ (P (f ))(k) ✳ ❱➻ ✈➟②✱ ❝→❝ ❦➳t q✉↔ ♥❤➟♥ ✤÷đ❝ ✤➣ ①➨t ♠ët sè tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ ❜➔✐ t♦→♥ ♥➔②✳ ✣â ❧➔ ❝→❝ ❞↕♥❣✿ [f n (f −1)m ](k) ✈ỵ✐ f ❧➔ ❤➔♠ ♥❣✉②➯♥ ✭①❡♠ ❬✺✹❪✮✱ (f n )(k) ✈ỵ✐ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✭①❡♠ ❬✶✵❪✮✳ ❈❤ó♥❣ tỉ✐ ✤➣ ❣✐↔♠ ❜ỵt ❦❤â ❦❤➠♥ ♥➔② ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❞↕♥❣ (P d (f ))(k) ❚ø ✤â ✈➔ ❞ò♥❣ ❝→❝ ❦✐➸✉ t÷ì♥❣ tü ❝õ❛ ✣à♥❤ ỵ tự ú tổ ữủ ỵ õ ởt t q t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ữỡ ợ tỹ ỵ tự ❝õ❛ ❘✐tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ♥❤✐➲✉ ❜✐➳♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t✧✳ ❚r♦♥❣ ❈❤÷ì♥❣ ✷✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ❱➜♥ ✤➲ ✷✿ ❱➜♥ ✤➲ ①→❝ ✤à♥❤✱ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ tự q s tr trữớ ủ p ữợ õ ỵ tt tự ❝õ❛ ❈❤÷ì♥❣ ✷ ✤÷đ❝ ✈✐➳t ❞ü❛ tr➯♥ ❝→❝ ❜➔✐ ❜→♦ ❬✹❪✱ ❬✺❪✱ ❬✼❪✳ ◆❤÷ ✤➣ ✤➲ ❝➟♣ ✤➳♥ ð tr➯♥✱ ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝ô♥❣ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❝â ❝→❝ ❦➳t q✉↔ t❤ó ✈à tr♦♥❣ tr÷í♥❣ ❤đ♣ p✲❛❞✐❝✳ ❚r♦♥❣ ❬✸✶❪✱ ❑❤♦→✐✱ ❆♥ ✈➔ ▲❛✐ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❞↕♥❣ (f n )(k) ✈➔ ♥❤➟♥ ✤÷đ❝ ❦➳t q✉↔✿ ♥➳✉ (f n )(k) ✈➔ (g n )(k) ♥❤➟♥ ❝❤✉♥❣ ❣✐→ trà ✶ ❝â t➼♥❤ ❜ë✐ ✈ỵ✐ f, g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✈➔ n, k ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ t❤ä❛ ♠➣♥ n ≥ 3k + t❤➻ f ✈➔ g s❛✐ ❦❤→❝ ♥❤❛✉ ♠ët ❝➠♥ ❜➟❝ n ❝õ❛ ✤ì♥ ✈à✳ ❚ø ✤â✱ ❜➔✐ t♦→♥ t❤ù ♥❤➜t ✤➦t r❛ tr♦♥❣ ❈❤÷ì♥❣ ✷ ❧➔✿ t❤❛② ✈➻ ①➨t ❝→❝ ❤➔♠ f, g ✱ ❝❤ó♥❣ tỉ✐ ①❡♠ ①➨t ❝→❝ t♦→♥ tû ✈✐ ♣❤➙♥ ❞↕♥❣ (P n (f ))(k) ✈➔ (Qn (g))(k) ♥❤➟♥ ❝ò♥❣ ♠ët ❣✐→ trà✱ ð ✤â P, Q ❧➔ ❝→❝ ✤❛ t❤ù❝ ❦✐➸✉ ❋❡r♠❛t✲❲❛r✐♥❣✳ ❚ø ✤â✱ ❝❤ó♥❣ tổ tt ữủ ỵ ỵ ❧➔ ♠ët ❦➳t q✉↔ ✈➲ ✈➜♥ ✤➲ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✈➔ ✤❛ tự õ ú ỵ r n 3k + tr ỵ tèt ❤ì♥ ✤✐➲✉ ❦✐➺♥ t÷ì♥❣ ù♥❣ n ≥ 3k + tr♦♥❣ ❦➳t q✉↔ ❝õ❛ ❑❤♦→✐✲❆♥✲▲❛✐ ✭①❡♠ ❬✸✶❪✮✳ ❚r♦♥❣ ❬✹✾❪ ❨❛♥❣ ✤➣ ✤➦t r❛ ✈➜♥ ✤➲ s❛✉✿ ❧✐➺✉ ✤➥♥❣ t❤ù❝ f −1 (S) = g −1 (S) ✈ỵ✐ S = {−1, 1} ✤è✐ ✈ỵ✐ ❝→❝ ✤❛ t❤ù❝ ❝ò♥❣ ❜➟❝ f, g s➩ ❦➨♦ t❤❡♦ f = g ❤❛② ❧➔ f = −g ❄ ❈➙✉ ❤ä✐ ♥➔② ❝ơ♥❣ ✤➣ ✤÷đ❝ ❣✐↔✐ ✤→♣ tr♦♥❣ ❬✹✷❪✱ ❬✹✸❪✳ ❚ø ✤â✱ ❝➙✉ ❤ä✐ t❤ù ❤❛✐ ✤➦t r❛ tr♦♥❣ ❈❤÷ì♥❣ ✷ ❧➔✿ ❝❤♦ S, T ❧➔ ❝→❝ t➟♣ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ P (z), Q(z) t÷ì♥❣ ù♥❣ t❤➻ t❛ ❝â t❤➸ ❦➳t ❧✉➟♥ ❣➻ ✈➲ f, g ♥➳✉ Ef (S) = Eg (T )❄✳ ỵ ũ q ✤➣ ❣✐↔✐ ✤→♣ ❝❤♦ ❝➙✉ ❤ä✐ ✤➦t r❛ ✈➔ ❣â♣ ♣❤➛♥ tr↔ ❧í✐ ❈➙✉ ❤ä✐ ❝õ❛ ❈✳❈✳❨❛♥❣ tr♦♥❣ ❬✸✽❪✱ ❈➙✉ ❤ä✐ ❝õ❛ ❋✳P❛❦♦✈✐❝❤ tr♦♥❣ ❬✹✹❪ tr♦♥❣ tr÷í♥❣ ❤đ♣ p✲❛❞✐❝✳ ❚r♦♥❣ ❈❤÷ì♥❣ ✷ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ t❤✐➳t ❧➟♣ ✤÷đ❝ ❝→❝ ❦➳t q ỵ ởt ỵ tt t❤ù ❤❛✐ ❝❤♦ ♠ët ✈❡❝✲tì ❝→❝ ❤➔♠ ♥❣✉②➯♥ p✲❛❞✐❝✳ ✣à♥❤ ỵ t q t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ♥❤✐➲✉ ❜✐➳♥ p✲❛❞✐❝✳ ✼ ❈❤÷ì♥❣ õ t ỵ tự tt ✈➔ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ t➼❝❤ q✲s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t✧✳ ❚r♦♥❣ ❈❤÷ì♥❣ ✸ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ❱➜♥ ữợ õ ỵ tự ❘✐tt✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❈❤÷ì♥❣ ✸ ✤÷đ❝ ✈✐➳t ❞ü❛ tr➯♥ ❝→❝ ❜➔✐ ❜→♦ ❬✻❪✱ ❬✷✷❪✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝✱ ❝❤õ ✤➲ ♥➔② ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❣➛♥ ✤➙② ✈➔ ✤❛♥❣ ✤÷đ❝ t✐➳♣ tö❝ ❜ð✐ ❈✳❨✳❋❛♥❣✲▼✳▲✳❋❛♥❣ ✭❬✶✼❪✮✱ ■✳▲❛❤✐r✐ ✭❬✸✻❪✮✱ ▲❛✐♥❡✲❨❛♥❣ ✭❬✸✼❪✮✱ ▲✐✉✲❈❛♦ ✭❬✸✾❪✮✱ ❳✳❈✳◗✐✱ ▲✳❩✳❨❛♥❣✲❑✳▲✐✉ ✭❬✹✺❪✮✱ ❈✳❈✳❨❛♥❣ ✭❬✺✵❪✮✱ ❍✳❳✳❨✐ ✭❬✺✷❪✮✱✳✳✳ ❚✉② ♥❤✐➯♥✱ ❝→❝ ❦➳t q✉↔ ♠ỵ✐ ❝❤➾ ✤➲ ❝➟♣ ✤➳♥ ❧ỵ♣ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❝â ❜➟❝ ❤ú✉ ❤↕♥ ✤è✐ ✈ỵ✐ t➼❝❤ s❛✐ ♣❤➙♥ ❤♦➦❝ ❜➟❝ ❦❤ỉ♥❣ ✤è✐ ✈ỵ✐ t➼❝❤ q ✲s❛✐ ♣❤➙♥✳ ❘➜t ♥❤✐➲✉ ❦➳t q✉↔ t❤ó ✈à ❝ơ♥❣ ✤➣ ♥❤➟♥ ✤÷đ❝ ố ợ tr ởt trữớ ổst ✭①❡♠ ❬✾❪✱ ❬✶✻❪✱ ❬✷✼❪✱ ❬✷✽❪✱ ❬✸✵❪✱ ❬✹✶❪✮✳ ❑✳❇♦✉ss❛❢✱ ❆✳ ❊s❝❛ss✉t✱ ❏✳ ❖❥❡❞❛ ✭❬✶✶❪✮ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❝→❝ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝ ♠➔ f P (f ), g P (g) ❝ò♥❣ ♥❤➟♥ ♠ët ❤➔♠ ♥❤ä✳ ❚r♦♥❣ ❬✾❪✱ ❏✳✲P✳ ❇❡③✐✈✐♥✱ ❑✳ ❇♦✉ss❛❢ ✈➔ ❆✳ ❊s❝❛ss✉t✱ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ ✤↕♦ ❤➔♠ ♠ët ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❈❤÷ì♥❣ ✸ ❧➔ t❤✐➳t ❧➟♣ ❝→❝ ❦➳t q✉↔ ✤è✐ ✈ỵ✐ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ t➼❝❤ q ✲s❛✐ ♣❤➙♥ ❞↕♥❣ f n f m (qz + c)✱ ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✈➔ q ✲s❛✐ ♣❤➙♥ ❞↕♥❣ (f nm (z)f nd (qz + c))(k) ❱ô ❍♦➔✐ ❆♥✲P❤↕♠ ◆❣å❝ ❍♦❛ ❬✹❪✱ ❱ô ❍♦➔✐ ❆♥✲P❤↕♠ ◆❣å❝ ❍♦❛✲❍➔ ❍✉② ❑❤♦→✐ ❬✻❪✱ ❱ô ❍♦➔✐ ❆♥✲❍➔ ❍✉② ❑❤♦→✐ õ t q t ữợ ự ú ỵ r t q s t❤ù❝ ✈✐ ♣❤➙♥ ♥➯✉ tr➯♥ ❝❤÷❛ ✤÷đ❝ ✤➲ ❝➟♣ tr♦♥❣ trữớ ủ ự ỵ ộ ố ❤➺ ❣✐ú❛ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f ✈➔ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ f (qz + c) ❝â t❤➸ ❦❤ỉ♥❣ t❤✐➳t ❧➟♣ ✤÷đ❝ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝✳ ◆â ❝❤➾ t❤✐➳t ❧➟♣ ✤÷đ❝ tr♦♥❣ tr÷í♥❣ ❤đ♣ p✲❛❞✐❝ ❞♦ t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t ❝õ❛ ❝❤✉➞♥ p✲❛❞✐❝✳ ❉ò♥❣ ỵ t❤ù ❤❛✐ ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝✮ ✈➔ ❝→❝ ❇ê tt ú tổ t ữủ ỵ ỵ ỵ ✸✳✷✳✼ ❧➔ ♠ët ❦➳t q✉↔ ❝❤♦ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ t➼❝❤ q ✲s❛✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ p✲❛❞✐❝✳ ỵ ởt t q ❞✉② ♥❤➜t ❝õ❛ t➼❝❤ q ✲s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ p✲❛❞✐❝✳ ❈→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ →♥ ✤÷đ❝ ❜→♦ ❝→♦ t↕✐ ❍ë✐ t❤↔♦ q✉è❝ t➳ ✈➲ ❣✐↔✐ t➼❝❤ ♣❤ù❝ ✈➔ ù♥❣ ❞ö♥❣ ❧➛♥ t❤ù ✷✵ t↕✐ ❍➔ ◆ë✐ ♥❣➔② ✷✾✴✵✼✲✸✴✵✽✴✷✵✶✷❀ ❍ë✐ ♥❣❤à ❚♦→♥ ❤å❝ ♣❤è✐ ❤ñ♣ ❱✐➺t✲P❤→♣✱ ❍✉➳ ✷✵✲✷✹✴✵✽✴✷✵✶✷❀ ✣↕✐ ❤ë✐ ❚♦→♥ ❤å❝ ❱✐➺t ◆❛♠ ❧➛♥ t❤ù ✽✱ ◆❤❛ ❚r❛♥❣ ✶✵✲✶✹✴✵✽✴✷✵✶✸❀ ❍ë✐ ♥❣❤à ✣↕✐ sè✲ ❍➻♥❤ ❤å❝✲ ❚♦♣♦✱ ❇✉æ♥ ▼❛ ❚❤✉ët ♥❣➔② ✷✻✲✸✵✴✶✵✴✷✵✶✻❀ ❈→❝ ❙❡♠✐♥❛r ❝õ❛ ❇ë ♠ỉ♥ ●✐↔✐ t➼❝❤✱ ❦❤♦❛ ❚♦→♥ ✲ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥❀ ❈→❝ ✽ ❙❡♠✐♥❛r ❝õ❛ ♥❤â♠ ♥❣❤✐➯♥ ❝ù✉ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❚❤➠♥❣ ▲♦♥❣ ✈➔ tr÷í♥❣ ❈❛♦ ✤➥♥❣ ❍↔✐ ❉÷ì♥❣✳ ✽✺ ❚÷ì♥❣ tü✱ t❛ ❝â (n − m)T (r, g) ≤ 16T (r, g) + 12T (r, f ) − log r + O(1) ❇ð✐ ✈➟②✱ ❝â (n − m) T (r, f ) + T (r, g) ≤ 28 T (r, f ) + T (r, g) − log r + O(1), (n − m − 28) T (r, f ) + T (r, g) + log r ≤ O(1) ❉♦ n ≥ m + 28 t❛ ❣➦♣ ♠➙✉ t❤✉➝♥✳ A.B = tù❝ ❧➔ f n f m (qz + c).g n g m (qz + c) = 1✳ rữớ ủ l ợ ln+m = g A = B tù❝ ❧➔ f n f m (qz + c) = g n g m (qz + c) ❚❤❡♦ ❇ê ✤➲ ✸✳✷✳✶✐✐✮ s✉② r❛ f = hg ✈ỵ✐ hn+m = ✤➲ ✸✳✷✳✶✐✮ s✉② r❛ f = rữớ ủ ỵ f, g ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ K, n, m ❧➔ l ✈ỵ✐ ln+m = 1, ❤♦➦❝ g f = hg ✈ỵ✐ hn+m = ♥➳✉ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ①↔② r❛✿ m = 1, n ≥ 13 ✈➔ f n f (qz + c) ✈➔ g n g(qz + c) ♥❤➟♥ ❝â t➼♥❤ ❜ë✐❀ m = 1, n ≥ 25 ✈➔ f n f (qz + c) ✈➔ g n g(qz + c) ♥❤➟♥ ❦❤æ♥❣ t➼♥❤ ❜ë✐❀ m ≥ 2, n ≥ m + 16 ✈➔ f n f m (qz + c) ✈➔ g n g m (qz + c) ♥❤➟♥ ❝â t➼♥❤ ❜ë✐❀ m ≥ 2, n ≥ m + 28 ✈➔ f n f m (qz + c) ✈➔ g n g m (qz + c) ♥❤➟♥ ❦❤æ♥❣ t➼♥❤ ❜ë✐✳ ❤❛✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ q, c ∈ K, |q| = 1✳ ❑❤✐ ✤â f = ✶✳ ✷✳ ✸✳ ✹✳ ỵ tự tt ❞✉② ♥❤➜t ❝õ❛ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ❇ê ✤➲ ✸✳✸✳✶✳ ❈❤♦ q, c ∈ K ✈ỵ✐ |q| = 1, n, m, d, k số ữỡ ợ n > 2k + 1, m > d ❑❤✐ ✤â ✶✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ f nm f nd (qz + c) (k) g nm g nd (qz + c) (k) =1 ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ (f, g) ✷✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ f nm f nd (qz + c) (k) = g nm g nd (qz + c) (k) ✽✻ ❝â ♥❣❤✐➺♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ (f, g) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f = hg ✈ỵ✐ h ∈ K ✈➔ hn(m+d) = ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t A = (f nm (z)f nd (qz + c))(k) , B = (g nm (z)g nd (qz + c))(k) ✱ A B C = f m (z)f d (qz + c), D = g m (z)(g d (qz + c), P = n−k , Q = n−k ❑❤✐ C D n (k) n−k n (k) n−k ✤â A = (C ) = C P, B = (D ) = D Q ✶✳ (f nm (z)f nd (qz + c))(k) (g nm (z)g nd (qz + c))(k) = (C n )(k) (Dn )(k) = ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ C = 0✱ C = ∞✱ D = 0✱ D = ∞✳ ●✐↔ sû r➡♥❣ C ❝â ❦❤ỉ♥❣ ✤✐➸♠✳ ●å✐ a ❧➔ ❦❤ỉ♥❣ ✤✐➸♠ ✈ỵ✐ ω(C, 0, a) = α✱ α ≥ 1✳ ❑❤✐ ✤â a ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ D ✈ỵ✐ ω(D, ∞, a) = β ✱ β ≥ s❛♦ ❝❤♦ nα − k = nβ + k ✈➔ k(m + d) + 16 > 2k + t❛ ❣➦♣ m−d ♠➙✉ t❤✉➝♥✳ ❇➡♥❣ ❝→❝❤ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü✱ t❛ ❝â D = 0✱ C = ∞✱ D = ∞✳ ❉♦ C, D ❦❤→❝ ❤➡♥❣✱ t❛ ❝ô♥❣ ❣➦♣ ♠➙✉ t❤✉➝♥✳ ✷✳ (f nm (z)f nd (qz + c))(k) = (g nm (z)g nd (qz + c))(k) , (C n )(k) = (Dn )(k) ❇ð✐ ✈➻ f, g ❦❤→❝ ❤➡♥❣✱ ✈➔ ❞♦ ❇ê ✤➲ ✸✳✶✳✺ t❛ t❤➜② C, D ❦❤→❝ ❤➡♥❣✳ ❉♦ ✤â C n = Dn + s, Dn = C n − s✱ ✈ỵ✐ s ❧➔ ♠ët ✤❛ t❤ù❝ ❜➟❝ < k ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ s ≡ 0✳ ●✐↔ sû s ≡ ❚❤➳ t❤➻ n(α − β) = 2k ✳ ❚ø ✤➙② ✈➔ ❞♦ n ≥ 2k + nT (r, D) = T (r, Dn ) + O(1) ≤ T (r, C n ) + T (r, s) + O(1) ≤ nT (r, C) + (k − 1) log r + O(1) ❚ø ✤➙② ✈➔ ❞♦ n ≥ 2k + k(m + d) + 16 > 2k + t❛ ♥❤➟♥ ✤÷đ❝ m−d k−1 1 < , T (r, D) ≤ T (r, C) + log r + O(1) n 2 Dn Cn ,G = ❉♦ C, D ❦❤→❝ ❤➡♥❣✱ t❛ ♥❤➟♥ ✤÷đ❝ ✣➦t F = s s F s = C n , nT (r, C) = T (r, C n ) ≤ T (r, F ) + T (r, s) + O(1) ≤ T (r, F ) + (k − 1) log r + O(1), nT (r, C) − (k − 1) log r ≤ T (r, F ) + O(1), 1 N1 (r, ) ≤ N1 (r, ) ≤ T (r, C) + O(1), F C 1 N1 (r, ) ≤ T (r, D) + O(1) ≤ T (r, C) + log r + O(1), F ✭✸✳✶✼✮ ✽✼ N1 (r, F ) ≤ N1 (r, C n ) + N1 (r, ) ≤ N1 (r, C) + (k − 1) log r + O(1) s ≤ T (r, C) + (k − 1) log r + O(1) ❚ø ✤➙② ✈➔ ❞♦ ❇ê ✤➲ ✷✳✶✳✾✱ ❞♦ ❝â F − = G ♥➯♥ t❛ s✉② r❛ nT (r, C) − (k − 1) log r + O(1) ≤ T (r, F ) 1 ) + N1 (r, F ) + N1 (r, ) − log r + O(1) F F −1 ≤ T (r, C) + T (r, C) + (k − 1) log r + N1 (r, ) − log r + O(1) G ≤ 2T (r, C) + (k − 1) log r + N1 (r, ) − log r + O(1) D ≤ 2T (r, C) + T (r, C) + log r + (k − 1) log r − log r + O(1) ❱➟②✱ t❛ ❝â ≤ N1 (r, (n − 3)T (r, C) − 2(k − 1) log r + log r ≤ O(1) ▼➦t ❦❤→❝✱ ❞♦ C ❦❤→❝ ❤➡♥❣ sè✱ t❛ ♥❤➟♥ ✤÷đ❝ T (r, C) ≥ log r + O(1) ❱➟② k(m + d) + 16 > (n−2k−1) log r+ log r ≤ O(1)✳ ❚ø ✤➙② ✈➔ ❞♦ n ≥ 2k+ m−d 2k + t❛ ❣➦♣ ♠ët ♠➙✉ t❤✉➝♥✳ ❱➟② s ≡ 0✳ ❉♦ ✈➟②✱ C n = Dn ✈➔ C = eD, f m (z)f d (qz + c) = eg m (z)g d (qz + c) ✈ỵ✐ f f (qz + c) en = 1✳ ✣➦t h = ✳ ●✐↔ sû h ❦❤→❝ ❤➡♥❣✳ ❑❤✐ ✤â h(qz + c) = g g(qz + c) ❦❤→❝ ❤➡♥❣ ✈➔ T (r, h(qz + c)) = T (r, h) + O(1), hm = mT (r, h) = T (r, hm ) + O(1) = T r, e , hd (qz + c) e + O(1) hd (qz + c) = dT (r, h(qz + c)) + O(1) = dT (r, h) + O(1) ❙✉② r❛ (m − d)T (r, h) = O(1) ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t m > d, h ❦❤→❝ ❤➡♥❣✳ ❱➟② h ❧➔ ❤➡♥❣✳ ❉♦ f m (z)f d (qz + c) = eg m (z)g d (qz + c), en = t❛ ❦➳t ❧✉➟♥ r➡♥❣ f = hg ✈ỵ✐ hm+d = e, hn(m+d) = ❱➟② ❇ê ữủ ự ỵ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ K, q, c ∈ K, |q| = 1, ❝❤♦ n, m, d, k ✱ ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ k(m + d) + 16 ❦✐➺♥ m > d ≥ 1, n ≥ 2k + ◆➳✉ (f nm (z)f nd (qz + c))(k) ✈➔ m−d nm nd (k) (g (z)g (qz + c)) ♥❤➟♥ ✶ ❈▼✱ t❤➻ f = hg ✈ỵ✐ hn(m+d) = 1✱ h ∈ K ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t A = (f nm (z)f nd (qz + c))(k) , B = (g nm (z)g nd (qz + c))(k) ✱ B A C = f m (z)f d (qz + c), D = g m (z)(g d (qz + c), P = n−k , Q = n−k ❑❤✐ C D ✤â A = (C n )(k) = C n−k P, B = (Dn )(k) = Dnk Q ú ỵ r N1 (r, A) + N1,(2 (r, A) = N2 (r, A), 1 ) + N1,(2 (r, ) = N2 (r, ), A A A N1 (r, B) + N1,(2 (r, B) = N2 (r, B), 1 N1 (r, ) + N1,(2 (r, ) = N2 (r, ) B B B ❑❤✐ ✤â✱ →♣ ❞ư♥❣ ❇ê ✤➲ ✸✳✶✳✹ ✤è✐ ✈ỵ✐ (C n )(k) ✱ (Dn )(k) t❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✳ N1 (r, ❚r÷í♥❣ ❤đ♣ ✶✳ T (r, A) ≤ N2 (r, A) + N2 (r, 1 ) + N2 (r, B) + N2 (r, ) − log r + O(1), A B T (r, B) ≤ N2 (r, A) + N2 (r, 1 ) + N2 (r, B) + N2 (r, ) − log r + O(1) A B ✭✸✳✶✽✮ ❚❛ t❤➜② r➡♥❣ ♥➳✉ a ❧➔ ♠ët ❝ü❝ A t C(a) = ợ à01 (a) ≥ A n + k ≥ ✈➔ ❞♦ ❇ê ✤➲ ✸✳✶✳✹ t❛ ❝â N1 (r, C) = N1 (r, f f (qz + c)) ≤ N1 (r, f ) + N1 (r, f (qz + c)) + O(1) ≤ T (r, f ) + T (r, f (qz + c)) + O(1) = 2T (r, f ) + O(1)✳ ❚÷ì♥❣ tü✱ N1 (r, C1 ) ≤ 2T (r, f ) + O(1) ❉♦ ✤â✱ t❤❡♦ ❇ê ✤➲ ✸✳✶✳✺ t❛ ❝â m d N2 (r, A) = 2N1 (r, C) ≤ 4T (r, f ) + O(1), 1 1 ) ≤ N2 (r, n−k ) + N (r, ) = 2N1 (r, ) + N (r, ) A C P C P ≤ 4T (r, f ) + N (r, ) + O(1) P ≤ 4T (r, f ) + k(m + d)T (r, f ) + kN1 (r, C) + O(1) N2 (r, ✽✾ ❚÷ì♥❣ tü✱ t❛ ❝ô♥❣ ❝â N2 (r, B) ≤ 4T (r, g) + O(1), 1 ) ≤ 4T (r, g) + N (r, ) + O(1) B Q ≤ 4T (r, g) + k(m + d)T (r, g) + kN1 (r, D) + O(1) N2 (r, ❚ø ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✈➔ ✭✸✳✶✽✮ t❛ ❝â T (r, A) ≤ 8(T (r, f ) + T (r, g)) + N (r, 1 ) + N (r, ) − log r + O(1) P Q ≤ 8(T (r, f )+T (r, g))+k(m+d)T (r, f )+kN1 (r, C)+N (r, )−log r+O(1), Q 1 ) + N (r, ) − log r + O(1) P Q ≤ 8(T (r, f )+T (r, g))+k(m+d)T (r, g)+kN1 (r, D)+N (r, )−log r+O(1), P 1 T (r, A) + T (r, B) ≤ 16(T (r, f ) + T (r, g)) + N (r, ) + N (r, ) P Q +k(m + d)(T (r, f ) + T (r, g)) + k(N1 (r, C) + N1 (r, D)) − log r + O(1) T (r, B) ≤ 8(T (r, f ) + T (r, g)) + N (r, ❉♦ ❇ê ✤➲ ✸✳✶✳✺ t❛ ❝â (n − 2k)(m − d)T (r, f ) + kN (r, C) + N (r, ) ≤ T (r, A) + O(1), P (n − 2k)(m − d)T (r, g) + kN (r, D) + N (r, ) ≤ T (r, B) + O(1) Q ❑➳t ❤ñ♣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤➙②✱ t❛ ♥❤➟♥ ✤÷đ❝ (n − 2k)(m − d)(T (r, f ) + T (r, g)) + k(N (r, C) + N (r, D))+ N (r, 1 ) + N (r, ) P Q ≤ (k(m + d) + 16)(T (r, f ) + T (r, g)) + k(N1 (r, C) + N (r, D)) + N (r, ) P ) − log r + O(1), Q [(n − 2k)(m − d) − (k(m + d) + 16)](T (r, f ) + T (r, g)) + log r ≤ O(1) +N (r, ✾✵ ❱➻ n ≥ 2k + ❚r÷í♥❣ ❤đ♣ ✷✳ k(m + d) + 16 ✱ t❛ ❣➦♣ ♠➙✉ t❤✉➝♥✳ m−d (f nm (z)f nd (qz + c))(k) (g nm (z)g nd (qz + c))(k) = (C n )(k) (Dn )(k) = ✣÷đ❝ s✉② r❛ tø ❇ê ✤➲ ✸✳✸✳✶✳✶✳ ❚r÷í♥❣ ❤đ♣ ✸✳ (f nm (z)f nd (qz + c))(k) = (g nm (z)g nd (qz + c))(k) , (C n )(k) = (Dn )(k) ❑➳t ❧✉➟♥ ❝õ❛ tr÷í♥❣ ❤đ♣ ♥➔② ✤÷đ❝ s✉② r❛ tứ ỵ ữủ ự ỵ f g ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ K, q, c ∈ K, |q| = 1, ✈➔ ❝❤♦ n, m, d, k ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ 2k(2m + 2d + 3) + 28 m > d ≥ 1, n ≥ 2k + ◆➳✉ (f nm (z)f nd (qz + c))(k) m−d nm nd (k) ✈➔ (g (z)g (qz + c)) ♥❤➟♥ ✶ ■▼✱ t❤➻ f = hg ✈ỵ✐ hn(m+d) = 1✱ h ∈ K ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ sû ❞ö♥❣ ❝→❝ ❦➼ ❤✐➺✉ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ỵ õ ✤è✐ ✈ỵ✐ (C n )(k) ✱ (Dn )(k) t❛ ①❡♠ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✳ ❚r÷í♥❣ ❤đ♣ ✶✳ 1 ) + N2 (r, B) + N2 (r, )+ A B 1 2(N1 (r, A) + N1 (r, )) + N1 (r, B) + N1 (r, ) − log r + O(1), A B 1 T (r, B) ≤ N2 (r, A) + N2 (r, ) + N2 (r, B) + N2 (r, )+ A B 1 2(N1 (r, B) + N1 (r, )) + N1 (r, A) + N1 (r, ) − log r + O(1) B A ❑➳t ❤ñ♣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ❝â T (r, A) ≤ N2 (r, A) + N2 (r, 1 ) + N2 (r, B) + N2 (r, )) A B 1 +3(N1 (r, A) + N1 (r, ) + N1 (r, B) + N1 (r, )) − log r + O(1) A B ❚ø ✤➙②✱ t÷ì♥❣ tü ♥❤÷ ❚r÷í♥❣ ❤đ♣ ✶✳ ❝õ❛ ỵ t õ T (r, A) + T (r, B) ≤ 2(N2 (r, A) + N2 (r, (n − 2k)(m − d)(T (r, f ) + T (r, g)) + k(N (r, C) + N (r, D)) ✾✶ 1 ) + N (r, ) ≤ T (r, A) + T (r, B) + O(1); ✭✸✳✶✾✮ P Q 1 T (r, A) + T (r, B) ≤ 16(T (r, f ) + T (r, g)) + N (r, ) + N (r, )+ P Q k(m + d)(T (r, f ) + T (r, g)) + k(N1 (r, C) + N1 (r, D)) + 3(N1 (r, A)+ +N (r, 1 ) + N1 (r, B) + N1 (r, )) − log r + O(1); A B N1 (r, A) ≤ 2T (r, f ) + O(1), N1 (r, ) ≤ 2T (r, f )+ A k(m + d + 2)T (r, f ) + O(1); N1 (r, B) ≤ 2T (r, g) + O(1), N1 (r, ) ≤ 2T (r, g) + k(m + d + 2)T (r, g) + O(1) B ❉♦ ✭✸✳✶✾✮✱ ✭✸✳✷✵✮✱ ✭✸✳✷✶✮ t❛ ❝â N1 (r, ✭✸✳✷✵✮ ✭✸✳✷✶✮ (n−2k)(m−d)(T (r, f )+T (r, g)) ≤ 16(T (r, f )+T (r, g))+k(m+d)(T (r, f ) +T (r, g))+3[2T (r, f )+2T (r, f )+k(m+d+2)T (r, f )+2T (r, g)+2T (r, g) +k(m + d + 2)T (r, g)] − log r + O(1) = (28 + 2k(2m + 2d + 3))(T (r, f ) + T (r, g)) − log r + O(1) ❚ø ✤â✱ s✉② r❛ [(n−2k)(m−d)−(28+2k(2m+2d+3))](T (r, f )+T (r, g))+4 log r ≤ O(1) 28 + 2k(2m + 2d + 3) ✳ m−d ❚❛ sû ❞ư♥❣ ❝→❝ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ ❝→❝ ❚r÷í♥❣ ❤đ♣ ✷ ✈➔ ✸ ❝õ❛ ✣à♥❤ ỵ ụ õ t f = hg ợ hn(m+d) = ỵ ữủ ự ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t n ≥ 2k + ❚r÷í♥❣ ❤đ♣ ✷ ✈➔ ❚r÷í♥❣ ❤đ♣ ✸ ✳ ✣à♥❤ ỵ f g ❤➻♥❤ ❦❤→❝ ❤➡♥❣ tr➯♥ K, q, c ∈ K, |q| = 1, ✈➔ ❝❤♦ n, m, d, k ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ m > d ≥ ❑❤✐ ✤â f = gh ✈ỵ✐ hn(m+d) = 1, h ∈ K ♥➳✉ ♠ët tr♦♥❣ ❤❛✐ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿ ✶✳ n ≥ 2k + k(m+d)+16 ✈➔ (f nm (z)f nd (qz + c))(k) ✈➔ (g nm (z)g nd (qz + c))(k) m−d ♥❤➟♥ ✶ ❈▼❀ ✷✳ n ≥ 2k + 2k(2m+2d+3)+28 ✈➔ (f nm (z)f nd (qz + c))(k) ✈➔ (g nm (z)g nd (qz + m−d c))(k) ♥❤➟♥ ✶ ■▼✳ ✾✷ ❑➳t ❧✉➟♥ ❝õ❛ ❈❤÷ì♥❣ ✸ ❚r♦♥❣ ❈❤÷ì♥❣ ✸✱ ❝❤ó♥❣ tỉ✐ ✤➣ t❤✐➳t ❧➟♣ ✤÷đ❝ ❝→❝ ❇ê ✤➲ ✸✳✷✳✶✱ ✸✳✸✳✶ ♥❤÷ ❧➔ ❤❛✐ ỵ tt tự t q s ♣❤➙♥ ❞↕♥❣ f n f m (qz + c) ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ s❛✐ ♣❤➙♥ ❞↕♥❣ (f nm f nd (qz + c))(k) , ð ✤â q, c ∈ K ✈ỵ✐ |q| = ✈➔ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ K ❈❤ó♥❣ tỉ✐ ❝ơ♥❣ ✤➣ t❤✐➳t ❧➟♣ ✤÷đ❝ ❤❛✐ ❦➳t q✉↔ ✈➲ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ t➼❝❤ q ✲s❛✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ s❛✐ ♣❤➙♥ õ tr õ ỵ ú ỵ r t q ữ õ tr tr÷í♥❣ ❤đ♣ ♣❤ù❝✳ ✾✸ ❑➳t ❧✉➟♥ ✈➔ ❦✐➳♥ ♥❣❤à ▲✉➟♥ →♥ ♥❣❤✐➯♥ ❝ù✉ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝ ✈➔ p✲❛❞✐❝✱ ①❡♠ ①➨t ❝→❝ tữỡ tỹ ỵ tt ố ợ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ t➼❝❤ q✲s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ ♠ët tr÷í♥❣ ❦❤ỉ♥❣ ❆❝s✐♠❡t✳ ◆❤ú♥❣ ❦➳t q✉↔ ❝❤➼♥❤ t ữủ ởt ỵ tữỡ tỹ ỵ tự tt ỵ ởt ỵ tữỡ tỹ ỵ tự t tt ỵ t ữủ ởt t q Bi U RSM ỵ ởt t q U RSM ỵ ởt t q t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❝❤♦ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥ ỵ t ỵ t❤ù ❤❛✐ ❝õ❛ ❘✐tt ❝❤♦ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✈❡❝✲tì tr ởt trữớ ổst ỵ ỵ t ữủ t q ✈➲ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ ❤➔♠ ✈➔ ✤❛ t❤ù❝ ỵ q ỵ t ữủ ỵ tt t❤ù ❤❛✐ ❝❤♦ t➼❝❤ q ✲s❛✐ ♣❤➙♥ ❞↕♥❣ f n f m (qz + c) ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ s❛✐ ♣❤➙♥ ❞↕♥❣ (f nm f nd (qz + c))(k) , ð ✤â q, c ∈ K ✈ỵ✐ |q| = ✈➔ f ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ K, ✈ỵ✐ K ❧➔ ♠ët tr÷í♥❣ ❦❤ỉ♥❣✲❆❝s✐♠❡t ✭❇ê ✤➲ ✸✳✷✳✶✱ ❇ê ✤➲ ✸✳✸✳✶✮✳ ❚❤✐➳t ❧➟♣ ✤÷đ❝ ❤❛✐ ❦➳t q✉↔ ✈➲ ❱➜♥ ✤➲ ❞✉② ♥❤➜t ❝❤♦ t➼❝❤ q ✲s❛✐ ♣❤➙♥ ✈➔ ✤❛ t❤ù❝ s õ tr ỵ ✸✳✸✳✹✮✳ ◆❤ú♥❣ ✈➜♥ ✤➲ ❝➛♥ t✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉✳ ✶✳ ❚✐➳♣ tư❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t÷ì♥❣ tü ❝õ❛ ❤❛✐ ✤à♥❤ ỵ tt ❞✉② ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝ ✈➔ ♣✲❛❞✐❝✳ ✷✳ ❚✐➳♣ tư❝ ự ự ỵ tt ✈➔♦ ❜➔✐ t♦→♥ ①→❝ ✤à♥❤ ❤➔♠ ✈➔ t➟♣ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t✳ ✾✹ ❉❛♥❤ ♠ư❝ ❈ỉ♥❣ tr➻♥❤ ❝õ❛ t→❝ ❣✐↔ ✤➣ ❝æ♥❣ ❜è ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➲ t➔✐ ✶✳ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✵✽✮✱ ✧ ❆♥ ❛♥❛❧♦❣✉❡ ♦❢ ▼❛s♦♥✬s t❤❡♦r❡♠ ❢♦r ♣✲ ❛❞✐❝ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s ✐♥ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✧✱ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥❝❡✱ ◆❳❇ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ❍➔ ◆ë✐✱ ✺✸✭✶✮✱ ♣♣✳ ✶✷✲✷✶✳ ✷✳ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✷✮✱ ✧❆ ✈❡rs✐♦♥ ♦❢ t❤❡ ❍❛②♠❛♥ ❝♦♥❥❡❝t✉r❡ ❢♦r ♣✲❛❞✐❝ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❞✐❢❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s✧✱ ■♥t❡r✲ r❛❝t✐♦♥s ❜❡t✇❡❡♥ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ❛♥❛❧②s✐s✱ ❙❝✐✳ ❚❡❝❤♥✐❝s P✉❜❧✳❍♦✉s❡✱ ❍❛♥♦✐✱ ♣♣✳ ✶✺✷✲✶✻✶✳ ✸✳ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛✱ ❛♥❞ ❍❛ ❍✉② ❑❤♦❛✐ ✭✷✵✶✼✮✱ ✧❱❛❧✉❡ s❤❛r✐♥❣ ♣r♦❜❧❡♠s ❢♦r ❞✐❢❢❡r❡♥t✐❛❧ ❛♥❞ ❞✐❢❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ♠❡r♦✲ ♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ✐♥ ❛ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ❢✐❡❧❞✧✱ ♣✲❆❞✐❝ ◆✉♠❜❡rs✱ ❯❧✲ tr❛♠❡tr✐❝ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✉♠❡ ✾✱ ■ss✉❡ ✶✱ ♣♣✳ ✶✕✶✹✳ ✹✳ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✼✮✱ ✧❖♥ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦❜❧❡♠ ♦❢ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②✲ ♥♦♠✐❛❧s✧✱ ❆♥♥❛❧❡s ❯♥✐✈✳❙❝✐✳❇✉❞❛♣❡st✱ ❙❡❝t✳ ❈♦♠♣✱ ✹✻✱ ♣♣✳✷✽✾✲✸✵✷✳ ✺✳ ❍❛ ❍✉② ❑❤♦❛✐✱ ❱✉ ❍♦❛✐ ❆♥✱ ❛♥❞ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✼✮✱ ✧❖♥ ❢✉♥❝✲ t✐♦♥❛❧ ❡q✉❛t✐♦♥s ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❆r❝❤✐✈ ❞❡r ▼❛t❤❡♠❛t✐❦✱ ❙♣r✐♥❣❡r ■♥t❡r♥❛t✐♦♥❛❧ P✉❜❧✐s❤✐♥❣✱ ❱♦❧✉♠❡ ✶✵✾✱ ■ss✉❡ ✻✱ ♣♣ ✺✸✾✕✺✹✾✳ ✻✳ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛✱ ❛♥❞ ❍❛ ❍✉② ❑❤♦❛✐ ✭❙✉❜♠✐t✮✱ ✧❯♥✐q✉❡✲ ♥❡ss t❤❡♦r❡♠s ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②✲ ♥♦♠✐❛❧✧✳ ✾✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❳✉➙♥ ▲❛✐ ✭✷✵✶✼✮✱ ❱➜♥ ✤➲ ①→❝ ✤à♥❤ ❤➔♠ ❦❤✐ ❤❛✐ ✤↕♦ ❤➔♠ ❝ò♥❣ ♥❤➟♥ ♠ët t➟♣✱ ▲✉➟♥ →♥ ❚✐➳♥ sÿ ❚♦→♥ ❤å❝✱ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❚❤→✐ ◆❣✉②➯♥✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❬✷❪ ▲➯ ◗✉❛♥❣ ◆✐♥❤ ✭✷✵✶✼✮✱ ❱➲ ①→❝ ✤à♥❤ ❤➔♠ ✈➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ q✉❛ ✤✐➲✉ ❦✐➺♥ ↔♥❤ ♥❣÷đ❝ ❝õ❛ t➟♣ ❤ñ♣ ✤✐➸♠✱ ▲✉➟♥ →♥ ❚✐➳♥ sÿ ❚♦→♥ ❤å❝✱ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❚❤→✐ ◆❣✉②➯♥✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❚❛ ❚❤✐ ❍♦❛✐ ❆♥✱ ◆❣✉②❡♥ ❚❤✐ ◆❣♦❝ ❉✐❡♣ ✭✷✵✶✸✮✱ ✧●❡♥✉s ♦♥❡ ❢❛❝t♦rs ♦❢ ❝✉r✈❡ ❞❡❢✐♥❡❞ ❜② s❡♣❛r❛t❡❞ ✈❛r✐❛❜❧❡ ♣♦❧②♥♦♠✐❛❧✧✱ ❏✳ ◆✉♠❜❡r ❚❤❡♦r②✱ ✶✸✸ ✭✽✮✱ ♣♣✳ ✷✻✶✻✲✷✻✸✹✳ ❬✹❪ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✷✮✱ ✧❆ ✈❡rs✐♦♥ ♦❢ t❤❡ ❍❛②♠❛♥ ❝♦♥✲ ❥❡❝t✉r❡ ❢♦r ♣✲❛❞✐❝ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❞✐❢❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s✧✱ ■♥t❡rr❛❝✲ t✐♦♥s ❜❡t✇❡❡♥ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ❛♥❛❧②s✐s✱ ❙❝✐✳ ❚❡❝❤♥✐❝s P✉❜❧✳❍♦✉s❡✱ ❍❛♥♦✐✱ ♣♣✳ ✶✺✷✲✶✻✶✳ ❬✺❪ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✼✮✱ ✧❖♥ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦❜❧❡♠ ♦❢ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②✲ ♥♦♠✐❛❧s✧✱ ❆♥♥❛❧❡s ❯♥✐✈✳❙❝✐✳❇✉❞❛♣❡st✱ ❙❡❝t✳ ❈♦♠♣✱ ✹✻✱ ♣♣✳✷✽✾✲✸✵✷✳ ❬✻❪ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛✱ ❛♥❞ ❍❛ ❍✉② ❑❤♦❛✐ ✭✷✵✶✼✮✱ ✧❱❛❧✉❡ s❤❛r✐♥❣ ♣r♦❜❧❡♠s ❢♦r ❞✐❢❢❡r❡♥t✐❛❧ ❛♥❞ ❞✐❢❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ♠❡r♦✲ ♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ✐♥ ❛ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ❢✐❡❧❞✧✱ ♣✲❆❞✐❝ ◆✉♠❜❡rs✱ ❯❧✲ tr❛♠❡tr✐❝ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✉♠❡ ✾✱ ■ss✉❡ ✶✱ ♣♣✳ ✶✕✶✹✳ ❬✼❪ ❱✉ ❍♦❛✐ ❆♥✱ P❤❛♠ ◆❣♦❝ ❍♦❛✱ ❛♥❞ ❍❛ ❍✉② ❑❤♦❛✐ ✭❙✉❜♠✐t✮✱ ✧❯♥✐q✉❡✲ ♥❡ss t❤❡♦r❡♠s ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②✲ ♥♦♠✐❛❧✧✳ ❬✽❪ ❱✉ ❍♦❛✐ ❆♥ ❛♥❞ ▲❡ ◗✉❛♥❣ ◆✐♥❤ ✭✷✵✶✻✮✱ ✧❖♥ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❋❡r♠❛t✲❲❛r✐♥❣ t②♣❡ ❢♦r ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ✈❡❝t♦r✐❛❧ ❡♥t✐r❡ ❢✉♥❝✲ t✐♦♥s✧✱ ❇✉❧❧✳❑♦r❡❛♥ ▼❛t❤✳❙♦❝✱ ✺✸✭✹✮✱ ♣♣✳ ✶✶✽✺✲✶✶✾✻✳ ✾✻ ❬✾❪ ❇❡③✐✈✐♥ ❏✳ P✳✱ ❇♦✉ss❛❢ ❑✳ ❛♥❞ ❊s❝❛ss✉t ❆✳ ✭✷✵✶✷✮✱ ✧❩❡r♦s ♦❢ t❤❡ ❞❡r✐✈❛✲ t✐✈❡ ♦❢ ❛ ♣✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥✧✱ ❇✉❧❧✳ ❙❝✐✳ ▼❛t❤➨♠❛t✐q✉❡s✱ ✶✸✻✭✽✮✱ ♣♣✳ ✽✸✾✕✽✹✼✳ ❬✶✵❪ ❇❤♦♦s♥✉r♠❛t❤ ❙✉❜❤❛s ❙✳ ❛♥❞ ❉②❛✈❛♥❛❧ ❘❡♥✉❦❛❞❡✈✐ ❙✳ ✭✷✵✵✼✮✱ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡✲s❤❛r✐♥❣ ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❈♦♠♣✉t❡rs ❛♥❞ ▼❛t❤❡♠❛t✐❝s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ✺✸✱ ♣♣✳ ✶✶✾✶✲✶✷✵✺✳ ❬✶✶❪ ❇♦✉ss❛❢ ❑✳ ✱ ❊s❝❛ss✉t ❆✳✱ ❖❥❡❞❛ ❏✳ ✭✷✵✶✷✮✱ ✧p✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝✲ t✐♦♥s (f )( ) P (f ), (g)( ) P (g) s❤❛r✐♥❣ ❛ s♠❛❧❧ ❢✉♥❝t✐♦♥✧✱ ❇✉❧❧✳ ❙❝✐✳ ♠❛t❤✱ ✶✸✻✱ ♣♣✳ ✶✼✷✲✷✵✵✳ ❬✶✷❪ ❇♦✉t❛❜❛❛ ❆✳ ✭✶✾✾✵✮✱ ✧❚❤✬❡♦r✐❡ ❞❡ ◆❡✈❛♥❧✐♥♥❛ ♣✲❛❞✐q✉❡✧✱ ▼❛♥✉s❝r✐♣t❛ ▼❛t❤✱ ✻✼✱ ♣♣✳ ✷✺✶✲✷✻✾✳ ❬✶✸❪ ❚✳ ❉✐♥❤ ✭✷✵✵✷✮✱ ✧❊♥s❡♠❜❧❡s ❞✬✉♥✐❝✐t❡✬ ♣♦✉r ❧❡s ♣♦❧♥♦♠❡s✧✱ ❊r❣♦❞✐❝ ❚❤❡♦r② ❉②♥❛♠ ❙②st❡♠s✱ ✷✷✭✶✮✱ ♣♣✳ ✶✼✶✲✶✽✻✳ ❬✶✹❪ ❚✳ ❉✐♥❤ ✭✷✵✵✺✮✱ ✧❉✐str✐❜✉t✐♦♥ ❞❡s ♣r➨✐♠❛❣❡s ❡t ❞❡s ♣♦✐♥ts ♣➨r✐♦❞✐q✉❡s ❞✬✉♥❡ ❝♦rr❡s♣♦♥❞❛♥❝❡ ♣♦❧②♥♦♠✐❛❧❡✧✱ ❇✉❧❧✳ ❙♦❝✳ ▼❛t❤✳ ❋r❛♥❝❡✱ ✶✸✸✭✸✮✱ ♣♣✳ ✸✻✸✕✸✾✹✳ ❬✶✺❪ ❊s❝❛ss✉t ❆✳ ❛♥❞ ❖❥❡❞❛ ❏✳ ✭✷✵✶✹✮✱ ✧❚❤❡ ♣✲❛❞✐❝ ❍❛②♠❛♥ ❈♦♥❥❡❝t✉r❡ ✇❤❡♥ ♥❂✷✧✱ ❈♦♠♣❧❡① ❱❛r✐❛❜❧❡ ❛♥❞ ❊❧❧✐♣t✐❝ ❊q✉❛t✐♦♥s✱ ✺✾✭✶✵✮✱ ♣♣✳ ✶✹✺✶✲✶✹✺✻✳ ❬✶✻❪ ❊s❝❛ss✉t ❆✳ ✭✷✵✶✺✮✱ ✧❱❛❧✉❡ ❉✐str✐❜✉t✐♦♥ ✐♥ ♣✲❛❞✐❝ ❆♥❛❧②s✐s✧✱ ❲♦r❧❞ ❙❝✐✳ P✉❜❧✳ ❈♦✳ Pt❡✱ ▲t❞✳ ❙✐♥❣❛♣♦r❡✳ ❬✶✼❪ ❋❛♥❣ ❈✳ ❨✳ ❛♥❞ ❋❛♥❣ ▼✳ ▲✳ ✭✷✵✵✷✮✱ ✧❯♥✐q✉❡♥❡ss ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝✲ t✐♦♥s ❛♥❞ ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧s✧✱ ❈♦♠♣✉t✳ ▼❛t❤✳ ❆♣♣❧✱ ✹✹ ✱ ♣♣✳ ✻✵✼✲ ✻✶✼✳ ❬✶✽❪ ❋❛♥❣ ▼✳ ▲✳ ✭✷✵✵✷✮✱ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡✲s❤❛r✐♥❣ ♦❢ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✧✱ ❈♦♠♣✉t✳ ▼❛t❤✳ ❆♣♣❧✱ ✹✹✱ ♣♣✳ ✽✷✸✲✽✸✶✳ ❚❤❡♦r② ❆♣♣❧✱ ✸✼✭✶✲✹✮✱ ♣♣✳ ✶✽✺✲✶✾✸✳ ❬✶✾❪ ❋✉❥✐♠♦t♦ ❍✳ ✭✷✵✵✵✮✱ ✧❖♥ ✉♥✐q✉❡♥❡ss ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s s❤❛r✐♥❣ ❢✐♥✐t❡ s❡ts✧✱ ❆♠❡r✳ ❏✳▼❛t❤✱ ✶✷✷✭✻✮✱ ♣♣✳ ✶✶✼✺ ✲ ✶✷✵✸✳ ❬✷✵❪ ❍❛②♠❛♥ ❲✳❑✳ ✭✶✾✻✹✮✱ ✧▼❡r♦♠♦r♣❤✐❝ ❋✉♥❝t✐♦♥s✧✱ ❖①❢♦r❞ ▼❛t❤❡♠❛t✲ ✐❝❛❧ ▼♦♥♦❣r❛♣❤s ❈❧❛r❡♥❞♦♥ Pr❡ss✱ ❖①❢♦r❞✳ ❬✷✶❪ ❍❛②♠❛♥ ❲✳❑✳ ✭✶✾✻✼✮✱ ✧❘❡s❡❛r❝❤ ♣r♦❜❧❡♠s ✐♥ ❋✉♥❝t✐♦♥ ❚❤❡♦r②✧✱ ❚❤❡ ❆t❤❧♦♥❡ Pr❡ss ❯♥✐✈❡rs✐t② ♦❢ ▲♦♥❞♦♥✱ ▲♦♥❞♦♥✳ ❬✷✷❪ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✵✽✮✱ ✧ ❆♥ ❛♥❛❧♦❣✉❡ ♦❢ ▼❛s♦♥✬s t❤❡♦r❡♠ ❢♦r ♣✲❛❞✐❝ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s ✐♥ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✧✱ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥❝❡✱ ◆❳❇ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ❍➔ ◆ë✐✱ ✺✸✭✶✮✱ ♣♣✳ ✶✷✲✷✶✳ ✾✼ ❬✷✸❪ ❍✉ P✳ ❈✳ ❛♥❞ ❨❛♥❣ ❈✳ ❈✳ ✭✶✾✾✾✮✱ ✧ ❆ ✉♥✐q✉❡ r❛♥❣❡ s❡t ❢♦r ♣✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ✇✐t❤ ✶✵ ❡❧❡♠❡♥ts✧✱ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠✐❝❛✳✱ ✷✹✱ ♣♣✳ ✾✺✲✶✵✽✳ ❬✷✹❪ ❍✉ P✳ ❈✳ ❛♥❞ ❨❛♥❣ ❈✳ ❈✳ ✭✷✵✵✵✮✱ ✧▼❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ♦✈❡r ♥♦♥✲ ❆r❝❤✐♠❡❞❡❛♥ ❢✐❡❧❞s✧✱ ❑❧✉✇❡r✳ ❬✷✺❪ ❍❛ ❍✉② ❑❤♦❛✐ ✭✶✾✽✸✮✱ ✧❖♥ ♣✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❉✉❦❡ ▼❛t❤✳ ❏✳✱ ✺✵✱♣♣✳ ✻✾✺✲✼✶✶✳ ❬✷✻❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ❱✉ ❍♦❛✐ ❆♥ ✭✷✵✵✸✮✱ ✧❱❛❧✉❡ ❞✐str✐❜✉t✐♦♥ ❢♦r ♣✲❛❞✐❝ ❤②♣❡rs✉r❢❛❝❡s✧✱ ❚❛✐✇❛♥❡s❡ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✼✭✶✮✱ ♣♣✳ ✺✶✲✻✼✳ ❬✷✼❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ❱✉ ❍♦❛✐ ❆♥ ✭✷✵✶✶✮✱ ✧❱❛❧✉❡ ❞✐str✐❜✉t✐♦♥ ♣r♦❜❧❡♠ ❢♦r ♣✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✧✱ ❆♥♥✳ ❋❛❝✳ ❙❝✳ ❚♦✉❧♦✉s❡✱ ❱♦❧✳ ❳❳ ✭◆♦✳ ❙♣❡❝✐❛❧✮✱ ♣♣✳✶✸✺✲✶✹✾✳ ❬✷✽❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ❱✉ ❍♦❛✐ ❆♥ ✭✷✵✶✷✮✱ ✧ ❱❛❧✉❡ s❤❛r✐♥❣ ♣r♦❜❧❡♠ ❢♦r p✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❞✐❢❢❡r❡♥❝❡ ♦♣❡r❛t♦rs ❛♥❞ ❞✐❢✲ ❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s✧✱ ❯❦r❛♥✐❛♥ ▼❛t❤✳ ❏✳✱ ✻✹✭✷✮✱ ♣♣✳ ✶✹✼✲✶✻✹✳ ❬✷✾❪ ❍❛ ❍✉② ❑❤♦❛✐✱ ❱✉ ❍♦❛✐ ❆♥✱ ❛♥❞ P❤❛♠ ◆❣♦❝ ❍♦❛ ✭✷✵✶✼✮✱ ✧❖♥ ❢✉♥❝✲ t✐♦♥❛❧ ❡q✉❛t✐♦♥s ❢♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❆r❝❤✐✈ ❞❡r ▼❛t❤❡♠❛t✐❦✱ ❙♣r✐♥❣❡r ■♥t❡r♥❛t✐♦♥❛❧ P✉❜❧✐s❤✐♥❣✱ ❱♦❧✉♠❡ ✶✵✾✱ ■ss✉❡ ✻✱ ♣♣ ✺✸✾✕✺✹✾✳ ❬✸✵❪ ❍❛ ❍✉② ❑❤♦❛✐✱ ❱✉ ❍♦❛✐ ❆♥ ❛♥❞ ◆❣✉②❡♥ ❳✉❛♥ ▲❛✐ ✭✷✵✶✷✮✱ ✧❱❛❧✉❡ s❤❛r✐♥❣ ♣r♦❜❧❡♠ ❛♥❞ ❯♥✐q✉❡♥❡ss ❢♦r p✲❛❞✐❝ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❆♥♥✳ ❯♥✐✈✳ ❙❝✐✳ ❇✉❞❛♣❡st✳✱ ❙❡❝t✳ ❈♦♠♣✳✱ ✸✽✱ ♣♣✳ ✼✶✲✾✷✳ ❬✸✶❪ ❍❛ ❍✉② ❑❤♦❛✐✱ ❱✉ ❍♦❛✐ ❆♥ ❛♥❞ ◆❣✉②❡♥ ❳✉❛♥ ▲❛✐ ✭✷✵✶✼✮✱ ✧❱❛❧✉❡✲ s❤❛r✐♥❣ ❛♥❞ ❯♥✐q✉❡♥❡ss ♣r♦❜❧❡♠s ❢♦r ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧s ✐♥ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✧✱ ❈♦♠♣❧❡① ❱❛r✐❛❜❧❡s ❛♥❞ ❊❧❧✐♣t✐❝ ❊q✉❛t✐♦♥s✱ ♣♣ ✶✲✶✼✳ ❬✸✷❪ ❍❛ ❍✉② ❑❤♦❛✐✱ ❱✉ ❍♦❛✐ ❆♥ ❛♥❞ ▲❡ ◗✉❛♥❣ ◆✐♥❤ ✭✷✵✶✹✮✱ ✧❯♥✐q✉❡✲ ♥❡ss t❤❡♦r❡♠s ❢♦r ❤♦❧♦♠♦r♣❤✐❝ ❝✉r✈❡s ✇✐t❤ ❍②♣❡rs✉r❢❛❝❡s ♦❢ ❋❡r♠❛t✲ ❲❛r✐♥❣ t②♣❡✧✱ ❈♦♠♣❧❡① ❆♥❛❧✳ ❖♣❡r✳ ❚❤❡♦r②✱ ❱♦❧✳✽✱ ◆♦✳✸✱ ♣♣✳ ✺✾✶✲✼✾✵✳ ❬✸✸❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ❚❛ ❚❤✐ ❍♦❛✐ ❆♥ ✭✷✵✵✶✮✱ ✧❖♥ ✉♥✐q✉❡♥❡ss ♣♦❧②✲ ♥♦♠✐❛❧s ❛♥❞ ❇✐✲❯❘❙ ❢♦r ♣✲❛❞✐❝ ▼❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❏✳ ◆✉♠❜❡r ❚❤❡♦r②✱ ✽✼✱ ♣♣✳ ✷✶✶✲✷✷✶✳ ❬✸✹❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ▼❛✐ ❱❛♥ ❚✉ ✭✷✵✵✹✮✱ ✧♣✲❛❞✐❝ ◆❡✈❛♥❧✐♥♥❛✲❈❛rt❛♥ ❚❤❡♦r❡♠✧✱ ■♥t❡r♥❛t✳ ❏✳ ▼❛t❤✱ ✻✭✶✾✾✺✮✱ ♣♣✳ ✼✶✾✲✼✸✶✳ ❬✸✺❪ ❍❛ ❍✉② ❑❤♦❛✐ ❛♥❞ ❨❛♥❣ ❈✳ ❈✳✱ ✧❖♥ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥ P (f ) = Q(g)✧✱ ❆❞✈❛♥❝❡s ✐♥ ❈♦♠♣❧❡① ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥✱ ❱❛❧✉❡ ❉✐s✲ tr✐❜✉t✐♦♥ ❚❤❡♦r② ❛♥❞ ❘❡❧❛t❡❞ ❚♦♣✐❝s✱ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡rs✱ ❉♦r❞r❡❝❤t✱ ❇♦st♦♥✱ ▲♦♥❞♦♥✱ ♣♣✳ ✷✵✶✲✷✵✽✳ ✾✽ ❬✸✻❪ ▲❛❤✐r✐ ■✳ ✭✶✾✾✼✮ ✱ ✧❯♥✐q✉❡♥❡ss ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛s ❣♦✈❡r♥❡❞ ❜② t❤❡✐r ❞✐❢❢❡r❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧s✧✱ ❨♦❦♦❤❛♠❛ ▼❛t❤✳ ❏✳✱ ✹✹✱ ♣♣✳ ✶✹✼✲ ✶✺✻✳ ❬✸✼❪ ▲❛✐♥❡ ■✳ ❛♥❞ ❨❛♥❣ ❈✳ ❈✳ ✭✷✵✵✼✮✱ ✧❱❛❧✉❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❞✐❢❢❡r❡♥❝❡ ♣♦❧②✲ ♥♦♠✐❛❧s✧✱ Pr♦❝✳ ❏❛♣❛♥✳ ❆❝❛❞✳✱ ❙❡r✳ ❆✱ ✽✸✭✽✮✱ ♣♣✳✶✹✽✲✶✺✶✳ ❬✸✽❪ ▲✐ P✳ ❛♥❞ ❨❛♥❣ ❈✳❈✳ ✭✷✵✵✹✮✱ ✧❙♦♠❡ ❋✉rt❤❡r ❘❡s✉❧ts ♦♥ t❤❡ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥ P (f ) = Q(g)✧✱ ❱❛❧✉❡ ❉✐str✐❜✉t✐♦♥ ❚❤❡♦r② ❛♥❞ ❘❡❧❛t❡❞ ❚♦♣✲ ✐❝s✱ ❆❞✈❛♥❝❡❞ ❈♦♠♣❧❡① ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥✱ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝✱ ❇♦st♦♥✱ ▼❆✱ ❱♦❧✳✸✱ ♣♣✳ ✷✶✾✲✷✸✶✳ ❬✸✾❪ ▲✐✉ ❑✳✱ ▲✐✉ ❳✳✱ ❈❛♦ ❚✳ ❇✳ ✭✷✵✶✶✮✱ ✧❱❛❧✉❡ ❞✐str✐❜✉t✐♦♥ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ❞✐❢❢❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧s✧✱ ❆❞✈✳ ❉✐❢❢❡r❡♥❝❡ ❊q✉✳✱ ❛rt✐❝❧❡ ■❉✷✸✹✷✶✺✱ ✶✷♣♣✳ ❬✹✵❪ ❑✳ ▼❛s✉❞❛ ❛♥❞ ❏✳ ◆♦❣✉❝❤✐ ✭✶✾✾✻✮✱ ✧❆ ❝♦♥str✉❝t✐♦♥ ♦❢ ❤②♣❡r❜♦❧✐❝ ❤②✲ ♣❡rs✉r❢❛❝❡ ♦❢ P N (C)✧✱ ▼❛t❤✳ ❆♥♥✳✱ ✸✵✹ ✱♣♣✳ ✸✸✾✲✸✻✷✳ ✶✽✱ ♣♣✳ ✹✸✼✲✹✺✵✳ ❬✹✶❪ ❖❥❡❞❛ ❏✳ ✭✷✵✵✽✮✱ ✧❍❛②♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ✐♥ ❛ p✲❛❞✐❝ ❢✐❡❧❞✧✱ ❚❛✐✇❛♥❡s❡ ❏✳ ▼❛t❤✳ ✶✷✭✾✮✱ ♣♣✳ ✷✷✾✺✲✷✸✶✸✳ ❬✹✷❪ ❖str♦✈s❦✐✐ ■✳✱ P❛❦♦✈✐t❝❤ ❋✳✱ ❩❛✐❞❡♥❜❡r❣ ▼✳ ✭✶✾✾✻✮✱ ✧❆ r❡♠❛r❦ ♦♥ ❝♦♠✲ ♣❧❡① ♣♦❧②♥♦♠✐❛❧s ♦❢ ❧❡❛st ❞❡✈✐❛t✐♦♥✧✱ ■♥t❡r♥❛t✳ ▼❛t❤✳ ❘❡s✳ ◆♦t✐❝❡s✱ ✶✹✱ ♣♣✳ ✻✾✾✕✼✵✸✳ ❬✹✸❪ P❛❦♦✈✐❝❤ ❋✳ ✭✷✵✵✽✮✱ ✧ ❖♥ ♣♦❧②♥♦♠✐❛❧s s❤❛r✐♥❣ ♣r❡✐♠❛❣❡s ♦❢ ❝♦♠♣❛❝t s❡ts✱ ❛♥❞ r❡❧❛t❡❞ q✉❡st✐♦♥s✧✱ ●❡♦♠✳ ❋✉♥❝t✳ ❆♥❛❧✱ ✶✽✭✶✮✱ ♣♣✳ ✶✻✸✲✶✽✸ ✳ ❬✹✹❪ P❛❦♦✈✐❝❤ ❋✳ ✭✷✵✶✵✮✱ ✧❖♥ t❤❡ ❡q✉❛t✐♦♥ P (f ) = Q(g), ✇❤❡r❡ P, Q ❛r❡ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ f, g ❛r❡ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✧✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳✱ ✶✸✷✭✻✮✱ ♣♣✳ ✶✺✾✶✲✶✻✵✼✳ ❬✹✺❪ ◗✐ ❳✳ ❈✳✱ ❨❛♥❣ ▲✳ ❩✳✱ ▲✐✉ ❑✳ ✭✷✵✶✵✮✱ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ♣❡r✐♦❞✐❝✐t② ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❝♦♥❝❡r♥✐♥❣ t❤❡ ❞✐❢❢❡r❡♥❝❡ ♦♣❡r❛t♦r✧✱ ❈♦♠♣✳ ▼❛t❤✳ ❆♣♣❧✳✱ ✻✵✭✻✮✱ ♣♣✳ ✶✼✸✾✲✶✼✹✻✳ ❬✹✻❪ ❘✐tt ❏✳ ✭✶✾✷✷✮✱ ✧Pr✐♠❡ ❛♥❞ ❝♦♠♣♦s✐t❡ ♣♦❧②♥♦♠✐❛❧s✧✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✷✸✭✶✮✱ ♣♣✳ ✺✶✲✻✻✳ ❬✹✼❪ ❘✉ ▼✳ ✭✷✵✵✶✮✱ ✧❆ ♥♦t❡ ♦♥ ♣✲❛❞✐❝ ◆❡✈❛♥❧✐♥♥❛ t❤❡♦r②✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✷✾✱ ♣♣✳ ✶✷✻✸✲✶✷✻✾✳ ❬✹✽❪ ❙✐✉ ❨✳❚✳✱ ❨❡✉♥❣ ❙✳❑✳ ✭✶✾✾✼✮✱ ✧❉❡❢❡❝ts ❢♦r ❛♠♣❧❡ ❞✐✈✐s♦rs ♦❢ ❆❜❡❧✐❛♥ ✈❛✲ r✐❡t✐❡s✱ ❙❝❤✇❛r③ ❧❡♠♠❛✱ ❛♥❞ ❤②♣❡r❜♦❧✐❝ ❤②♣❡rs✉r❢❛❝❡s ♦❢ ❧♦✇ ❞❡❣r❡❡s✧✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳✱ ✶✶✾✱ ♣♣✳ ✶✶✸✾✲✶✶✼✷✳ ✾✾ ❬✹✾❪ ❨❛♥❣ ❈✳ ✭✶✾✼✽✮✱ ✧❖♣❡♥ ♣r♦❜❧❡♠ ✐♥ ❈♦♠♣❧❡① ❛♥❛❧②s✐s✧✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❙✳❯✳◆✳❨✳❇r♦❝❦♣♦rt ❈♦♥❢✳ ♦♥ ❈♦♠♣❧❡① ❋✉♥❝t✐♦♥ ❚❤❡♦r②✱ ❏✉♥❡ ✼✕✾✱ ✶✾✼✻✱ ❊❞✐t❡❞ ❜② ❙❛♥❢♦r❞ ❙✳ ▼✐❧❧❡r✳ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ▼❛r❝❡❧ ❉❡❦❦❡r✱ ■♥❝✳✱ ◆❡✇ ❨♦r❦✲❇❛s❡❧✱ ❱♦❧✳✸✻✳ ❬✺✵❪ ❨❛♥❣ ❈✳❈✳ ✭✶✾✼✻✮✱ ✧❖♥ t✇♦ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✱ ✇❤✐❝❤ t♦❣❡t❤❡r ✇✐t❤ t❤❡✐r ❢✐rst ❞❡r✐✈❛t✐✈❡s ❤❛✈❡ t❤❡ s❛♠❡ ③❡r♦s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✺✻✱ ♣♣✳ ✶✲✻✳ ❬✺✶❪ ❨❛♥❣ ❈✳❈✳ ❛♥❞ ❍✉❛ ❳✳❍✳ ✭✶✾✾✼✮✱ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡✲s❤❛r✐♥❣ ♦❢ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❆♥♥✳ ❆❝❛❞✳ ❙❝✐✳ ❋❡♥♥✳ ▼❛t❤✳✱ ✷✷✱ ♣♣✳ ✸✾✺✲ ✹✵✻✳ ❬✺✷❪ ❨✐ ❍✳ ❳✳ ✭✶✾✾✵✮✱ ✧❆ q✉❡st✐♦♥ ♦❢ ❈✳ ❈✳ ❨❛♥❣ ♦♥ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✧✱ ❑♦❞❛✐ ▼❛t❤✳ ❏✳✱ ✶✸✱ ♣♣✳ ✸✾✲✹✻✳ ❬✺✸❪ ❨✐ ❍✳ ❳✳ ✭✶✾✾✺✮✱ ✧❚❤❡ ✉♥✐q✉❡ r❛♥❣❡ s❡ts ♦❢ ❡♥t✐r❡ ♦r ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✧✱ ❈♦♠♣❧❡① ❱❛r✐❛❜❧❡s ❚❤❡♦r② ❆♣♣❧✳✱ ✷✽✱ ♣♣✳ ✶✸✲✷✶✳ ❬✺✹❪ ❩❤❛♥❣ ❳✳❨✳✱ ▲✐♥ ❲✳❈✳ ✭✷✵✵✽✮✱ ✧❯♥✐q✉❡♥❡ss ❛♥❞ ✈❛❧✉❡✲s❤❛r✐♥❣ ♦❢ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✹✸✱ ♣♣✳ ✾✸✽✲✾✺✵✳ ... ❜➟❝ k ❝õ❛ f ợ k số ữỡ ổ ❣✐→ trà ✶ t❤➻ f ❧➔ ❤➔♠ ❤➡♥❣✳ ❍❛②♠❛♥ ❝ô♥❣ ✤÷❛ r❛ ❣✐↔ t❤✉②➳t s❛✉✳ ●✐↔n t❤✉②➳t ❍❛②♠❛♥✳ ❬✷✶❪ ◆➳✉ ♠ët ❤➔♠ ♥❣✉②➯♥ f t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (z)f (z) = ợ n số ữỡ ✈ỵ✐ ♠å✐ z... ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✷✳✸ ❝❤♦ ✭✶✳✶✾✮ t s r tỗ t số ổ h s❛♦ ❝❤♦ ψ = hϕ, tù❝ ❧➔ Rr−1 ◦ · · · ◦ R1 ◦ f = hDs−1 ◦ · · · ◦ D1 ◦ g ❚✐➳♣ tư❝ ♥❤÷ ✈➟②✱ t t r tỗ t số t ổ s❛♦ ❝❤♦ R1 ◦ f = tDs−r+1 ◦ · · · ◦... ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ tr➯♥ ỗ tớ õ ú t t q ự ỵ tt ✤➲ ✶✳ ❱➜♥ ✤➲ ✷✳ ✷✳ ▼ö❝ t✐➯✉ ❝õ❛ ❧✉➟♥ t ởt số ỵ tữỡ tỹ ỵ tt ố ợ ✈➔ ✤❛ t❤ù❝ ✈✐ ♣❤➙♥✱ ✤❛ t❤ù❝ s❛✐ ♣❤➙♥✱ ✤❛ t❤ù❝ q ✲s❛✐ ♣❤➙♥ tr♦♥❣ tr÷í♥❣