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P♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❡①tr❡♠❛❧ ♣r♦❜❧❡♠s ✐♥ ❞✐s❝r❡t❡ ❣❡♦♠❡tr② ♦♥ ❊✉❝❧✐❞❡❛♥ s♣❤❡r❡ ◆✐❦♦❧❛✐ ❆ . ❑✉❦❧✐♥ 30 . 08 . 2017 ✶

m ✖ ❞✐♠❡♥s✐♦♥ ; � m q ( k ) 1 q ( k ) 2 , q 1 , q 2 ∈ R m ; m ≥ 2 , � q 1 , q 2 � = k =1 q ∈ R m � � � S m − 1 = � � q, q � − 1 = 0 , e m = (0 , 0 , . . . , 0 , 1) ∈ S m − 1 ; n ✖ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ; ( S m − 1 ) n = S m − 1 × · · · × S m − 1 ; n ≥ 2 , h : [ − 1 , 1] → ( −∞ , ∞ ] ✖ ❧♦✇❡r s❡♠✐❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ; � h ( � q i , q j � ) : ( S m − 1 ) n → ( −∞ , ∞ ] . W h ( q 1 , q 2 , . . . , q n ) = 1 ≤ i<j ≤ n ✷

❚❤♦♠s♦♥ ♣r♦❜❧❡♠ P♦s❡❞ ❜② ❏ . ❏ . ❚❤♦♠s♦♥ (1904) ❢♦r m = 3 . � (2 − 2 t ) − 1 / 2 , t ∈ [ − 1 , 1) , φ ( t ) = ∞ , t = 1 . � W φ ( q 1 , q 2 , . . . , q n ) | q 1 , q 2 , . . . , q n ∈ S 2 � ω φ = min = ? ✸

❚❤♦♠s♦♥ ♣r♦❜❧❡♠ n = 2 n = 3 n = 4 n = 5 n = 6 n = 12 ❘ . ❊ . ❙❝❤✇❛rt③ (2010) ❱ . ❆ . ❨✉❞✐♥ (1993) ◆ . ◆ . ❆♥❞r❡❡✈ (1996) ✹

❚❤♦♠s♦♥ ♣r♦❜❧❡♠ ❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛t✐♦♥ T ⊂ [ − 1 , 1) ✖ ✜♥✐t❡ s❡t ; h T ✖ ✉♥✐✈❛r✐❛t❡ ♣♦❧✐♥♦♠✐❛❧ ✇✐t❤ : • h T ( t ) = φ ( t ) , t ∈ T ; • h ′ T ( t ) = φ ′ ( t ) , t ∈ T \ {− 1 } ; • deg( h T ) = | T | + | T \ {− 1 }| − 1 . ❈❧❛✐♠ ✭❱ . ❆ . ❨✉❞✐♥ & ◆ . ◆ . ❆♥❞r❡❡✈✮ h T ( t ) ≤ φ ( t ) , t ∈ [ − 1 , 1]; W h T ≤ W φ ♦♥ ( S 2 ) n , ω h T ≤ ω φ ; � W h T ( q 1 , q 2 , . . . , q n ) | q 1 , q 2 , . . . , q n ∈ S 2 � min = ? ✺

❚❤♦♠s♦♥ ♣r♦❜❧❡♠ n = 2 n = 3 n = 4 T = {− 1 T = {− 1 T = {− 1 } 2 } 3 } deg( h T ) = 0 deg( h T ) = 1 deg( h T ) = 1 n = 5 n = 6 n = 12 √ √ 5 5 T = {− 1 , − 1 2 , 0 } T = {− 1 , 0 } T = {− 1 , − 5 , 5 } deg( h T ) = 4 deg( h T ) = 2 deg( h T ) = 4 ✻

P❛❝❦✐♥❣ ♣r♦❜❧❡♠ s ∈ [ − 1 , 1); τ ( m, s ) = max { n | ∃ q 1 , q 2 , . . . , q n ∈ S m − 1 : � q i , q j � ≤ s, i � = j } . �� � − 1 � � 2 R 2 s = 1 − 2 ; R = 1 − s − 1 . 1 + R ❑✐ss✐♥❣ ♥✉♠❜❡r ♣r♦❜❧❡♠ τ ( m ) = τ ( m, 1 / 2) . ✼

❑✐ss✐♥❣ ♥✉♠❜❡r ♣r♦❜❧❡♠ τ (2) = 6; ❑ . ❙❝❤ütt❡ & ❇ . ▲ . ✈❛♥ ❞❡r ❲❛❡r❞❡♥ (1953): τ (3) = 12; ❱ . ■ . ▲❡✈❡♥s❤t❡✐♥ ; ❆ . ▼ . ❖❞❧②③❦♦ & ◆ . ❏ . ❆ . ❙❧♦❛♥❡ (1978): τ (8) = 240 , τ (24) = 196560; ❖ . ❘ . ▼✉s✐♥ (2003): τ (4) = 24; 40 ≤ τ (5) ≤ 44 ( ❍ . ❉ . ▼✐tt❡❧♠❛♥ & ❋ . ❱❛❧❧❡♥t✐♥ , 2009 , ❜❛s❡❞ ♦♥ ✇♦r❦ ♦❢ ❈ . ❇❛❝❤♦❝ & ❋ . ❱❛❧❧❡♥t✐♥ , 2006) . ✽

P❛❝❦✐♥❣ ♣r♦❜❧❡♠ ❚❤❡♦r❡♠ ❋♦r n ≥ 3 t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t : • τ ( m, s ) < n ; • t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧ h s✉❝❤ t❤❛t ✶ h ( t ) ≤ 0 , t ∈ [ − 1 , s ]; ✷ W h > 0 ♦♥ ( S m − 1 ) n . ❈♦♥❥❡❝t✉r❡ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦❧②♥♦♠✐❛❧ ✐s s✉✐t❛❜❧❡ ❢♦r m = 3 , s = 1 / 2 , n = 13: � � 2 � � 2 � � t + 3 t + 1 t − 1 h ( t ) = ( t + 1) . 5 5 2 ✾

P♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❋♦r ❣✐✈❡♥ m ≥ 2 , n ≥ 2 ❛♥❞ ✉♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧ h ✜♥❞ � W h ( q 1 , q 2 , . . . , q n ) | q 1 , q 2 , . . . , q n ∈ S m − 1 � ω h = min ; � W h ( q 1 , q 2 , . . . , q n ) = h ( � q i , q j � ) . 1 ≤ i<j ≤ n ◆♦t❡ t❤❛t W h ✐s ♠✉❧t✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧ ♦❢ mn ✈❛r✐❛❜❧❡s ✇✐t❤ deg( W h ) = 2 deg( h ) . ❈♦♥❞✐t✐♦♥s q 1 , q 2 , . . . , q n ∈ S m − 1 ❛r❡ ♣♦❧②♥♦♠✐❛❧ t♦♦ : � q j , q j � − 1 = 0 , 1 ≤ j ≤ n. ❲❡ ❝❛♥ ✜① ♦♥❡ ♣♦✐♥t : � W h ( q 1 , q 2 , . . . , q n − 1 , e m ) | q 1 , q 2 , . . . , q n − 1 ∈ S m − 1 � ω h = min . ✶✵

❙❡♠✐❞❡✜♥✐t❡ ♣r♦❣r❛♠♠✐♥❣ ( ❙❉P ) A 0 , A 1 , . . . , A p ✖ ❣✐✈❡♥ s②♠♠❡tr✐❝ ♠❛tr✐❝❡s ♦❢ s✐③❡ l × l ; c 1 , c 2 , . . . , c p ∈ R ;  max X • A 0 ;   X • A i = c i , 1 � i � p ;   X � 0; l l � � X • Y = X ij Y ij . i =1 j =1 ✶✶

❙✉♠ ♦❢ sq✉❛r❡s ❈❧❛✐♠ ▲❡t A ❜❡ ❛ ❝♦♠♠✉t❛t✐✈❡ R ✲❛❧❣❡❜r❛ , f ∈ A . ❋♦r ❛ ❧✐♥❡❛r s✉❜s♣❛❝❡ L = span { b 1 , . . . , b l } ⊂ A t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t : (1) r � f 2 f = k , f 1 , . . . , f r ∈ L ; k =1 (2) l l � � f = Y ij b i b j i =1 j =1 ❢♦r s♦♠❡ ♠❛tr✐① Y � 0 ♦❢ s✐③❡ l × l. ✶✷

❈♦♥✈❡① r❡❧❛①❛t✐♦♥ P♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ � W h ( q 1 , q 2 , . . . , q n − 1 , e m ) | q 1 , q 2 , . . . , q n − 1 ∈ S m − 1 � ω h = min ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❙❉P ♣r♦❜❧❡♠s ( d ≥ deg( h )):   � ω h ( d ) = sup γ, γ ∈ R ;    n − 1 � ( Q + j − Q − W h ( q 1 , . . . , q n − 1 , e m ) − γ = Q 0 + j ) · ( � q j , q j � − 1);   j =1   Q ± j ✐s s✉♠ ♦❢ sq✉❛r❡s ♦❢ ❡❧❡♠❡♥ts ♦❢ A ≤ d , 0 ≤ j ≤ n − 1 . A = R [ q ( k ) j ] 1 ≤ j ≤ n − 1 , 1 ≤ k ≤ m ; A ≤ d = { f ∈ A | deg( f ) ≤ d } . ω h ( d ) ≤ ω h . � ✶✸

❲❡ ❝❛♥ ✉s❡ ●rö❜♥❡r ❜❛s❡s ( ❲ . ●rö❜♥❡r , ❇ . ❇✉❝❤❜❡r❣❡r , 1965):  ω h ( d ) = sup γ, γ ∈ R ;   W h ( q 1 , . . . , q n − 1 , e m ) − γ = Q 0 (mod I );   Q 0 ✐s s✉♠ ♦❢ sq✉❛r❡s ♦❢ ❡❧❡♠❡♥ts ♦❢ ( A / I ) ≤ d ;    �  n − 1 � � I = f j · ( � q j , q j � − 1) � f j ∈ A  ⊂ A ,  j =1 A / I ✐s q✉♦t✐❡♥t ❛❧❣❡❜r❛ , ( A / I ) ≤ d ✐s ✐♠❛❣❡ ♦❢ A ≤ d ❜② q✉♦t✐❡♥t ♠❛♣ . ω h ( d ) ≤ ω h ( d ) ≤ ω h . � ❙❝❤♠ü❞❣❡♥ P♦s✐t✐✈st❡❧❧❡♥s❛t③ ( ❑ . ❙❝❤♠ü❞❣❡♥ , 1991) ω h ( d ) ր ω h , d → ∞ . ✶✹

❈♦♠♣❧❡①✐t② ❚❤❡ ❝♦♠♣❧❡①✐t② ( s✐③❡ ♦❢ ♠❛tr✐❝❡s ) ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❞❡♣❡♥❞s ♦♥ dim ( A ≤ d ) = (( n − 1) m + d )! (( n − 1) m )! d ! ≍ d ( n − 1) m , d → ∞ ; dim (( A / I ) ≤ d ) ≍ d ( n − 1)( m − 1) , d → ∞ . ✶✺

❙②♠♠❡tr② ❚❤❡ ♣♦❧②♥♦♠✐❛❧ � W h ( q 1 , q 2 , . . . , q n − 1 , e m ) , W h ( q 1 , . . . , q n ) = h ( � q i , q j � ) , 1 ≤ i<j ≤ n ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r : • ❢♦r σ ∈ S( n − 1) ( ❜✐❥❡❝t✐♦♥ ♦❢ t❤❡ s❡t { 1 , 2 , . . . , n − 1 } ): W h ( q σ (1) , q σ (2) , . . . , q σ ( n − 1) , e m ) = W h ( q 1 , q 2 , . . . , q n − 1 , e m ); • ❢♦r ρ ∈ O( m − 1) ( ♦rt❤♦❣♦♥❛❧ ♠❛tr✐① ) ✇❡ ❤❛✈❡ W h ( ρq 1 , ρq 2 , . . . , ρq n − 1 , e m ) = W h ( q 1 , q 2 , . . . , q n − 1 , e m ) . ❙♦ t❤❡ ♣♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❣r♦✉♣ G = S( n − 1) × O( m − 1) . ✶✻

■rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ■rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❣r♦✉♣s S( n − 1) ❛♥❞ O( m − 1) ❛r❡ ❛❜s♦❧✉t❡❧② ✐rr❡❞✉❝✐❜❧❡ , s♦ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ G ❛r❡ ❥✉st t❡♥s♦r ♣r♦❞✉❝ts ♦❢ S( n − 1) ✬s ❛♥❞ O( m − 1) ✬s ♦♥❡s✳ • ■rr❡♣s ♦❢ S( n − 1) ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❛s ❙♣❡❝❤t ♠♦❞✉❧❡s ( ❲ . ❙♣❡❝❤t , 1935); • ✐rr❡♣s ♦❢ O(2) ❛r❡ s✐♥❡s✲❝♦s✐♥❡s + ❞❡t✲r❡♣r❡s❡♥t❛t✐♦♥ ; • ✐rr❡♣s ♦❢ O(3) ❛r❡ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s + ❞❡t✲r❡♣r❡s❡♥t❛t✐♦♥s ; • · · · ✶✼