On Reflexible Polynomials Aleksander Malni c University of - PowerPoint PPT Presentation

On Reflexible Polynomials Aleksander Malniˇ c University of Ljubljana and University of Primorska Joint work with Boˇ stjan Kuzman and Primoˇ z Potoˇ cnik Graphs, groups, and more: celebrating Brian Alspach’s 80th and Dragan Maruˇ sˇ c’s 65th birthdays Koper, Slovenia May 28 – June 1, 2018 1 / 11

Reflexible polynomials 2 / 11

Reflexible polynomials f ( x ) = a 0 + a 1 x + . . . + a k x k ∈ F [ x ] is reflexible if ∃ λ ∈ F ∗ ∀ i : type (1) λ a k − i = a i (1) ∃ λ ∈ F ∗ ∀ i : ( − 1) i a i type (2) λ a k − i = (2) 2 / 11

Reflexible polynomials f ( x ) = a 0 + a 1 x + . . . + a k x k ∈ F [ x ] is reflexible if ∃ λ ∈ F ∗ ∀ i : type (1) λ a k − i = a i (1) ∃ λ ∈ F ∗ ∀ i : ( − 1) i a i type (2) λ a k − i = (2) type (1) ⇒ λ = ± 1 2 / 11

Reflexible polynomials f ( x ) = a 0 + a 1 x + . . . + a k x k ∈ F [ x ] is reflexible if ∃ λ ∈ F ∗ ∀ i : type (1) λ a k − i = a i (1) ∃ λ ∈ F ∗ ∀ i : ( − 1) i a i type (2) λ a k − i = (2) type (1) ⇒ λ = ± 1 4 + 2 x + 3 x 2 + x 3 ∈ Z 5 [ x ] λ = − 1, type (1) 2 / 11

Reflexible polynomials f ( x ) = a 0 + a 1 x + . . . + a k x k ∈ F [ x ] is reflexible if ∃ λ ∈ F ∗ ∀ i : type (1) λ a k − i = a i (1) ∃ λ ∈ F ∗ ∀ i : ( − 1) i a i type (2) λ a k − i = (2) type (1) ⇒ λ = ± 1 4 + 2 x + 3 x 2 + x 3 ∈ Z 5 [ x ] λ = − 1, type (1) λ = 1 self-reciprocal, palindromic, Gorenstein polynomials 2 / 11

Reflexible polynomials f ( x ) = a 0 + a 1 x + . . . + a k x k ∈ F [ x ] is reflexible if ∃ λ ∈ F ∗ ∀ i : type (1) λ a k − i = a i (1) ∃ λ ∈ F ∗ ∀ i : ( − 1) i a i type (2) λ a k − i = (2) type (1) ⇒ λ = ± 1 4 + 2 x + 3 x 2 + x 3 ∈ Z 5 [ x ] λ = − 1, type (1) λ = 1 self-reciprocal, palindromic, Gorenstein polynomials 2 / 11

Reflexible polynomials f ( x ) = a 0 + a 1 x + . . . + a k x k ∈ F [ x ] is reflexible if ∃ λ ∈ F ∗ ∀ i : type (1) λ a k − i = a i (1) ∃ λ ∈ F ∗ ∀ i : ( − 1) i a i type (2) λ a k − i = (2) type (1) ⇒ λ = ± 1 4 + 2 x + 3 x 2 + x 3 ∈ Z 5 [ x ] λ = − 1, type (1) λ = 1 self-reciprocal, palindromic, Gorenstein polynomials over Z p : criptography, sequences, subfields in alg. closures 2 / 11

Reflexible polynomials f ( x ) = a 0 + a 1 x + . . . + a k x k ∈ F [ x ] is reflexible if ∃ λ ∈ F ∗ ∀ i : type (1) λ a k − i = a i (1) ∃ λ ∈ F ∗ ∀ i : ( − 1) i a i type (2) λ a k − i = (2) type (1) ⇒ λ = ± 1 4 + 2 x + 3 x 2 + x 3 ∈ Z 5 [ x ] λ = − 1, type (1) λ = 1 self-reciprocal, palindromic, Gorenstein polynomials over Z p : criptography, sequences, subfields in alg. closures over Q , C : cyclotomic polynomials are self-reciprocal 2 / 11

Reflexible polynomials f ( x ) = a 0 + a 1 x + . . . + a k x k ∈ F [ x ] is reflexible if ∃ λ ∈ F ∗ ∀ i : type (1) λ a k − i = a i (1) ∃ λ ∈ F ∗ ∀ i : ( − 1) i a i type (2) λ a k − i = (2) type (1) ⇒ λ = ± 1 4 + 2 x + 3 x 2 + x 3 ∈ Z 5 [ x ] λ = − 1, type (1) λ = 1 self-reciprocal, palindromic, Gorenstein polynomials over Z p : criptography, sequences, subfields in alg. closures over Q , C : cyclotomic polynomials are self-reciprocal irr. over Q / Z : char. poly. of auto. of certain unimodular latices 2 / 11

Reflexible polynomials f ( x ) = a 0 + a 1 x + . . . + a k x k ∈ F [ x ] is reflexible if ∃ λ ∈ F ∗ ∀ i : type (1) λ a k − i = a i (1) ∃ λ ∈ F ∗ ∀ i : ( − 1) i a i type (2) λ a k − i = (2) type (1) ⇒ λ = ± 1 4 + 2 x + 3 x 2 + x 3 ∈ Z 5 [ x ] λ = − 1, type (1) λ = 1 self-reciprocal, palindromic, Gorenstein polynomials over Z p : criptography, sequences, subfields in alg. closures over Q , C : cyclotomic polynomials are self-reciprocal irr. over Q / Z : char. poly. of auto. of certain unimodular latices type (2) ⇒ λ 2 = ( − 1) k k odd : λ 2 = − 1 and F = Z p , p ≡ 1 mod 4 k even : λ = ± 1, 2 / 11

Reflexible polynomials f ( x ) = a 0 + a 1 x + . . . + a k x k ∈ F [ x ] is reflexible if ∃ λ ∈ F ∗ ∀ i : type (1) λ a k − i = a i (1) ∃ λ ∈ F ∗ ∀ i : ( − 1) i a i type (2) λ a k − i = (2) type (1) ⇒ λ = ± 1 4 + 2 x + 3 x 2 + x 3 ∈ Z 5 [ x ] λ = − 1, type (1) λ = 1 self-reciprocal, palindromic, Gorenstein polynomials over Z p : criptography, sequences, subfields in alg. closures over Q , C : cyclotomic polynomials are self-reciprocal irr. over Q / Z : char. poly. of auto. of certain unimodular latices type (2) ⇒ λ 2 = ( − 1) k k odd : λ 2 = − 1 and F = Z p , p ≡ 1 mod 4 k even : λ = ± 1, 3 + 4 x + 2 x 2 + x 3 ∈ Z Z 5 [ x ], λ = 3, type (2) 2 / 11

Our motivation: symmetries of arc-transitive graphs 3 / 11

Our motivation: symmetries of arc-transitive graphs 4-val graphs with arc-transitive G ≤ Aut (Γ), not semi-simple 3 / 11

Our motivation: symmetries of arc-transitive graphs 4-val graphs with arc-transitive G ≤ Aut (Γ), not semi-simple first systematic approach by Gardiner and Praeger, 94 Praeger’s normal reduction Recursive factorization by N min ⊳ G 3 / 11

Our motivation: symmetries of arc-transitive graphs 4-val graphs with arc-transitive G ≤ Aut (Γ), not semi-simple first systematic approach by Gardiner and Praeger, 94 Praeger’s normal reduction Recursive factorization by N min ⊳ G Z r Classify Γ when Γ / Z p = K 1 , K 2 , C n 3 / 11

Our motivation: symmetries of arc-transitive graphs 4-val graphs with arc-transitive G ≤ Aut (Γ), not semi-simple first systematic approach by Gardiner and Praeger, 94 Praeger’s normal reduction Recursive factorization by N min ⊳ G Z r Classify Γ when Γ / Z p = K 1 , K 2 , C n Z r Completely solved, except for Γ / Z p = C n and p odd 3 / 11

p -coverings Γ → C (2) Z r Minimal Z n 4 / 11

p -coverings Γ → C (2) Z r Minimal Z n 4 / 11

p -coverings Γ → C (2) Z r Minimal Z n Classify minimal VT and ET elementary abelian covers of C (2) n 4 / 11

p -coverings Γ → C (2) Z r Minimal Z n Classify minimal VT and ET elementary abelian covers of C (2) n M, Maruˇ siˇ c, Potoˇ cnik, Elementary abelian covers, JACO, 2004. 4 / 11

p = C (2) Z r Z r Results: Γ / Z n , where Z p min ⊳ H : VT and ET 5 / 11

p = C (2) Z r Z r Results: Γ / Z n , where Z p min ⊳ H : VT and ET Thm 1. All minimal graphs Γ arise from cyclic or negacyclic codes. 5 / 11

p = C (2) Z r Z r Results: Γ / Z n , where Z p min ⊳ H : VT and ET Thm 1. All minimal graphs Γ arise from cyclic or negacyclic codes.  α 0 . . . α m 0 . . . · · · 0  . ... .   0 α 0 . . . α m .   .  ... ... ...  .   .   Z r × n   M g ( x ) = ∈ Z   p  .  ... ... ... .   .    .  ... .   . α 0 . . . α m 0   0 · · · · · · 0 α 0 . . . α m matrix associated with a proper divisor g ( x ) | x n ± 1, deg( g ( x )) = n − r 5 / 11

p = C (2) Z r Z r Results: Γ / Z n , where Z p min ⊳ H : VT and ET Thm 1. All minimal graphs Γ arise from cyclic or negacyclic codes.  α 0 . . . α m 0 . . . · · · 0  . ... .   0 α 0 . . . α m .   .  ... ... ...  .   .   Z r × n   M g ( x ) = ∈ Z   p  .  ... ... ... .   .    .  ... .   . α 0 . . . α m 0   0 · · · · · · 0 α 0 . . . α m matrix associated with a proper divisor g ( x ) | x n ± 1, deg( g ( x )) = n − r Z r Γ = Γ g ( x ) has vertex set Z p × Z Z n and ( v , j ) ∼ ( v ± u j +1 , j + 1) 5 / 11

Starting with a proper divisor g ( x ) | x n ± 1 6 / 11

Starting with a proper divisor g ( x ) | x n ± 1 Thm 2. Γ g ( x ) is at least VT and ET. 6 / 11

Starting with a proper divisor g ( x ) | x n ± 1 Thm 2. Γ g ( x ) is at least VT and ET. Lifted groups preserve the degree of symmetry (Djokovi´ c, 74) ⇒ Consider M – the largest group that lifts. When is M AT? 6 / 11

Starting with a proper divisor g ( x ) | x n ± 1 Thm 2. Γ g ( x ) is at least VT and ET. Lifted groups preserve the degree of symmetry (Djokovi´ c, 74) ⇒ Consider M – the largest group that lifts. When is M AT? g ( x ) = g d ( x d ) d maximal: g d ( x ) : reduced polynomial 6 / 11

Starting with a proper divisor g ( x ) | x n ± 1 Thm 2. Γ g ( x ) is at least VT and ET. Lifted groups preserve the degree of symmetry (Djokovi´ c, 74) ⇒ Consider M – the largest group that lifts. When is M AT? g ( x ) = g d ( x d ) d maximal: g d ( x ) : reduced polynomial p , and M acts on V ( C (2) Lemma. d | n and d | r = dim Z r n ) with kernel Z d 2 . 6 / 11

Starting with a proper divisor g ( x ) | x n ± 1 Thm 2. Γ g ( x ) is at least VT and ET. Lifted groups preserve the degree of symmetry (Djokovi´ c, 74) ⇒ Consider M – the largest group that lifts. When is M AT? g ( x ) = g d ( x d ) d maximal: g d ( x ) : reduced polynomial p , and M acts on V ( C (2) Lemma. d | n and d | r = dim Z r n ) with kernel Z d 2 . Thm 3. M , the largest group that lifts is AT ⇔ g d ( x ) is reflexible. 6 / 11