Matching Tag: irreducible-polynomials
Let S = { x 2 + c 1 , … , x 2 + c r } S=\{x^2+c_1,\dots,x^2+c_r\} S = { x 2 + c 1 , … , x 2 + c r } c 1 , … , c r ∈ Q c_1,\dots,c_r\in\mathbb{Q} c 1 , … , c r ∈ Q r ≥ 2 r\geq2 r ≥ 2 M S M_S M S S S S
Let α , β ∈ F q \alpha,\beta\in\mathbb{F}_q α , β ∈ F q λ ∈ F ‾ p \lambda\in\overline{\mathbb F}_p λ ∈ F p f λ ( z ) = z d + λ f_\lambda(z)=z^d+\lambda f λ ( z ) = z d + λ
Let f c ( z ) = z 2 + c f_c(z)=z^2+c f c ( z ) = z 2 + c m , n m,n m , n m ∣ n m\mid n m ∣ n Δ n , m \Delta_{n,m} Δ n , m m = n m=n m = n Φ k c y c \Phi^{\mathrm{cyc}}_k Φ k cyc k k k
Let f ( x ) ∈ Z [ x ] f(x)\in\mathbb{Z}[x] f ( x ) ∈ Z [ x ] n ≥ 1 n\geq 1 n ≥ 1 L ( f − g ) = ∑ j = 0 n ∣ b j − a j ∣ L(f-g)=\sum_{j=0}^{n}|b_j-a_j| L ( f − g ) = ∑ j = 0 n ∣ b j − a j ∣ f ( x ) = ∑ j = 0 n a j x j f(x)=\sum_{j=0}^{n}a_jx^j f ( x ) = ∑ j = 0 n a j x j g ( x ) = ∑ j = 0 n b j x j g(x)=\sum_{j=0}^{n}b_jx^j g ( x ) = ∑ j = 0 n b j x j
Let f f f F q \mathbb F_q F q n ≥ 3 n\ge 3 n ≥ 3 Q c Q_c Q c D D D n n n D ≡ 2 ( m o d 4 ) D\equiv 2\pmod 4 D ≡ 2 ( mod 4 )
Throughout, let ( f , g ) (f,g) ( f , g ) F q \mathbb{F}_q F q p p p f f f deg g ≥ 2 \deg g\geq 2 deg g ≥ 2 N f , g ( n ) N_{f,g}(n) N f , g ( n )
Throughout, let ( f , g ) (f,g) ( f , g ) F q \mathbb{F}_q F q p p p f f f deg g g r e a t e r t h a n o r e q u a l t o 2 \deg g greater than or equal to 2 deg g g r e a t er t han or e q u a l t o 2
Let q q q n ≥ 1 n\geq 1 n ≥ 1 q q q n n n T T T ( g , h ) ∈ F q [ X ] 2 (g,h)\in\mathbb{F}_q[X]^2 ( g , h ) ∈ F q [ X ] 2
Prescribed-coefficients enumeration conjecture. There exist ω 1 , … , ω N ∈ Z ‾ \omega_1,\ldots,\omega_N\in\overline{\mathbb{Z}} ω 1 , … , ω N ∈ Z q \sqrt q q υ 1 , … , υ N ∈ Z \upsilon_1,\ldots,\upsilon_N\in\mathbb{Z} υ 1 , … , υ N ∈ Z
Let q q q k , l k,l k , l c F k ( q ) c\mathcal{F}_k(q) c F k ( q ) k k k c F q c\mathbb{F}_q c F q
A monic cubic with depth-one emergent reducibility is a monic polynomial f ( x ) ∈ Z [ x ] f(x)\in\mathbb{Z}[x] f ( x ) ∈ Z [ x ] 3 3 3 f ∘ f f\circ f f ∘ f
Let F q \mathbb{F}_q F q F F F S ⊂ F q [ x ] \mathcal{S}\subset\mathbb{F}_q[x] S ⊂ F q [ x ] P ∈ S P\in\mathcal{S} P ∈ S
