The Number of Irreducible Polynomials of Even Degree
over F2 with the First Four Coefficients Given
B. Omidi Koma
School of Mathematics and Statistics
Carleton University
[email protected]
July 22, 2010
B. Omidi Koma ()
Irreducible Polynomials
July 22, 2010
1 / 34
1
Introduction
Definitions and Background
Previous Results
2
Generalizing Möbius Inversion Formula
3
Computing F (n, t1 , t2 , t3 , t4 )
4
The Formula For N(n, t1 , t2 , t3 , t4 )
5
Cosets of F2l in F2n
6
An Approximation of N(n, t1 , t2 , t3 , t4 )
7
Some Outputs of Our Approximation
8
Future Directions
9
References
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
July
Given
22, 2010
Trace and 2Constan
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Definitions and Background
For β ∈ F2n the k th trace is denoted by Tk (β), and defined as
Tk (β) =
X
β i1 β i2 · · · β ik .
0≤i1 <i2 <···<ik ≤n−1
F (n, t1 , · · · , tr ) : the number of β ∈ F2n where Ti (β) = ti , 1 ≤ i ≤ r .
Let f (x) = x n + an−1 x n−1 + · · · + a1 x + a0 be a polynomial of degree
n over F2 .
an−i is the i th trace of f , written as Ti (f ) = an−i , for 1 ≤ i ≤ n.
P(n, t1 , · · · , tr ) : the set of irreducible polynomials of degree n over F2 ,
where an−i = ti ∈ F2 . Let N(n, t1 , · · · , tr ) = |P(n, t1 , · · · , tr )|.
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
July
Given
22, 2010
Trace and 3Constan
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Previous Results
Let N(n, q) be the number of monic irreducible polynomials of degree
n over Fq [x]. Then
qn =
X
dN(n, q).
d|n
By Möbius inversion formula we have
N(n, q) =
n
1X
1 X n d
µ
q =
µ(d)q d .
n
d
n
d|n
d|n
The number of irreducible polynomials with the first coefficient prescribed
is given by Carlitz (1952).
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
July
Given
22, 2010
Trace and 4Constan
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Let n be an even integer, and a ≡ b (mod 4) be shortened to a ≡ b.
Then Cattell, Miers, Ruskey, Serra, and Sawada, (2003) proved that
nN(n, 1, 0) =
nN(n, 1, 1) =
nN(n, 0, 0) =
nN(n, 0, 1) =
B. Omidi Koma ()
X
d|n
X
d≡1
d≡3
X
d|n
X
d≡1
d≡3
µ(d)F (n/d, 1, 0) +
µ(d)F (n/d, 1, 1),
d|n
µ(d)F (n/d, 1, 1) +
µ(d)F (n/d, 1, 0),
d|n
X
µ(d)F (n/d, 0, 0) −
X
µ(d)F (n/d, 0, 1) −
n
X
µ(d)2 2d −1 ,
X
µ(d)2 2d −1 .
d|n
d|n,n/deven
d odd
d odd
d|n
d|n,n/deven
d odd
d odd
n
The Number of Irreducible Polynomials of Degree n over Fq with
July
Given
22, 2010
Trace and 5Constan
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Also F (n, t1 , t2 ) = 2n−2 + G (n, t1 , t2 ), and for n = 2m we have
Table: Values of G (n, t1 , t2 )
m (mod 4)
0
1
2
3
(0, 0)
−2m−1
0
2m−1
0
(0, 1)
2m−1
0
−2m−1
0
(1, 0)
0
−2m−1
0
2m−1
(1, 1)
0
2m−1
0
−2m−1
The number of irreducible polynomials of even degree n over F2 with given
first three coefficients is considered by Yucas and Mullen (2004) . For odd
degree n Fitzgerald and Yucas (2003) consider the same problem.
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
July
Given
22, 2010
Trace and 6Constan
/ 34
Generalizing Möbius Inversion Formula
For our study we proved the following generalization of Möbius Inversion.
Theorem
Suppose a ≡ b (mod 8) be written as a ≡ b. Let A(n), B(n), C (n), D(n),
α(n), β(n), γ(n), and δ(n) be functions defined on N. Then
A(n) =
B(n) =
X “n” X “n” X “n” X “n”
β
γ
δ
+
+
+
,
α
d
d
d
d
d |n
d |n
d |n
d |n
d≡1
d≡3
d≡5
d≡7
X “n” X “n” X “n” X “n”
+
+
+
,
α
δ
γ
β
d
d
d
d
d |n
d |n
d |n
d |n
d≡1
d≡3
d≡5
d≡7
X “n” X “n” X “n” X “n”
C (n) =
γ
+
+
+
,
δ
α
β
d
d
d
d
d |n
d |n
d |n
d |n
d≡1
d≡3
d≡5
d≡7
X “n” X “n” X “n” X “n”
+
+
+
,
γ
β
α
D(n) =
δ
d
d
d
d
d |n
d |n
d |n
d |n
d≡1
d≡3
d≡5
d≡7
if and only if
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
July
Given
22, 2010
Trace and 7Constan
/ 34
Theorem
(Countinued)
α(n) =
X
µ(d)A
“n”
d
+
µ(d)B
“n”
d
+
µ(d)C
“n”
d
+
d |n
δ(n) =
X
µ(d)A
d
+
µ(d)D
d |n
d≡1
d
+
X
d |n
d≡3
“n”
d
+
X
µ(d)D
d
+
X
µ(d)D
d
+
µ(d)C
d
+
d |n
d≡5
d
,
X
µ(d)C
“n”
d
,
µ(d)B
“n”
d
,
µ(d)A
“n”
.
d≡7
µ(d)A
“n”
d
+
X
d |n
d≡5
X
“n”
d |n
d |n
“n”
µ(d)D
d≡7
“n”
d≡5
“n”
X
d |n
d |n
d≡3
“n”
µ(d)C
d≡5
“n”
d |n
d≡1
X
X
X
d |n
d≡3
d≡1
γ(n) =
d
+
d |n
d |n
X
“n”
d≡3
d≡1
β(n) =
µ(d)B
d |n
d |n
X
X
d≡7
µ(d)B
“n”
d
+
X
d
d |n
d≡7
In general, we proved that when modulo is 2j , where j > 3, we have 2j−1
functions in terms of some other 2j−1 functions. For our case, j = 3.
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
July
Given
22, 2010
Trace and 8Constan
/ 34
Computing F (n, t1, t2 , t3, t4)
Lemma
Let β ∈ F2n and f ∈ F2 [x] of degree n/d be the minimal polynomial of β.
Then the i th trace of β is the coefficient of x n−i in f d , or Ti (β) = Ti (f d ),
where 1 ≤ i ≤ n.
Lemma
Let d ≥ 1 be an integer, and f ∈ F2 [x]. Then
(i) T1 (f d ) = dT1 (f );
d
d
(ii) T2 (f ) =
T1 (f ) + dT2 (f );
2
d
(iii) T3 (f d ) =
T1 (f ) + dT3 (f );
3
d
d
d
(iv) T4 (f ) =
T1 (f ) +
T2 (f ) + dT4 (f ).
4
2
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
July
Given
22, 2010
Trace and 9Constan
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Lemma
Let d be a given integer such that d ≥ 1. For 0 ≤ j ≤ 7 we shorten d ≡ j
(mod 8) as d ≡ j. If f ∈ F2 [x] is a given polynomial, then for 0 ≤ j ≤ 7
the coefficients Ti (f d ) where i = 1, 2, 3, 4 are given in the following table.
Table: Different coefficients of f d
d ≡i
d ≡0
d ≡1
d ≡2
d ≡3
d ≡4
d ≡5
d ≡6
d ≡7
T1 (f d )
0
T1 (f )
0
T1 (f )
0
T1 (f )
0
T1 (f )
B. Omidi Koma ()
T2 (f d )
0
T2 (f )
T1 (f )
T1 (f ) + T2 (f )
0
T2 (f )
T1 (f )
T1 (f ) + T2 (f )
T3 (f d )
0
T3 (f )
0
T1 (f ) + T3 (f )
0
T3 (f )
0
T1 (f ) + T3 (f )
T4 (f d )
0
T4 (f )
T2 (f )
T2 (f ) + T4 (f )
T1 (f )
T1 (f ) + T4 (f )
T1 (f ) + T2 (f )
T1 (f ) + T2 (f ) + T4 (f )
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and10Constan
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Theorem
Let n be an even positive integer and ti ∈ F2 , for 1 ≤ i ≤ 4. Also let
a ≡ b (mod 8) be shortened as a ≡ b. Then the number of elements
β ∈ F2n with prescribed first four traces Ti (β) = ti can be given by
7 [
X
n
F (n, t1 , t2 , t3 , t4 ) =
. |Si | ,
d
i =0
d|n
d≡i
where the sets S0 , . . . , S7 are defined as:
S0 = {f ∈ P(n/d) : ti = 0, i = 1, 2, 3, 4},
S1 = {f ∈ P(n/d) : Ti (f ) = ti , i = 1, 2, 3, 4},
S2 = {f ∈ P(n/d) : t1 = t3 = 0, T1 (f ) = t2 , T2 (f ) = t4 },
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and11Constan
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Theorem
(Continued)
S3 = {f ∈ P(n/d) : T1 (f ) = t1 , T1 (f ) + Ti (f ) = ti , i = 2, 3,
T2 (f ) + T4 (f ) = t4 },
S4 = {f ∈ P(n/d) : ti = 0, i = 1, 2, 3, T1 (f ) = t4 },
S5 = {f ∈ P(n/d) : Ti (f ) = ti , i = 1, 2, 3, T1 (f ) + T4 (f ) = t4 },
S6 = {f ∈ P(n/d) : t1 = t3 = 0, T1 (f ) = t2 , T1 (f ) + T2 (f ) = t4 },
S7 = {f ∈ P(n/d) : T1 (f ) = t1 , T1 (f ) + Ti (f ) = ti , i = 2, 3,
T1 (f ) + T2 (f ) + T4 (f ) = t4 }.
We use the last theorem and our generalization of Möbius Inversion
formula to find the number N(n, t1 , t2 , t3 , t4 ).
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and12Constan
/ 34
The Formula For N(n, t1, t2 , t3, t4)
Theorem
For even degree n, different values of N(n, t1 , t2 , t3 , t4 ) are given as
(i) nN(n, 1, 1, 1, 0)
=
X
µ(d)F (n/d, 1, 1, 1, 0) +
d |n
+
d≡1
d≡3
X
X
µ(d)F (n/d, 1, 1, 1, 1) +
X
d≡7
µ(d)F (n/d, 1, 0, 0, 1) +
d |n
X
X
B. Omidi Koma ()
µ(d)F (n/d, 1, 1, 1, 1),
d≡7
µ(d)F (n/d, 1, 1, 1, 1) +
d |n
+
X
d |n
d≡5
=
µ(d)F (n/d, 1, 1, 1, 0)
d≡3
µ(d)F (n/d, 1, 0, 0, 0) +
d |n
nN(n, 1, 1, 1, 1)
X
d |n
d≡1
+
µ(d)F (n/d, 1, 0, 0, 1),
d |n
d≡5
=
µ(d)F (n/d, 1, 0, 0, 0)
d |n
d |n
nN(n, 1, 0, 0, 1)
X
X
µ(d)F (n/d, 1, 0, 0, 1)
d |n
d≡1
d≡3
X
X
µ(d)F (n/d, 1, 1, 1, 0) +
d |n
d |n
d≡5
d≡7
µ(d)F (n/d, 1, 0, 0, 0),
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and13Constan
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Theorem
(Continued)
nN(n, 1, 0, 0, 0)
=
X
µ(d)F (n/d, 1, 0, 0, 0) +
d |n
+
d≡1
d≡3
X
X
µ(d)F (n/d, 1, 0, 0, 1) +
µ(d)F (n/d, 1, 1, 1, 0),
d |n
d≡7
d≡5
=
µ(d)F (n/d, 1, 1, 1, 1)
d |n
d |n
(ii) nN(n, 0, 0, 1, 0)
X
X
µ(d)F (n/d, 0, 0, 1, 0) ,
d |n
d odd
(iii) nN(n, 0, 0, 1, 1)
=
X
µ(d)F (n/d, 0, 0, 1, 1) ,
d |n
d odd
(iv ) nN(n, 1, 1, 0, 0)
=
X
µ(d)F (n/d, 1, 1, 0, 0) +
d |n
B. Omidi Koma ()
X
µ(d)F (n/d, 1, 0, 1, 0)
d |n
d≡1
+
X
d≡3
µ(d)F (n/d, 1, 1, 0, 1) +
X
d |n
d |n
d≡5
d≡7
µ(d)F (n/d, 1, 0, 1, 1),
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and14Constan
/ 34
Theorem
(Continued)
nN(n, 1, 0, 1, 1)
=
X
µ(d)F (n/d, 1, 0, 1, 1) +
d |n
+
d≡1
d≡3
X
X
µ(d)F (n/d, 1, 0, 1, 0) +
X
d≡7
µ(d)F (n/d, 1, 1, 0, 1) +
d |n
X
X
B. Omidi Koma ()
X
µ(d)F (n/d, 1, 1, 0, 1)
d |n
d≡1
X
µ(d)F (n/d, 1, 0, 1, 0),
d≡7
µ(d)F (n/d, 1, 0, 1, 0) +
d |n
+
X
d |n
d≡5
=
µ(d)F (n/d, 1, 0, 1, 1)
d≡3
µ(d)F (n/d, 1, 1, 0, 0) +
d |n
nN(n, 1, 0, 1, 0)
X
d |n
d≡1
+
µ(d)F (n/d, 1, 1, 0, 1),
d |n
d≡5
=
µ(d)F (n/d, 1, 1, 0, 0)
d |n
d |n
nN(n, 1, 1, 0, 1)
X
d≡3
µ(d)F (n/d, 1, 0, 1, 1) +
X
d |n
d |n
d≡5
d≡7
µ(d)F (n/d, 1, 1, 0, 0).
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and15Constan
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Theorem
(Continued) In the following cases, a ≡ b represents a ≡ b (mod 4).
(v ) nN(n, 0, 1, 1, 1)
=
X
µ(d)F (n/d, 0, 1, 1, 1) +
=
d≡1
d≡3
X
X
µ(d)F (n/d, 0, 1, 1, 0) +
d |n
d≡1
(vi) nN(n, 0, 0, 0, 0)
=
X
d≡3
µ(d)F (n/d, 0, 0, 0, 0) −
=
d |n
d odd
B. Omidi Koma ()
X
µ(d)F (n/2d, 0, 0),
d |n, n even
d
d odd
X
µ(d)F (n/d, 0, 1, 1, 1),
d |n
d |n
(vii) nN(n, 0, 0, 0, 1)
µ(d)F (n/d, 0, 1, 1, 0),
d |n
d |n
nN(n, 0, 1, 1, 0)
X
d odd
µ(d)F (n/d, 0, 0, 0, 1) −
X
µ(d)F (n/2d, 0, 0),
d |n, n even
d
d odd
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and16Constan
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Theorem
(Continued) and finally,
(viii) nN(n, 0, 1, 0, 0)
=
X
µ(d)F (n/d, 0, 1, 0, 0) −
d≡1
+
d≡1
µ(d)F (n/d, 0, 1, 0, 1) −
d≡3
nN(n, 0, 1, 0, 1)
=
µ(d)F (n/d, 0, 1, 0, 1) −
d |n
d≡3
B. Omidi Koma ()
X
µ(d)F (n/2d, 1, 1)
d |n, n even
d
d≡1
+
µ(d)F (n/2d, 1, 1),
d≡3
d |n
X
X
d |n, n even
d
d |n
X
µ(d)F (n/2d, 1, 0)
d |n, n even
d
d |n
X
X
d≡1
µ(d)F (n/d, 0, 1, 0, 0) −
X
µ(d)F (n/2d, 1, 0).
d |n, n even
d
d≡3
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and17Constan
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Theorem
(Continued) and finally,
(viii) nN(n, 0, 1, 0, 0)
=
X
µ(d)F (n/d, 0, 1, 0, 0) −
d≡1
+
d≡1
µ(d)F (n/d, 0, 1, 0, 1) −
d≡3
nN(n, 0, 1, 0, 1)
=
µ(d)F (n/d, 0, 1, 0, 1) −
d |n
d≡3
B. Omidi Koma ()
X
µ(d)F (n/2d, 1, 1)
d |n, n even
d
d≡1
+
µ(d)F (n/2d, 1, 1),
d≡3
d |n
X
X
d |n, n even
d
d |n
X
µ(d)F (n/2d, 1, 0)
d |n, n even
d
d |n
X
X
d≡1
µ(d)F (n/d, 0, 1, 0, 0) −
X
µ(d)F (n/2d, 1, 0).
d |n, n even
d
d≡3
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and18Constan
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Cosets of F2l in F2n
Let n = 4l . For a given α ∈ F2l the linear functional Lα : F2l → F2 is
defined as Lα (β) = T2l (αβ), for all β ∈ F2l . We define
W0 = Ker(T2l ) = {β ∈ F2l | T2l (β) = 0},
and Wi (γ) = {β ∈ F2l | Ti (β + γ) = Ti (β) + Ti (γ)}, for i = 1, . . . , 4.
For γ ∈ F2n there exist three possibilities for Wi (γ), where i = 2, 3, 4.
Either Wi (γ) = F2l , or Wi (γ) = W0 , or Wi (γ) is a hyperplane different
that W0 . In total there exist 27 different cases. We proved that some of
these cases are possible. We divide them into three groups, based on W2 .
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and19Constan
/ 34
Table: Group one, W2 (γ) = F2l
PP
PP W3
F 2l
W0
W3 6= W0 , F2l
PP
W4
P
P
F 2l
Cat A N/A
N/A
W0
Cat A1 N/A
N/A
W4 6= W0 , F2l
Cat A2 N/A
N/A
Table: Group two, W2 (γ) = W0
PP
PP W3
F 2l
W0
W3 6= W0 , F2l
PP
W4
P
P
F 2l
N/A
N/A
N/A
W0
Cat B1 Cat C1
N/A
W4 6= W0 , F2l
Cat B2 Cat C2
N/A
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and20Constan
/ 34
Table: Group three, W2 (γ) 6= W0 , F2l
PP
PP W3
F 2l
PP
W4
P
P
F 2l
N/A
W0
Cat F1
W4 6= W0 , F2l
Cat F2
W0
W3 6= W0 , F2l
N/A
Cat D1
Cat D2
N/A
Cat E1
Cat E2
For each category we have some conditions on γ ∈ F2n . For example, a
l
2l
3l
coset γ + F2l is in D1 iff γ̂ = γ + γ 2 + γ 2 + γ 2 6= 1 and either
(i ) T2l (γ̂) = 0 and γ̂ 4 = γ̂ + 1, or
(ii ) T2l (γ̂) = 1 and γ̂ 4 = γ̂ 2 + γ̂ + 1.
B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and21Constan
/ 34
For γ ∈ F2n the coset γ + F2l is in one of the 13 categories. Then we
can compute different traces of an element γ + β from the coset γ + F2l ,
where β ∈ F2l . For example if the coset γ + F2l is in category D1 , the
traces of an element γ + β are given in the following table.
Table: Traces in Category D1
```
``` Ti (β + γ)
T1 (β + γ) T2 (β + γ) T3 (β + γ)
```
β ∈ F2l
``
β ∈ W2 ∩ W0
β ∈ W2 \ (W2 ∩ W0 )
β ∈ W0 \ (W2 ∩ W0 )
β ∈ F2l \ (W2 ∪ W0 )
B. Omidi Koma ()
T1 (γ)
T1 (γ)
T1 (γ)
T1 (γ)
T2 (γ)
T2 (γ)
T2 (γ) + 1
T2 (γ) + 1
T3 (γ)
T3 (γ) + 1
T3 (γ)
T3 (γ) + 1
T4 (β + γ)
T4 (γ)
T4 (γ)
T4 (γ)
T4 (γ)
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
22, 2010
Trace and22Constan
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In different categories Ti (γ + β), 1 ≤ i ≤ 4, is in terms of Ti (γ), which is
either zero or one. To complete the study, we need to know in each
category for how many of the elements γ we have Ti (γ) = 0, and
Ti (γ) = 1. We expect that Ti (γ) depends on m, or l , where n = 2m = 4l .
Once the conditions are obtained one is able to compute the the exact
value of N(n, t1 , t2 , t3 , t4 ), where n = 4l . Similar results to these studies
are then needed for the case n = 4l + 2.
B. Omidi Koma ()
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An Approximation of N(n, t1, t2, t3, t4 )
We have 16 cases for (t1 , t2 , t3 , t4 ). In our main theorem these 16 cases
are divided to 8 groups, and each group has some of the cases of
(t1 , t2 , t3 , t4 ) that are connected to each other. Then for different groups
the formulas for the number N(n, t1 , t2 , t3 , t4 ) are given in terms of
′
′
′
′
′
′
′
′
F (n/d, t1 , t2 , t3 , t4 )’s where d | n, and (t1 , t2 , t3 , t4 ) is from the same
group as (t1 , t2 , t3 , t4 ). Assume that
′
′
′
′
F (n/d, t1 , t2 , t3 , t4 ) =
B. Omidi Koma ()
n
2 d −4 if dn ≥ 4 ,
0
otherwise.
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We let n = 2k0 p1 k1 · · · ps ks , where k0 , . . . , ks ≥ 1, and p1 , . . . , ps are odd
prime divisors of n. Assume that S1 = {p1 , . . . , ps }, and for 2 ≤ q ≤ s let
Sq = {d : d | n, d is the product of exactly q distinct odd primes pi ∈ S1 }
For the first 12 cases of (t1 , t2 , t3 , t4 ) which are in groups (i ) to (v ), the
number N(n, t1 , t2 , t3 , t4 ) can be given as
N(n, t1 , t2 , t3 , t4 ) =
B. Omidi Koma ()

1  n−4
2
+
n
s X
X
q=1 d∈Sq

[n/d ≥ 4](−1)q 2n/d −4  .
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For (t1 , t2 , t3 , t4 ) from groups (vi ) we have (t1 , t2 , t3 , t4 ) = (0, 0, 0, 0) and


s
1  n−4 X X
N(n, 0, 0, 0, 0) =
[n/d ≥ 4](−1)q 2n/d −4 
2
+
n
q=1 d∈Sq


s X
X
1
(−1)q F (n/2d, 0, 0) .
(F (n/d, 0, 0) +
−
n
q=1 d∈Sq
In group (vii ) we have (t1 , t2 , t3 , t4 ) = (0, 0, 0, 1) and


s X
X
1  n−4
N(n, 0, 0, 0, 1) =
2
+
[n/d ≥ 4](−1)q 2n/d −4 
n
q=1 d∈Sq


s X
X
1
(−1)q F (n/2d, 0, 1) .
(F (n/d, 0, 1) +
−
n
q=1 d∈Sq
B. Omidi Koma ()
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Finally in group (viii ), if (t1 , t2 , t3 , t4 ) = (0, 1, 0, 0) then the estimate for
N(n, t1 , t2 , t3 , t4 ) can be given as


s
1  n−4 X X
(−1)q [n/d ≥ 4]2n/d −4 − F (n/2, 1, 0)
2
+
n
q=1 d∈Sq


s X
X
1
(−1)q ([d ≡ 1]F (n/2d , 1, 0) + [d ≡ 3]F (n/2d , 1, 1)) ,
−
n
q=1 d∈Sq
and if (t1 , t2 , t3 , t4 ) = (0, 1, 0, 1) the estimate for N(n, t1 , t2 , t3 , t4 ) is


s X
X
1  n−4
(−1)q [n/d ≥ 4]2n/d −4 − F (n/2, 1, 1)
2
+
n
q=1 d∈Sq


s X
X
1
(−1)q ([d ≡ 1]F (n/2d , 1, 1) + [d ≡ 3]F (n/2d , 1, 0)) .
−
n
q=1 d∈Sq
B. Omidi Koma ()
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Table: Different values of N(16, t1 , t2 , t3 , t4 ), where ti = 0, 1.
Case No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Total
B. Omidi Koma ()
(t1 , t2 , t3 , t4 )
(1, 1, 1, 0)
(1, 0, 0, 1)
(1, 1, 1, 1)
(1, 0, 0, 0)
(0, 0, 1, 0)
(0, 0, 1, 1)
(1, 1, 0, 0)
(1, 0, 1, 1)
(1, 1, 0, 1)
(1, 0, 1, 0)
(0, 1, 1, 1)
(0, 1, 1, 0)
(0, 0, 0, 0)
(0, 0, 0, 1)
(0, 1, 0, 0)
(0, 1, 0, 1)
our estimate
256
256
256
256
256
256
256
256
256
256
256
256
252.5
251.5
252
252
4080
N(16, t1 , t2 , t3 , t4 )
260
260
252
252
256
256
256
256
256
256
264
264
240
256
248
248
4080
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Table: Different values of N(18, t1 , t2 , t3 , t4 ), where ti = 0, 1.
Case No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Total
B. Omidi Koma ()
(t1 , t2 , t3 , t4 )
(1, 1, 1, 0)
(1, 0, 0, 1)
(1, 1, 1, 1)
(1, 0, 0, 0)
(0, 0, 1, 0)
(0, 0, 1, 1)
(1, 1, 0, 0)
(1, 0, 1, 1)
(1, 1, 0, 1)
(1, 0, 1, 0)
(0, 1, 1, 1)
(0, 1, 1, 0)
(0, 0, 0, 0)
(0, 0, 0, 1)
(0, 1, 0, 0)
(0, 1, 0, 1)
our estimate
910
910
910
910
910
910
910
910
910
910
910
910
906.89
903.5
906.44
906.44
14543.27
N(18, t1 , t2 , t3 , t4 )
917
896
917
896
921
892
927
917
893
917
921
892
913
900
913
900
14532
The Number of Irreducible Polynomials of Degree n over Fq with
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Future Directions
(1) A natural way to extend this problem is completing the counting
argument to find N(n, t1 , t2 , t3 , t4 ).
(2) Studying the number N(n, t1 , t2 , t3 , t4 ), where n is an odd number.
(3) Change the finite field to a general field Fq
(4) Studying the number N(n, t1 , . . . , tk ), where 4 ≤ k ≤ n.
B. Omidi Koma ()
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Thank You
B. Omidi Koma ()
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B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
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B. Omidi Koma ()
The Number of Irreducible Polynomials of Degree n over Fq with
JulyGiven
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