the hansen-mullen conjectureusers.math.uoc.gr/~gkapet/files/sabanci_talk_1.pdf · 2016. 11. 27. ·...

The Hansen-Mullen Conjecture

Giorgos Kapetanakis

Sabanci University

October 11, 2016

Giorgos Kapetanakis (Sabanci) The Hansen-Mullen Conjecture October 11, 2016 1 / 45

1 Introduction

2 Wan’s proofPreliminariesWeighted sumsCompletion of the proof

3 The HMC for self-reciprocal irreducible polynomialsMotivationPreliminariesWeighted sumCompletion of the proofThe restriction k ≤ n/2Further research

4 Other extensions of the HMC

5 Different approaches


Introduction

Let q be a prime power. We denote by Fq the finite field of qelements.

A generator of the multiplicative group F∗q is called primitive.

A polynomial of degree n of Fq[X] is called primitive if one (henceall) of its roots is a primitive element of Fqn .

Primitive polynomials are irreducible, but not vice versa.


Introduction

The properties of irreducible polynomials over Fq have proved to be ofgreat theoretical and practical interest. Such properties, that have beeninvestigated, include

primitivity,

normality,

having certain coefficients fixed to given values and

combinations of those properties with other interesting ones.

One such result is the following.

Theorem (Cohen (1990))

Let n ≥ 2 and t ∈ Fq with t 6= 0 if n = 2 or if n = 3 and q = 4. Thenthere exists a primitive polynomial of degree n over Fq with trace t.


Introduction

Based on the above and on computational evidence, Hansen and Mullen(1992) stated two important conjectures.

Conjecture (Hansen-Mullen (1992))

Let a ∈ Fq, let n ≥ 2 and 0 ≤ j < n. Then there exists an irreduciblepolynomial f(X) = Xn +

∑n−1k=0 akX

k ∈ Fq[X] with aj = a except when

j = a = 0 or

q = 2m, n = 2, j = 1, and a = 0.


Introduction

Conjecture (Hansen-Mullen (1992))

Let a ∈ Fq, let n ≥ 2 and 0 ≤ j < n. Then there exists a primitivepolynomial f(X) = Xn +

∑n−1k=0 akX

k ∈ Fq[X] with aj = a except when

j = 0 and a 6= (−1)nα, where α ∈ Fq is a primitive element;

n = 2, j = 1, and a = 0;

q = 4, n = 3, j = 2, and a = 0;

q = 4, n = 3, j = 1, and a = 0;

q = 2, n = 4, j = 2, and a = 1.

The exceptions are genuine.

Both conjectures are established.


Introduction

Cohen (1990) proved the HMC (Hansen-Mullen Conjecture) forj = n− 1.

In order to further support their claim, Hansen and Mullen (1992)proved their conjecture for j = 0.

Wan (1997) proved the HMC for irreducible polynomials if n ≥ 36 orq > 19 with the help of character sums.

Soon, Ham and Mullen (1998) completed the proof, by providingexamples for the remaining cases with the help of computers.


Introduction

Cohen (1990) proved the HMC for j = n− 1.

In order to further support their claim, Hansen and Mullen (1992)proved their conjecture for j = 0.

The work of Cohen (2000) and Cohen and Huczynska (2003) provethe HMC for j = 1.

Han (1996) and Cohen and Mills (2003) proved the HMC forj = n− 2 and n ≥ 5.

The work of Fan and Han (2004), Mills (2004) and Cohen and King(2004), established the HMC for j = n− 3 and n ≥ 7.


Introduction

Fan and Han (2003) used p-adic methods to asymptotically prove theHMC for primitive polynomials for fixed n and q large enough, exceptwhen n/2 ≤ j ≤ n/2 + 1.

Fan and Han (2004) proved the HMC for primitive polynomials foreven q and odd n ≥ 7, by using character sums, p-adic methods,sieving techniques and computer searches.

Cohen (2006) proved the HMC for primitive polynomials for n ≥ 9, byusing character sums, p-adic methods, sieving techniques andcomputer searches.

Two years later, Cohen and Presern (2008) completed the proof ofthe Hansen-Mullen conjecture for primitive polynomials.


Introduction

The main idea behind the techniques of most authors dates back to thework of Carlitz (1952) and still remains popular. Roughly, this method is:

1 Express the characteristic or a characteristic-like function for apolynomial with the desired properties with help of characters.

2 With the the help of character sum estimates or calculations, thisleads to a lower bound of the number of polynomials with the desiredproperties.

3 This implies sufficient asymptotic conditions for the existence of thedesired polynomial.

4 These conditions could be relaxed (i.e. with sieving) and/or give riseto effective results.

5 If desirable, one can we deal with the remaining cases with acase-by-case approach.

Nonetheless, there are also other approaches of the HMC.


Wan’s proof

Now, we will present Wan’s proof for the HMC for irreducible polynomials.

D. Wan.Generators and irreducible polynomials over finite fields.Math. Comp., 66(219):1195–1212, 1997.


Wan’s proof Preliminaries

Definition

Let G be a finite abelian group. A character of G is a grouphomomorphism G→ C∗, where C∗ stands for the multiplicative group ofC. The characters of G form a group under multiplication, which isisomorphic to G. This group is called the dual of G and denoted by G.Furthermore, the character χ0 : G→ C∗, where χ0(g) = 1 for all g ∈ G, iscalled the trivial character of G. Finally, by χ we denote the inverse of χ.

A character or exponential sum is a sum that involves characters.



The simplest, albeit very important, form of character sum is the following.

Lemma (Orthogonality relations)

Let χ be a non-trivial character of a group G and g a non-trivial elementof G. Then ∑

x∈Gχ(x) = 0 and

∑χ∈G

χ(g) = 0.

The orthogonality relations are true for arbitrary group G.



Definition

Given some F ∈ Fq[X], a Dirichlet character modulo F is a functionχ : Fq[X]→ C∗, such that

1 χ(G+ FH) = χ(G),

2 χ(GH) = χ(G)χ(H) and

3 χ(G) 6= 0 ⇐⇒ (G,F ) = 1,

for every G,H ∈ Fq[X].

Dirichlet characters were originally defined over Z,

Dirichlet characters modulo F are essentially the characters of(Fq[X]/F Fq[X])∗, extended to zero.



For χ a Dirichlet character modulo M ∈ Fq[X] and n ∈ N set

cn(χ) :=∑d|n

n

d

∑P monic irreducible

deg(P )=n/d

χ(P )d.

In particular the logarithmic derivative of the Dirichlet L-function of χ(multiplied by u) is

∑∞n=0 cn(χ)un. It follows from the Riemann

Hypothesis from function fields, proven by Weil (1948), that:

Theorem (Weil)

Let χ be a Dirichlet character modulo M . Then

1 If χ 6= χ0 then|cn(χ)| ≤ (deg(M)− 1)q

n2 .

2 If χ 6= χ0 and χ(F∗q) = 1, then

|1 + cn(χ)| ≤ (deg(M)− 2)qn2 .



For H ∈ Fq[X], we define the von Mangoldt function as

Λ(H) =

deg(P ), if H is a power of the irreducible P ,

1, if H = 1,

0, otherwise.

It follows directly from the definition of Λ, that

cn(χ) =∑

H monicdeg(H)=n

Λ(H)χ(H).



By using Weil’s theorem and this characterization for cn(χ), one can seethat the following character sum estimates hold.

Corollary

Let χ be a Dirichlet character modulo M . Then

1 If χ 6= χ0 then∣∣∣∣∣∣∣∣∑

P monic irreducibledeg(P )=d

χ(P )

∣∣∣∣∣∣∣∣ ≤1

d(deg(M) + 1)qd/2.

2 If χ 6= χ0 and χ(F∗q) = 1, then∣∣∣∣∣∣∣∣1

d+

∑P monic irreducible

deg(P )=d

χ(P )

∣∣∣∣∣∣∣∣ ≤1

ddeg(M)qd/2.



Similarly, we can estimate character sums that run through irreduciblepolynomials with constant term 1 (not necessarily monic).

Corollary

Let χ be a non-trivial Dirichlet character modulo M , such that χ(F∗q) = 1,then ∣∣∣∣∣∣∣∣

∑P irreducible with constant term 1

deg(P )=d

χ(P )

∣∣∣∣∣∣∣∣ ≤1

ddeg(M)qd/2.


Wan’s proof Weighted sums

From now on, fix n, 1 ≤ j < n and a ∈ Fq and define the followingweighted sum

wa(n, j) :=∑

H monic of degree j

Λ(aH)∑

P≡aH (mod Xj+1)P monic irreducible of degree n

1.

If wa(n, j) 6= 0, then there exists a monic irreducible of degree n with itsj-th coefficient prescribed to a.



From the orthogonality relations, we have that

wa(n, j) =∑H

Λ(aH)∑P

1

qj(q − 1)

∑χ

χ(P )χ(aH)

=1

qj(q − 1)

∑χ

(∑P

χ(P )

)(∑H

Λ(aH)χ(aH)

),

where we note that for a 6= 0,∣∣∣∣∣∑H

Λ(aH)χ(aH)

∣∣∣∣∣ =

∣∣∣∣∣∑H

Λ(H)χ(H)

∣∣∣∣∣ ,while the case a = 0 requires special, but similar, treatment.



We then separate the term of χ0 and apply the previous results and weeventually get.

Proposition

We have ∣∣∣∣wa(n, j)− πnq − 1

∣∣∣∣ < 1

n(j2 + 2j)q(n+j)/2.

The above, combined with known lower bounds for πn, implies.

Corollary

Let a ∈ Fq, n ≥ 3 and 1 ≤ j < n. If

qn−j ≥ (q − 1)2(j + 1)4,

then there exists an irreducible P ∈ Fq[X], with deg(P ) = n and Pj = a,where Pj is the coefficient of Xj .



The latter is effective for j ≤ n/2. For larger values of j, we consider thereciprocal of a monic irreducible, i.e. an irreducible with constant termequal to 1. Let Gj−1 be the set of polynomials of Fq[X] of degree j − 1with constant term equal to 1 and fix a ∈ Fq, n and 1 ≤ j < n and define

Wa(n, j) :=∑

H∈Gj−1

Λ(H)∑

P irreducible of degree nand constant term equal to 1P≡H+aXj (mod Xj+1)

1.

If Wa(n, j) 6= 0, then there exists an irreducible P ∈ Fq[X] of degree nwith constant term equal to 1 and Pj = a. This means that its reciprocal,i.e. PR := XnP (1/X), is a monic irreducible of degree n with PRn−j = a.



With a similar treatment as with wa(n, j), we show that

Proposition

If j > 1, then ∣∣∣∣Wa(n, j)−πnq

∣∣∣∣ < 1

nj2q(n+j−1)/2.

Again, by using known bounds for πn, the above implies the following.

Corollary

Let a ∈ Fq and 1 ≤ j < n. If

qn−j−1 ≥ (j + 1)4,

then there exists an irreducible P ∈ Fq[X] with constant term equal to 1and its j-th coefficient fixed to a.


Wan’s proof Completion of the proof

The latter is effective for j ≤ n/2, but since this implies the existence of amonic irreducible polynomial of degree n with the coefficient of Xn−j

prescribed, it is actually effective for j ≥ n/2.By putting everything together, we end up with the following

Corollary

Let a ∈ Fq, n ≥ 3 and 1 ≤ j < n. If either

qn−j ≥ (q − 1)2(j + 1)4 or qj−1 ≥ (n− j + 1)4,

then there exists a monic irreducible P ∈ Fq[X] of degree n with thecoefficient of Xj equal to a.

A few computations verify that at least one of the requirements of theabove is true for any value of j if q > 19 or m ≥ 36, thus.

Theorem (Wan)

If q > 19 or m ≥ 36, then the HMC for irreducible polynomials is true.


The HMC for self-reciprocal irreducible polynomials

Next, we will present the work of Garefalakis and K. on the HMC forself-reciprocal irreducible polynomials.

T. Garefalakis and G. Kapetanakis.On the Hansen-Mullen conjecture for self-reciprocal irreduciblepolynomials.Finite Fields Appl., 18(4):832–841, 2012.

T. Garefalakis and G. Kapetanakis.A note on the Hansen-Mullen conjecture for self-reciprocal irreduciblepolynomials.Finite Fields Appl., 35(C):61–63, 2015.


The HMC for self-reciprocal irreducible polynomials Motivation

Given a polynomial Q ∈ Fq[X], its reciprocal QR is defined as

QR(X) = Xdeg(Q)Q(1/X).

One class of polynomials that has been intensively investigated is that ofself-reciprocal irreducible polynomials, that is, irreducible polynomials thatsatisfy QR(X) = Q(X). Besides their theoretical interest, self-reciprocalirreducible polynomials have been useful in applications, and in particularin the construction of error-correcting codes.


The HMC for self-reciprocal irreducible polynomials Motivation

It is natural to expect that self-reciprocal monic irreduciblepolynomials over finite fields, with some coefficient fixed, exist. Here,we restrict ourselves to the case where q is odd and prove that thereexists a self-reciprocal irreducible monic polynomial over Fq, of degree2n with its k-th coefficient prescribed, provided that

qn−k−1

2 ≥ 16

5k(k + 5) +

1

2.

With this result in mind, we show that we can prescribe the k-thcoefficient of a self-reciprocal irreducible polynomial of degree 2n,provided that k ≤ bn/2c, with a small number of genuine exceptions.

Our proof is based on an estimate of a weighted sum, similar to theone that Wan considers, character sums and Carlitz’s (1967)characterization of self-reciprocal irreducible monic polynomials.


The HMC for self-reciprocal irreducible polynomials Preliminaries

Carlitz (1967) characterized self-reciprocal irreducible monicpolynomials over Fq (srimp): Q is a srimp iff

Q(X) = XnP (X +X−1)

for some monic irreducible P of degree n, such that ψ(P ) = −1,where ψ(P ) = (P |X2 − 4), the Jacobi symbol of P modulo X2 − 4.

We compute

Q(X) =

n∑i=0

i∑j=0

(i

j

)PiX

n−i+2j ,

which imples

Qk =∑

0≤j≤kk−j∈2Z

(n− jk−j2

)Pn−j .



In the last expression, Qk is written in terms of the large degreecoefficients of P . In order to express in terms of the low degreecoefficients of a polynomial, we define.

P = XnP (4X−1)

and prove that Pi = 4n−iPn−i and ψ(P ) = −εψ(P ), where ε = ±1,depending on q and n.

It follows that

Qk =∑

0≤j≤kk−j∈2Z

(n− jk−j2

)4jPj =

k∑j=0

δjPj ,

where

δj =

{(n−jk−j2

)4j , if k − j ≡ 0 (mod 2),

0, if k − j ≡ 1 (mod 2).



We define τn,k : Gk → Fq, H 7→∑k

j=0 δjHj , whereGk := {H ∈ Fq[X] : deg(H) ≤ k and H0 = 1}. We have proved:

Proposition

Let n ≥ 2, 1 ≤ k ≤ n, and a ∈ Fq. Suppose there exists an irreducibleP ∈ Fq[X] with P0 = 1, such that ψ(P ) = ε and P ≡ H (mod Xk+1) forsome H ∈ Gk with τn,k(H) = a. Then there exists a srimp Q, of degree2n, with Qk = a.

Next, we need to correlate the inverse image of τn,k with Gk−1. In thisdirection, we prove.

Proposition

Let a ∈ Fq, n ≥ 2 and 1 ≤ k ≤ n. Let F =∑k

i=0 FiXi ∈ Fq[X], with

F0 = 1 and Fi = δk−iδ−1k , 1 ≤ i ≤ k − 1, and Fk = δ−1k (δ0 − a). Then

the map τ−1n,k(a)→ Gk−1 : H 7→ HF (mod Xk+1) is a bijection.


The HMC for self-reciprocal irreducible polynomials Weighted sum

Let n ≥ 2, 1 ≤ k ≤ n and a ∈ Fq. Inspired by Wan’s work (1997) weintroduce the following weighted sum.

wa(n, k) =∑

H∈τ−1n,k(a)

Λ(FH)∑

ψ(P )=ε, P≡H (mod Xk+1)

1.

It is clear that if wa(n, k) > 0, then there exists some P such that P ≡ H(mod Xk+1) for some H ∈ Gk, with τn,k(H) = a and ψ(P ) = ε. Thenthere exists a srimp Q, of degree 2n with Qk = a.



Let U be the subgroup of (Fq[X]/Xk+1Fq[X])∗ that contains classes ofpolynomials with constant term equal to 1. The set Gk−1 is a set ofrepresentatives of U . Further, the group of characters of U consistsexactly of those characters of (Fq[X]/Xk+1Fq[X])∗ that are trivial on Fq,that is, U = {χ ∈ (Fq[X]/Xk+1Fq[X])∗ : χ(F∗q) = 1}. Using theseobservations and the orthogonality relations, we get that

wa(n, k) =1

qk

∑χ∈U

∑P∈Jnψ(P )=ε

χ(P )∑

H∈τ−1n,k(a)

Λ(FH)χ(H)

=1

qk

∑χ∈U

∑P∈Jnψ(P )=ε

χ(P )χ(G)∑

H∈Gk−1

Λ(H)χ(H),

where G is the inverse of F modulo Xk+1 and Jn stands for the set ofirreducible polynomials of degree n and constant term equal to 1.



We separate the term that corresponds to χ0 and we get∣∣∣∣∣∣wa(n, k)− πq(n, ε)

qk

∑H∈Gk−1

Λ(H)

∣∣∣∣∣∣ ≤1

qk

∑χ 6=χo

∣∣∣∣∣∣∑

P∈Jn, ψ(P )=ε

χ(P )

∣∣∣∣∣∣∣∣∣∣∣∣∑

H∈Gk−1

Λ(H)χ(H)

∣∣∣∣∣∣ ,where πq(n, ε) = |{P irreducible of degree n : ψ(P ) = ε}|.


The HMC for self-reciprocal irreducible polynomials Completion of the proof

We will also use the following result.

Theorem (Garefalakis (2011))

Let ψ(P ) = (P |X2 − 4) be the Jacobi symbol of P modulo X2 − 4 and χbe a non-trivial Dirichlet character modulo Xk+1, where k ≥ 1. Thefollowing bounds hold:

1 For every n ∈ N, n ≥ 2,∣∣∣∣∣∣∑

P monic irreducible, ψ(P )=−1

χ(P )

∣∣∣∣∣∣ ≤ k+5n q

n2 .

2 For every n ∈ N, n ≥ 2, n odd,∣∣∣∣∣∣∑

P monic irreducible, ψ(P )=1

χ(P )

∣∣∣∣∣∣ ≤ k+5n q

n2 .



The latter, along with other results we have already seen yield:

∑H∈Gk−1

Λ(H) =qk − 1

q − 1,

∣∣∣∣∣∣∣∣∑

deg(H)=nH0=1

Λ(H)χ(H)

∣∣∣∣∣∣∣∣ ≤ 1 + kqn2 , for n ≥ 1.

∣∣∣∣∣∣∣∣∑

P irreducibleP0=1, ψ(P )=ε

χ(P )

∣∣∣∣∣∣∣∣ ≤k+5n q

n2 , for n ≥ 2,

where χ is a non-trivial Dirichlet character modulo Xk+1, such thatχ(F∗q) = 1 and either ε = −1, or ε = 1 and n is odd.



All of the above combined give:

Theorem

Let n ≥ 2, 1 ≤ k ≤ n, and a ∈ Fq. There exists a srimp Q ∈ Fq[X], ofdegree 2n with Qk = a if the following bound holds.

πq(n,−1) ≥ k(k + 5)

n(√q + 1)q

n+k2 .



Further, Carlitz (1967) computed πq(n, ε) and those computations imply∣∣∣∣πq(n,−1)− qn

2n

∣∣∣∣ ≤ 1

2n

q

q − 1q

n3 ,

hence:

Theorem

Let n ≥ 2, 1 ≤ k ≤ n, and a ∈ Fq. There exists a srimp Q ∈ Fq[X], ofdegree 2n with Qk = a if the following bound holds.

qn−k−1

2 ≥ 16

5k(k + 5) +

1

2.


The HMC for self-reciprocal irreducible polynomials The restriction k ≤ n/2

Our final step is to content ourselves for k ≤ n/2 and solve the resultingproblem. Using the theory developed earlier, we conclude that there existsa srimp over Fq of degree 2n with its k-th coefficient prescribed, if

πq(n,−1) >bn/2c(bn/2c+ 5)

n(√q + 1)(qbn/2c/2 − 1)qn/2.

This bound is always true for n ≥ 27. For n < 27 this bound is satisfiedfor the pairs (q, n) described below

n 3 4 5 6 7 8 9 10q ≥ 149 ≥ 839 ≥ 37 ≥ 59 ≥ 17 ≥ 23 ≥ 11 ≥ 13

n 11 12 13 14 15 16 17 18q ≥ 9 ≥ 9 ≥ 7 ≥ 7 ≥ 5 ≥ 7 ≥ 5 ≥ 5

n 19 20 21 22 23 24 25 26q ≥ 5 ≥ 5 ≥ 5 ≥ 5 ≥ 5 ≥ 5 ≥ 3 ≥ 5


The HMC for self-reciprocal irreducible polynomials The restriction k ≤ n/2

Summing up, we have theoretically proved that:

Corollary

If n ≥ 3 an integer and q a power of an odd prime, then there exists asrimp of degree 2n such that any of its bn/2c low degree coefficients isprescribed, if either n ≥ 27 or q ≥ 839.

For the remaining cases, computers searches have been employed. Thecomputer results, combined with the above imply the following.

Theorem

Let n ≥ 3 an integer and q a power of an odd prime. If k ≤ n/2 anda ∈ Fq, then there exists a srimp of degree 2n such that any of its k-thcoefficient is prescribed to a, unless

1 q = 3, n = 3, a = 0 and k = 1 or

2 q = 3, n = 4, a = 0 and k = 2.


The HMC for self-reciprocal irreducible polynomials Further research

1 Can we extend this to k ≥ n/2?

2 What about even q?


Other extensions of the HMC

Now, we present some of the extensions of the HMC for irreduciblepolynomials that various authors have considered.

Many authors have considered prescribing the norn and the trace ofan irreducible polynomial (Wan (1997), Car (1999), Moisio (2008)and Omidi Koma, Panario and Wang (2010)).

Cohen (2005) proved that there exists an irreducible polynomial ofdegree n with its first k and last m coefficients prescribed whenever2 ≤ k +m ≤ n/3.

Garefalakis (2008) considered consecutive zero coefficients and heproved that for any c with 0 < c < 1 and any positive integer n suchthat (1− 3c)n ≥ 2 + 8 logq n, there exists an irreducible polynomialof degree n over Fq with any bcnc consecutive coefficients (otherthan the first or last) equal to 0.

Panario and Tzanakis (2012), Pollack (2013) and Ha (2016)considered prescribing multiple coefficients of an irreduciblepolynomial.



Similarly, much effort has been given in extending the HMC for primitivepolynomials.

Many authors have considerd prescribing the norm and trace of aprimitive polynomial (Cohen and Hachenberger (1999, 2000), Cohen(2000, 2012), Cohen and Huczynska (2003), Fan and Wang (2009)).

Chou and Cohen (2001) fixed the coefficient of X and the trace to 0,given that n ≥ 5.

There are several results about prescribing the first two (Han (1996,2009), Bao (1997), Cohen and Mills (2003), Cohen and King (2004)and Shuqin and Wenbao (2004)) and the first three (Cohen and King(2004), Fan and Han (2004) and Mills (2004)) coefficients of aprimitive polynomial.

Cohen and King (2004) and Fan and Han (2004) provided asymptoticresults about the existence of primitives with their first b(n− 1)/2ccoefficients prescribed.

Cohen (2004) proved that the first m, where m ≤ n/3, coefficients ofa primitive polynomial can be prescribed.



Another setting that has been investigated is prescribing coefficients ofprimitive and normal polynomials, where normal polynomials are thosewhose roots constitute a normal basis, that is a basis of the form{x, xq, . . . , xqn−1}.

Fan and Wang (2009) proved that there exists a primitive normalpolynomial with a prescribed coefficient when n ≥ 15.

Several efforts have been made in fixing the norm and trace of aprimitive normal polynomial (Cohen (2000), Cohen and Hachenberger(2000) and Cohen and Huczynska (2003)).

Fan and Wang (2009) proved the existence of a primitive normalpolynomial with its first two coefficients prescribed if n ≥ 5.

Asymptotic results about prescribing the last bn/2c (Fan and Han(2007)) and the first bn/2c (Fan (2009)) have also been established.


Different approaches

Finally, it is worth mentioning that recently several different approacheshave emerged.

Ha [Finite Fields Appl. 40:10–25, 2016] introduced number-theoreticmethods of Bourgain (2015) that enabled him to prescribe rcoefficients, given that r ≤ (1/4− ε)n, for any ε > 0. This workimproved previous work of Pollack (2013), where r ≤ (1/4− ε)

√n.

Tuxanidy and Wang (A new proof of the Hansen-Mullen irreducibilityconjecture, arXiv:1604.0423 [math.NT], unpublished. Irreduciblepolynomials with prescribed sums of coefficients, unpublished) havealso introduced a new promising method to the area.


Thank You!


the hansen-mullen conjectureusers.math.uoc.gr/~gkapet/files/sabanci_talk_1.pdf · 2016. 11. 27. ·...

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