[Paper Review] A new proof of the Hansen-Mullen irreducibility conjecture

Aleksandr Tuxanidy, Qiang Wang|arXiv (Cornell University)|Apr 14, 2016

Coding theory and cryptography4 references3 citations

TL;DR

This paper presents a novel proof of the Hansen-Mullen irreducibility conjecture using the least period of the discrete Fourier transform (DFT) of characteristic elementary symmetric functions over finite fields. By establishing that a key function related to coefficient generation has maximal least period except for known exceptions, the authors provide an elementary, unified proof of the existence of monic irreducible polynomials of degree $ n $ over $ \mathbb{F}_q $ with any prescribed coefficient (except in small exceptional cases), resolving a long-standing conjecture via a new analytic-structural approach.

ABSTRACT

We give a new proof of the Hansen-Mullen irreducibility conjecture. The proof relies on an application of a (seemingly new) sufficient condition for the existence of elements of degree $n$ in the support of functions on finite fields. This connection to irreducible polynomials is made via the least period of the discrete Fourier transform (DFT) of functions with values in finite fields. We exploit this relation and prove, in an elementary fashion, that a relevant function related to the DFT of characteristic elementary symmetric functions (which produce the coefficients of characteristic polynomials) has a sufficiently large least period (except for some genuine exceptions). This bears a sharp contrast to previous techniques in literature employed to tackle existence of irreducible polynomials with prescribed coefficients.

Motivation & Objective

• To provide a new, elementary proof of the Hansen-Mullen irreducibility conjecture, which asserts the existence of irreducible polynomials with any one coefficient prescribed to any value in $ \mathbb{F}_q $.
• To unify the treatment of all cases—including small $ q $ and $ n $, previously handled computationally—through a structural analysis of function periods.
• To establish a connection between the existence of irreducible polynomials with prescribed coefficients and the least period of DFT-related functions on finite fields.
• To offer a conceptual alternative to traditional analytic methods relying on character sums and Weil’s bound, which face limitations in extending beyond $ n/2 $ prescribed coefficients.

Proposed method

• The authors analyze the least period of a function $ \Delta_{w,c} $ derived from the DFT of characteristic elementary symmetric functions that generate polynomial coefficients.
• They use $ q $-adic digit representations and permutation arguments to show that if a period $ r $ of $ \Delta_{w,c} $ is not maximal, then a contradiction arises unless all digits of $ r $ are equal or $ r = (q^n - 1)/2 $.
• The proof leverages $ q $-symmetry and properties of tensor powers of the function $ \delta_{n/2} $, showing that $ \delta_{n/2}^{\otimes(q-1)} $ is $ q $-symmetric and thus invariant under digit permutations.
• By contradiction, they demonstrate that non-maximal periods lead to inconsistent digit sum conditions, forcing the least period to be $ q^n - 1 $, the maximum possible.
• The method relies on structural number-theoretic properties of $ q $-adic expansions and permutation invariance, avoiding character sum estimates common in prior work.
• The key insight is that maximal least period of $ \Delta_{w,c} $ implies the existence of elements of degree $ n $ in the support, which corresponds to irreducible polynomials with prescribed coefficients.

Experimental results

Research questions

• RQ1 Can the Hansen-Mullen irreducibility conjecture be proven using a method that avoids character sum estimates and Weil’s bound?
• RQ2 What structural properties of the DFT of symmetric functions over finite fields determine the existence of irreducible polynomials with prescribed coefficients?
• RQ3 Is there a uniform, elementary explanation for all exceptional cases in the Hansen-Mullen conjecture, including small $ q $ and $ n $?
• RQ4 Can the least period of a function related to polynomial coefficients be used as a sufficient condition for the existence of irreducible polynomials of degree $ n $?

Key findings

• The least period of the function $ \Delta_{w,c} $, derived from the DFT of elementary symmetric functions, is shown to be $ q^n - 1 $ for all $ n > 2 $, except in the case $ q $ odd and $ w = n/2 $, where $ r = (q^n - 1)/2 $ is possible.
• For $ n = 2 $ and $ q $ odd, the least period $ r $ must satisfy $ r > q - 1 $, which rules out all smaller periods and forces maximality.
• The proof establishes that $ \Delta_{w,c} $ has maximal least period $ q^n - 1 $, implying the existence of elements of degree $ n $ in its support, which corresponds to irreducible polynomials with prescribed coefficients.
• The method provides a unified explanation for all cases of the Hansen-Mullen conjecture, including the small cases previously verified computationally, without relying on asymptotic estimates or character sum bounds.
• The result confirms the existence of monic irreducible polynomials of degree $ n $ over $ \mathbb{F}_q $ with any coefficient $ [x^{n-w}]P(x) = c $, except for the known exceptions: $ (n,w,c) = (2,1,0) $ when $ q $ is odd, and $ w = n $ with $ c = 0 $.
• The proof is elementary and structural, contrasting with prior analytic methods, and offers a new framework for studying irreducibility with prescribed coefficients via function periods and $ q $-adic digit symmetry.