Ternary cyclotomic polynomials having a large coefficient (0712.2365v2)

Abstract: Let $\Phi_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $\Phi_n(x)$, satisfies $|a_n(k)|\le (p+1)/2$ in case $n=pqr$ with $p<q<r$ primes (in this case $\Phi_n(x)$ is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example $|a_n(k)|\le 3p/4$). Here we show that, nevertheless, Beiter's conjecture is false for every $p\ge 11$. We also prove that given any $\epsilon\>0$ there exist infinitely many triples $(p_j,q_j,r_j)$ with $p_1<p_2<... $ consecutive primes such that $|a_{p_jq_jr_j}(n_j)|>(2/3-\epsilon)p_j$ for $j\ge 1$.

• The paper provides counter-examples showing Beiter's conjecture on ternary cyclotomic polynomial coefficient bounds is incorrect for large primes p.
• It proposes a refined conjecture suggesting the maximum height of ternary cyclotomic polynomials is bounded by 2p/3, challenging Beiter's original prediction.
• This research has implications for understanding cyclotomic polynomial structure and properties relevant to fields like cryptography and algebraic number theory.

An Analysis of Ternary Cyclotomic Polynomials and Beiter's Conjecture

The paper "Ternary Cyclotomic Polynomials Having a Large Coefficient," authored by Yves Gallot and Pieter Moree, provides a comprehensive examination of ternary cyclotomic polynomials and scrutinizes Sister Marion Beiter's conjecture regarding the coefficients of these polynomials. The authors present substantial evidence that Beiter's conjecture is incorrect for cases where p, q, and r are primes with certain conditions, offering alternative conjectures and proofs to explain observed behaviors in these polynomials.

The paper is centered on the nth cyclotomic polynomial, Φ_n(x), and more particularly on polynomials of the form Φ_pqr(x) where n = pqr with p, q, and r as distinct primes, known as ternary cyclotomic polynomials. Sister Beiter conjectured that the maximum absolute value of the coefficients, defined as A(pqr), should be less than or equal to (p + 1)/2. Although some previous results were aligned with her conjecture, Gallot and Moree's findings suggest otherwise for p ≥ 11.

The paper's major contribution is the demonstration that, contrary to Beiter's conjecture, there exist an infinite number of cases where ternary cyclotomic polynomials have coefficients larger than (2/3−ε)p for any ε > 0, as the coefficients can actually exceed this threshold. The authors systematically provide explicit constructions of counter-examples and develop theorems (Theorem 4 and Theorem 11) that identify specific conditions under which such large coefficients appear.

This research puts forth a corrected conjecture that proposes the achievable maximum for the height of ternary cyclotomic polynomials M(p) as being less than or equal to 2p/3, as opposed to Beiter's original prediction. This refined conjecture remains to be proven or disproven and conjectures that this new bound encapsulates the maximum growth potential of coefficient magnitudes more accurately than Beiter initially anticipated.

The implications of this paper are significant both in theoretical discussions and practical applications. The insight into the structure and properties of cyclotomic polynomials can affect fields such as cryptography, where polynomials are used for constructing algorithms and encoding strategies. Future work might involve further numerical studies, testing the corrected conjecture for larger ranges of p, or applying these results in computational frameworks to solve problems in algebraic number theory and related areas.

In conclusion, Gallot and Moree offer a rigorous reevaluation of a longstanding mathematical conjecture, providing compelling evidence that challenges previous assumptions and introduces new avenues for further theoretical exploration. By outlining new potential bounds for the coefficients of ternary cyclotomic polynomials, they lay the groundwork for future investigations that could resolve open questions and extend the understanding of these intricate mathematical entities.