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On Gauss-Kraitchik formula for cyclotomic polynomials via symmetric functions

Tomohiro Yamada

TL;DR

This paper addresses explicit upper bounds for the coefficients arising in the Gauss-Kraïtchik decomposition of cyclotomic polynomials, focusing on the auxiliary polynomials $\Psi_d(X)$ and $\Xi_d(X)$ in the identity $4\Phi_d(X)=\Psi_d(X)^2-D\Xi_d(X)^2$ with $D=(-1)^{(d-1)/2} d$. The authors develop a symmetric-function approach, leveraging the Girard-Newton identity and a simple specialization $P_m(X)=X(X-1)\cdots(X-m+1)/m!$ to express coefficients $u_{d,n}$ and consequently bound $a_{d,n}$ and $b_{d,n}$. They obtain explicit upper bounds $|a_{d,n}+b_{d,n}\sqrt{D}| \leq \frac{F_{d,n}(F_{d,n}+1)\cdots (F_{d,n}+n-1)}{n!}$ and $|a_{d,n}|+|b_{d,n}|\sqrt{d} \leq \frac{G_{d,n}(G_{d,n}+1)\cdots (G_{d,n}+n-1)}{n!}$, along with Stirling-based corollaries and a precise asymptotic bound for the ratio $|\Xi_d(x)/\Psi_d(x)|$ for large $x$. These results yield quantitative control over the arithmetic properties of Gauss-Kraïtchik components and have potential applications to divisor sums and related number-theoretic questions.

Abstract

We give explicit upper bounds for coefficients of polynomials appearing in Gauss-Kraïtchik formula for cyclotomic polynomials. We use a certain relation between elementary symmetric polynomials and power sums polynomials.

On Gauss-Kraitchik formula for cyclotomic polynomials via symmetric functions

TL;DR

This paper addresses explicit upper bounds for the coefficients arising in the Gauss-Kraïtchik decomposition of cyclotomic polynomials, focusing on the auxiliary polynomials and in the identity with . The authors develop a symmetric-function approach, leveraging the Girard-Newton identity and a simple specialization to express coefficients and consequently bound and . They obtain explicit upper bounds and , along with Stirling-based corollaries and a precise asymptotic bound for the ratio for large . These results yield quantitative control over the arithmetic properties of Gauss-Kraïtchik components and have potential applications to divisor sums and related number-theoretic questions.

Abstract

We give explicit upper bounds for coefficients of polynomials appearing in Gauss-Kraïtchik formula for cyclotomic polynomials. We use a certain relation between elementary symmetric polynomials and power sums polynomials.
Paper Structure (4 sections, 5 theorems, 47 equations)

This paper contains 4 sections, 5 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Power sums of roots of unity
  3. Elementary symmetric polynomials and power sums polynomials
  4. Proof of the Theorems

Key Result

Theorem 1

Let $d\geq 5$ be an odd squarefree integer, $0\leq n\leq d^\prime$, and $F=F_{d, n}=\max\{\varphi(f)/2: f\mid d, 1<f\leq n\}\cup\{\left\lvert1+\sqrt{D}\right\rvert/2\}$. Then, and, putting $G_{d, n}=\max\{\varphi(f)/2: f\mid d, 1<f\leq n\}\cup\{\left\lvert1+\sqrt{d}\right\rvert/2\}$,

Theorems & Definitions (7)

  • Theorem 1
  • Corollary
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof