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On the discriminants of truncated logarithmic polynomials

John Cullinan, Rylan Gajek-Leonard

Abstract

We provide evidence for a conjecture of Yamamura that the truncated logarithmic polynomials \[ F_n(x) = 1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n} \] have Galois group $S_n$ for all $n \geq 1$.

On the discriminants of truncated logarithmic polynomials

Abstract

We provide evidence for a conjecture of Yamamura that the truncated logarithmic polynomials have Galois group for all .
Paper Structure (5 sections, 8 theorems, 27 equations)

This paper contains 5 sections, 8 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Commentary on Discriminants, Resultants, and Arithmetic
  3. Preliminary Results
  4. The Case $n=mq$
  5. Examples and Conclusions

Key Result

Theorem 7

If $n \equiv 0,2,3 \pmod{4}$, or if $n \equiv 1 \pmod{4}$ and is the odd power of a prime, then $\operatorname{disc}(F_n) \not \in \mathbf{Q}^{\times 2}$.

Theorems & Definitions (16)

  • Example 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Remark 10
  • Proposition 11
  • proof
  • Theorem 12
  • proof
  • Corollary 13
  • ...and 6 more