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A factorisation theory for generalised power series and omnific integers

Sonia L'Innocente, Vincenzo Mantova

Abstract

We prove that in every ring of generalised power series with non-positive real exponents and coefficients in a field of characteristic zero, every series admits a factorisation into finitely many irreducibles of infinite support, the number of which can be bounded in terms of the order type of the series, and a unique product, up to multiplication by a unit, of factors of finite support. We deduce analogous results for the ring of omnific integers within Conway's surreal numbers, using a suitable notion of infinite product. In turn, we solve Gonshor's conjecture that the omnific integer $ω^{\sqrt{2}} + ω+ 1$ is prime. We also exhibit new classes of irreducible and prime generalised power series and omnific integers, generalising previous work of Berarducci and Pitteloud.

A factorisation theory for generalised power series and omnific integers

Abstract

We prove that in every ring of generalised power series with non-positive real exponents and coefficients in a field of characteristic zero, every series admits a factorisation into finitely many irreducibles of infinite support, the number of which can be bounded in terms of the order type of the series, and a unique product, up to multiplication by a unit, of factors of finite support. We deduce analogous results for the ring of omnific integers within Conway's surreal numbers, using a suitable notion of infinite product. In turn, we solve Gonshor's conjecture that the omnific integer is prime. We also exhibit new classes of irreducible and prime generalised power series and omnific integers, generalising previous work of Berarducci and Pitteloud.

Paper Structure

This paper contains 52 sections, 116 theorems, 116 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Conway's conjectures on omnific integers
  3. Generalised power series
  4. Previous results
  5. Complete factorisations
  6. A new valuation
  7. New criteria for irreducibility and primality
  8. Preliminaries
  9. Generalised power series
  10. Ordinal arithmetic
  11. Surreal numbers
  12. Archimedean classes
  13. Pre-Schreier domains and GCD domains
  14. Ritt's theorem
  15. Berarducci's order value
  16. ...and 37 more sections

Key Result

Theorem A

Let $b \in {\mathbf{K}((\mathbb{R}^{\leq 0}))}$ with $b \neq 0$. Then there are $r \in \mathbb{R}^{\leq 0}$, $n \in \mathbb{N}$, and $c_1, \dots, c_n \in {\mathbf{K}((\mathbb{R}^{\leq 0}))}$ such that $b = t^rc_1 \cdots c_n$, where each $c_i$ satisfies exactly one of the following: and the supports of the $c_i$'s satisfying item:main-KRR-finite-irred are pairwise $\mathbb{Q}$-linearly independent

Figures (1)

  • Figure 1: The tree of surreal numbers.

Theorems & Definitions (290)

  • Conjecture 1.1.1: Con1976
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition
  • Theorem D
  • Theorem E
  • Theorem F
  • Remark 2.1.2
  • Corollary 2.1.4
  • ...and 280 more