Irreducibility over the Max-Min Semiring

TL;DR

This paper investigates the irreducibility of polynomials and power series over the max-min semiring, showing that almost all are irreducible and confirming a 2011 conjecture, using combinatorial, probabilistic, and measure-theoretic methods.

Contribution

It extends the concept of irreducibility to the max-min semiring and proves that almost all such polynomials and power series are irreducible, generalizing previous results.

Findings

Almost all polynomials are irreducible over the max-min semiring.

Almost all power series are asymptotically irreducible over the max-min semiring.

The results confirm a 2011 conjecture by Applegate, Le Brun, and Sloane.

Abstract

For sets $A, B\subset \mathbb N$, their sumset is $A + B := \{a+b: a\in A, b\in B\}$. If we cannot write a set $C$ as $C = A+B$ with $|A|, |B|\geq 2$, then we say that $C$ is $\textit{irreducible}$. The question of whether a given set $C$ is irreducible arises naturally in additive combinatorics. Equivalently, we can formulate this question as one about the irreducibility of boolean polynomials, which has been discussed in previous work by K. H. Kim and F. W. Roush (2005) and Y. Shitov (2014). We prove results about the irreducibility of polynomials and power series over the max-min semiring, a natural generalization of the boolean polynomials. We use combinatorial and probabilistic methods to prove that almost all polynomials are irreducible over the max-min semiring, generalizing work of Y. Shitov (2014) and proving a 2011 conjecture by D. L. Applegate, M. Le Brun, and N. J. A. Sloane. Furthermore, we use measure-theoretic methods and apply Borel's result on normal numbers to prove that almost all power series are asymptotically irreducible over the max-min semiring. This result generalizes work of E. Wirsing (1953).