极大极小半环上的不可约性
Irreducibility over the Max-Min Semiring
摘要：
对于集合$A, B\subset \mathbb N$，它们的和集是$A + B := \{a+b: a\in A, b\in B\}$。如果我们不能用$|A|, |B|\geq 2$把一个集合$C$写成$C = A+B$，那么我们说$C$是$\textit{irreducible}$。一个给定的集合$C$是否不可约的问题在加法组合学中自然出现。等价地，我们可以将这个问题表述为关于布尔多项式的不可约性问题，K. H. Kim和F. W. Zeroh（2005）和Y. Shitov（2014）.证明了极大极小半环上多项式和幂级数的不可约性，极大极小半环是布尔多项式的自然推广。本文利用组合和概率方法证明了极大极小半环上几乎所有多项式都是不可约的，推广了Y。Shitov（2014）和D. L. Applegate，M. Le Brun和N. J. A.斯隆此外，我们还利用测度论的方法和Borel关于正规数的结果证明了几乎所有的幂级数在极大极小半环上都是渐近不可约的。这一结果推广了E. Wirsing（1953）.
英文摘要：
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).
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