通过单项表示分解：加权自动机的算术刻画与歧义性
Factoring through monomial representations: arithmetic characterizations and ambiguity of weighted automata
AI总结：
本文利用单位方程等丢番图数论结果刻画可经单项或分块三角表示分解的群与半群，并建立可逆 WFA 的算术性质与歧义复杂性层级的完整对应及可判定性。
摘要：
我们通过算术性质刻画了分别可通过单项表示以及对角块为单项的分块三角表示分解的群表示。对于由可逆变换给出的半群表示，我们也得到了类似结果。这些刻画使用了丢番图数论中关于单位方程的结果（特征零下由 Evertse、van der Poorten 和 Schlickewei 给出，正特征下由 Derksen 和 Masser 给出）。\n\n专门针对特征零下有限生成群，我们的一个主要定理重新得到了 Corvaja、Demeio、Rapinchuk、Ren 和 Zannier 近期一项类似刻画的改进版本，该刻画受到有界生成（BG）性质研究的启发。在正特征下，我们得到了线性 BG 群的刻画，重新得到了 Abért、Lubotzky 和 Pyber 在 2003 年的一个定理。\n\n我们的动机来自域上的加权有限自动机（WFA）。对于可逆 WFA，我们证明 M-歧义性、有限歧义性和多项式歧义性由算术性质刻画。我们发现了算术性质与基于歧义性的 WFA 复杂性层级之间的完整对应。在可逆情形下，这是 Bell 与第二作者近期关于无歧义 WFA 刻画结果的深远推广，该结果解决了 Reutenauer 在 1979 年提出的一个猜想。作为推论，利用有限生成矩阵半群的（线性）Zariski 闭包的可计算性，M-歧义性、有限歧义性和多项式歧义性对可逆 WFA 是算法可判定的。
英文摘要：
We characterize group representations that factor through monomial representations, respectively, block-triangular representations with monomial diagonal blocks, by arithmetic properties. Similar results are obtained for semigroup representations by invertible transformations. The characterizations use results on unit equations from Diophantine number theory (by Evertse, van der Poorten, and Schlickewei in characteristic zero, and by Derksen and Masser in positive characteristic). Specialized to finitely generated groups in characteristic zero, one of our main theorems recovers an improvement of a very recent similar characterization by Corvaja, Demeio, Rapinchuk, Ren, and Zannier that was motivated by the study of the bounded generation (BG) property. In positive characteristic, we get a characterization of linear BG groups, recovering a theorem of Abért, Lubotzky, and Pyber from 2003. Our motivation comes from weighted finite automata (WFA) over a field. For invertible WFA we show that $M$-ambiguity, finite ambiguity, and polynomial ambiguity are characterized by arithmetic properties. We discover a full correspondence between arithmetic properties and a complexity hierarchy for WFA based on ambiguity. In the invertible case, this is a far-reaching generalization of a recent result by Bell and the second author, characterizing unambiguous WFA, that resolved a 1979 conjecture of Reutenauer. As a consequence, using the computability of the (linear) Zariski closure of a finitely generated matrix semigroup, the $M$-ambiguity, finite ambiguity, and polynomial ambiguity properties are algorithmically decidable for invertible WFA.
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