Factoring through monomial representations: arithmetic characterizations and ambiguity of weighted automata

Abstract: 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 Ab\'ert, 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.