Abstract
Given a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.
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Acknowledgements
I thank Tobias Fritz for providing useful feedback. This work was supported by the ÚNKP-19-4 New National Excellence Program of the Ministry for Innovation and Technology and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. I acknowledge support from the Hungarian National Research, Development and Innovation Office (NKFIH) within the Quantum Technology National Excellence Program (Project Nr. 2017-1.2.1-NKP-2017-00001) and via the research grants K124152, KH129601.
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Vrana, P. A Generalization of Strassen’s Theorem on Preordered Semirings. Order 39, 209–228 (2022). https://doi.org/10.1007/s11083-021-09570-7
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DOI: https://doi.org/10.1007/s11083-021-09570-7