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arXiv · 2112.05949

Abstract Vergleichsstellens\"atze for preordered semifields and semirings II

Tobias Fritz
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Abstract

The present paper continues our foundational work on real algebra with preordered commutative semifields and semirings. We prove two abstract Vergleichsstellens\"atze for preordered commutative semirings of polynomial growth. These generalize the results of Part I by no longer assuming $1 \ge 0$. Such a generalization comes with substantial technical complications: our Vergleichsstellens\"atze now also need to take into account infinitesimal information encoded in the form of monotone derivations in addition to the monotone homomorphisms to the nonnegative reals and tropical reals. The auxiliary technical results we develop along the way include surprising implications between inequalities in preordered semifields and a type classification for multiplicatively Archimedean fully preordered semifields. Among other applications, two companion papers use these results in order to derive limit new results in probability and information theory; one on asymptotics of random walks on topological abelian groups, and the other on the asymptotics of matrix majorization.

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Tobias Fritz. 2021-12-11. Abstract Vergleichsstellens\"atze for preordered semifields and semirings II. https://arxiv.org/abs/2112.05949

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