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In the early 1990’s, Kim and Roush developed path methods for establishing strong shift equivalence (SSE) of positive matrices over a dense subring U of R. This paper gives a detailed, unified and generalized presentation of these path methods. New arguments which address arbitrary dense subrings U of R are used to show that for any dense subring U of R, positive matrices over U which have just...

An expansion of a definably complete field either defines a discrete subring, or the image of every definable discrete set under every definable map is nowhere dense. As an application we show a definable version of Lebesgue’s differentiation theorem.

The notion of an amenable equivalence relation was introduced by Zimmer in the course of his analysis of orbit equivalence relations in ergodic theory (see [12]). Recently it played an important role in Monod’s striking family of examples of nonamenable groups which do not contain nonabelian free subgroups. If A is a subring of R, define H(A) to be the group of all piecewise PSL2(A) homeomorphi...

Given matrices A and B shift equivalent over a dense subring R of R, with A primitive, we show that B is strong shift equivalent over R to a primitive matrix. This result shows that the weak form of the Generalized Spectral Conjecture for primitive matrices implies the strong form. The foundation of this work is the recent result that for any ring R, the group NK1(R) of algebraic K-theory class...

We show how tropical varieties of ideals I E K[x] over a field K with non-trivial valuation can always be traced back to tropical varieties of ideals π−1I E RJtK[x] over some dense subring R in its ring of integers. Moreover, for homogeneous ideals, we present algorithms on how the latter can be computed in finite time, provided that π−1I is generated by elements in R[t, x]. While doing so, we ...

We show that every henselian valued field L of residue characteristic 0 admits a proper subfield K which is dense in L. We present conditions under which this can be taken such that L|K is transcendental and K is henselian. These results are of interest for the investigation of integer parts of ordered fields. We present examples of real closed fields which are larger than the quotient fields o...

In this paper, we construct subring homomorphic encryption scheme that is a homomorphic encryption scheme built on the decomposition ring, which is a subring of cyclotomic ring. In the scheme, each plaintext slot contains an integer in Zpl , rather than an element of GF(p) as in conventional homomorphic encryption schemes on cyclotomic rings. Our benchmark results indicate that the subring homo...