A BSS machine is δ-uniform if it does not use exact tests; such machines are equivalent (modulo parameters) to Type 2 Turing machines. We define a notion of closure related to Turing machines for archimedean fields, and show that such fields admit nontrivial δ-uniformly decidable sets iff they are not Turing closed. Then, the partially ordered set of Turing closed fields is proved isomorphic to...

1.5∗ (A.Ostaszewski, email 9-92) Consider Telgarsky’s game G(T ) where T ⊆ 2. Player I plays a countable cover of T Player II chooses onesay Xn. Player I wins iff ∩{cl(Xn) : n ∈ ω} ⊆ T . It is known that (a) Player I has winning strategy iff T is analytic. (b) If there exists A an analytic subset of cl(T ) not Borel separated from T , then Player II has a winning strategy. Is the converse of (b...

We define an invariant β(M) of a finite volume hyperbolic 3manifoldM in the Bloch group B(C) and show it is determined by the simplex parameters of any degree one ideal triangulation of M . We show β(M) lies in a subgroup of B(C) of finite Q-rank determined by the invariant trace field of M . Moreover, the Chern-Simons invariant of M is determined modulo rationals by β(M). This leads to a simpl...

Let (S*, k(S2)) be a knot formed by spinning a polyhedral arc a. about the standard 2-sphere S" in the 3-sphere S3. Then the second homotopy group of S"—k(S2) as a Z^-module is isomorphic to each of the following: (1) The fundamental ideal modulo the left ideal generated by a—\, where a is the image in ■¡r1(Si—k(S2)) of a generator of (2) The first homology group of the kernel of ^(SP-hiS1))-* ...

In this paper, we will show a reduction of ideal arithmetic, or more generally, of arithmetic of ZZ{modules of full rank in orders of number elds to problems of linear algebra over ZZ=mZZ, where m is a possibly composite integer. The problems of linear algebra over ZZ=mZZ will be solved directly, instead of either \reducing" them to problems of linear algebra over ZZ or factoring m and working ...

We deene an invariant (M) of a nite volume hyperbolic 3-manifold M in the Bloch group B(C) and show it is determined by the simplex parameters of any degree one ideal triangulation of M. We show (M) lies in a subgroup of B(C) of nite Q-rank determined by the invariant trace eld of M. Moreover, the Chern-Simons invariant of M is determined modulo ra-tionals by (M). This leads to a simplicial for...

We consider the congruence a P s i=1 b i h i mod I where h 1 ; : : : ; hs are given modulo a zero dimensional ideal I. We give two polynomial time algorithms for determining a Grr obner basis, relative to an arbitrary term order, of the module M of solutions of the congruence, and, in particular, for nding its minimal element. These are based on a generalization of an algorithm of Faug ere et a...

We notice that two combinatorial interpretations of the well-known Catalan numbers Cn = (2n)!/n!(n+1)! naturally give rise to a recursion for Cn. This recursion is ideal for the study of the congruences of Cn modulo 2 r, which attracted a lot of interest recently. We present short proofs of some known results, and improve Liu and Yeh’s recent classification of Cn modulo 2 r. The equivalence Cn ...