512 research outputs found
Some new results on decidability for elementary algebra and geometry
We carry out a systematic study of decidability for theories of (a) real
vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces,
Banach spaces and metric spaces, all formalised using a 2-sorted first-order
language. The theories for list (a) turn out to be decidable while the theories
for list (b) are not even arithmetical: the theory of 2-dimensional Banach
spaces, for example, has the same many-one degree as the set of truths of
second-order arithmetic.
We find that the purely universal and purely existential fragments of the
theory of normed spaces are decidable, as is the AE fragment of the theory of
metric spaces. These results are sharp of their type: reductions of Hilbert's
10th problem show that the EA fragments for metric and normed spaces and the AE
fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs
of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3
The model theory of Commutative Near Vector Spaces
In this paper we study near vector spaces over a commutative from a model
theoretic point of view. In this context we show regular near vector spaces are
in fact vector spaces. We find that near vector spaces are not first order
axiomatisable, but that finite block near vector spaces are. In the latter case
we establish quantifier elimination, and that the theory is controlled by which
elements of the pointwise additive closure of are automorphisms of the near
vector space
Separably closed fields and contractive Ore modules
We consider valued fields with a distinguished contractive map as valued
modules over the Ore ring of difference operators. We prove quantifier
elimination for separably closed valued fields with the Frobenius map, in the
pure module language augmented with functions yielding components for a p-basis
and a chain of subgroups indexed by the valuation group
Parametric Presburger arithmetic: complexity of counting and quantifier elimination
We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1,âŠ,tk. A formula in this language defines a parametric set StâZd as t varies in Zk, and we examine the counting function |St| as a function of t. For a single parameter, it is known that |St| can be expressed as an eventual quasiâpolynomial (there is a period m such that, for sufficiently large t, the function is polynomial on each of the residue classes mod m). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming Pâ NP) we construct a parametric set St1,t2 such that |St1,t2| is not even polynomialâtime computable on input (t1,t2). In contrast, for parametric sets StâZd with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that |St| is always polynomialâtime computable in the size of t, and in fact can be represented using the gcd and similar functions.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/151911/1/malq201800068_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/151911/2/malq201800068.pd
Software Engineering and Complexity in Effective Algebraic Geometry
We introduce the notion of a robust parameterized arithmetic circuit for the
evaluation of algebraic families of multivariate polynomials. Based on this
notion, we present a computation model, adapted to Scientific Computing, which
captures all known branching parsimonious symbolic algorithms in effective
Algebraic Geometry. We justify this model by arguments from Software
Engineering. Finally we exhibit a class of simple elimination problems of
effective Algebraic Geometry which require exponential time to be solved by
branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with
arXiv:1201.434
Equivariant Zariski Structures
A new class of noncommutative -algebras (for an algebraically closed
field) is defined and shown to contain some important examples of quantum
groups. To each such algebra, a first order theory is assigned describing
models of a suitable corresponding geometric space. Model-theoretic results for
these geometric structures are established (uncountable categoricity,
quantifier elimination to the level of existential formulas) and that an
appropriate dimension theory exists, making them Zariski structures
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