A wild model of linear arithmetic and discretely ordered modules

Glivický, Petr; Pudlák, Pavel

doi:10.1002/malq.201600012

Cited by 3 publications

(4 citation statements)

References 6 publications

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“…We have not expected that any such thing would be possible. On the other hand, with regard to the main results in [3], it could be interesting to have a closer look at the behaviour of prime sequences also in the rings R τ1,τ2 = {f ∈ Q[x, y] | f (τ 1 , τ 2 ) ∈ Z} where τ 1 , τ 2 ∈ Z and f (τ 1 , τ 2 ) is evaluated in p∈P Q p . This might be a much harder task though.…”

Section: Pids With Cofinal Set Of Twin Primes and Much Morementioning

confidence: 99%

See 1 more Smart Citation

Controlling distribution of prime sequences in discretely ordered principal ideal subrings of $\mathbb Q[x]$

Glivická¹,

Ester²,

Šaroch³

2022

Preprint

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We show how to construct discretely ordered principal ideal subrings of Q[x] with various types of prime behaviour. Given any set D consisting of finite strictly increasing sequences (d 1 , d 2 , . . . , d l ) of positive integers such that, for each prime integer p, the set {pZ, d 1 +pZ, . . . , d l +pZ} does not contain all the cosets modulo p, we can stipulate to have, for each (d 1 , . . . , d l ) ∈ D, a cofinal set of progressions (f, f + d 1 , . . . , f + d l ) of prime elements in our PID Rτ . Moreover, we can simultaneously guarantee that each positive prime g ∈ Rτ \ N is either in a prescribed progression as above or there is no other prime h in Rτ such that g − h ∈ Z.The study of various types of finite sequences in prime numbers is a firm and celebrated part of the field of number theory. The twin prime conjecture or the Green-Tao theorem, [5], are just two of many universally known problems in the area. Given a strictly increasing sequence d = (d 1 , d 2 , . . . , d l ) of positive integers such that a, a + d 1 , a + d 2 , . . . , a + d l are prime numbers for infinitely many a ∈ P, it easily follows that, for each prime number p, the set {pZ, d 1 +pZ, d 2 +pZ, . . . , d l +pZ} does not contain all the cosets modulo p. This is a simple necessary condition for the existence of cofinal sequence of (finite) prime progressions with differences prescribed by d.In [4], the authors defined and studied certain discretely ordered subrings R τ of Q[x] where Z[x] ⊆ R τ and τ is a fixed parameter from the ring of profinite integers (see the preliminaries section for definitions). Among other things, they showed that all these subrings are quasi-Euclidean (also known as ω-stage Euclidean) but none is actually Euclidean, cf. [4, Theorems 3.2 and 3.9]. Somewhat more surprisingly, they also proved that many of these subrings are actually PIDs. In particular, there are 2 ω pairwise non-isomorphic PIDs among the subrings of Q[x] according to [4, Proposition 3.4].The aim of this paper is to have a closer look at these discretely ordered PIDs with particular emphasis on the behaviour of finite progressions of primes. We demonstrate a way how to construct a profinite integer τ such that primes are rather sparse in the PID R τ , i.e. for each two distinct primes f, g ∈ R τ \ Z the difference f − g does not belong to Z. On the other hand, our main result, Theorem 4.3, shows in particular that there is a τ ∈ Z such that R τ is a PID and the above mentioned simple necessary condition for the existence of a cofinal sequence of prime progressions in R τ is actually sufficient for each d. All our results are rather selfcontained and elementary and should be therefore accessible to a wide mathematical audience.

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Section: Pids With Cofinal Set Of Twin Primes and Much Morementioning

confidence: 99%

“…Since Z[x] ⊆ R τ holds for each τ ∈ Z, all the rings R τ are, in particular, faithful Z[x]-modules. More-or-less following Glivický's Ph.D. thesis [2] (see also [3]), we define a first order theory ZLa in the language L = (+,…”

Section: Appendix a Logical Connectionsmentioning

confidence: 99%

Controlling distribution of prime sequences in discretely ordered principal ideal subrings of $\mathbb Q[x]$

Glivická¹,

Ester²,

Šaroch³

2022

Preprint

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show abstract

“…It remains to reduce the three parameters

p, q, M

to two. To do this, we shall adapt a trick of Glivický and Pudlák . Their context is slightly different from ours in that they use nonstandard integers rather than parameters that range over

double-struckZ

, and that their results involve computability rather than complexity.…”

Section: Proof Of Theorem and Its Corollariesmentioning

confidence: 99%

“…In each case, we leverage the main result of Nguyen and Pak which yields a 3‐parametric Σ 2 PA formula, and then show how this can be reduced to a 2‐parametric Σ 2 PA formula whose points are equally “hard” to count (modulo polynomial‐time reductions). The first reduction we present, in § , uses a trick due to Glivický and Pudlák to encode multiplication by three different integers using multiplication by only two integers, and this reduction has the advantage of not increasing the number of free variables in the formula. Next, in § we present a more general counting‐reduction technique which is less ad hoc and reduces any k ‐parametric PA formula to a 2‐parametric PA formula with the same number of quantifier alternations; the idea here is a little more transparent than in § , but it has the disadvantage of introducing many more new free and quantified variables to the formula, so we consider that it is interesting to present both reductions.…”

Section: Introductionmentioning

confidence: 99%

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

Bogart

Goodrick

Nguyen

et al. 2019

Mathematical Logic Qtrly

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We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1,…,tk. A formula in this language defines a parametric set Sboldt⊆double-struckZd as boldt varies in Zk, and we examine the counting function false|Sboldtfalse| as a function of t. For a single parameter, it is known that false|Stfalse| can be expressed as an eventual quasi‐polynomial (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 P≠NP) we construct a parametric set St1,t2 such that false|St1,t2false| is not even polynomial‐time computable on input (t1,t2). In contrast, for parametric sets Sboldt⊆double-struckZd with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that false|Sboldtfalse| is always polynomial‐time computable in the size of t, and in fact can be represented using the gcd and similar functions.

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Controlling distribution of prime sequences in discretely ordered principal ideal subrings of ℚ[𝕩]

Glivická

Ester

Šaroch

2023

Proc. Amer. Math. Soc.

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We show how to construct discretely ordered principal ideal subrings of Q [ x ] \mathbb {Q}[x] with various types of prime behaviour. Given any set D \mathcal D consisting of finite strictly increasing sequences ( d 1 , d 2 , … , d l ) (d_1,d_2,\dots , d_l) of positive integers such that, for each prime integer p p , the set { p Z , d 1 + p Z , … , d l + p Z } \{p\mathbb {Z}, d_1+p\mathbb {Z},\dots , d_l+p\mathbb {Z}\} does not contain all the cosets modulo p p , we can stipulate to have, for each ( d 1 , … , d l ) ∈ D (d_1,\dots , d_l)\in \mathcal D , a cofinal set of progressions ( f , f + d 1 , … , f + d l ) (f, f+d_1, \dots , f+d_l) of prime elements in our principal ideal domain R τ R_\tau . Moreover, we can simultaneously guarantee that each positive prime g ∈ R τ ∖ N g\in R_\tau \setminus \mathbb {N} is either in a prescribed progression as above or there is no other prime h h in R τ R_\tau such that g − h ∈ Z g-h\in \mathbb {Z} . Finally, all the principal ideal domains we thus construct are non-Euclidean and isomorphic to subrings of the ring Z ^ \widehat {\mathbb {Z}} of profinite integers.

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A wild model of linear arithmetic and discretely ordered modules

Cited by 3 publications

References 6 publications

Controlling distribution of prime sequences in discretely ordered principal ideal subrings of $\mathbb Q[x]$

Controlling distribution of prime sequences in discretely ordered principal ideal subrings of $\mathbb Q[x]$

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

Controlling distribution of prime sequences in discretely ordered principal ideal subrings of ℚ[𝕩]

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