Interpolation Domains

Cahen, Paul-Jean; Chabert, Jean-Luc; Frisch, Sophie

doi:10.1006/jabr.1999.8151

Cited by 23 publications

(13 citation statements)

References 5 publications

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“…Partial results in this direction, but sometimes over more general rings and sometimes in the language of iterated differences, were obtained in several other papers, e.g. in [2,3,4,5,7,8,9,10,11,12,13,14,17], and in the books [1,15]. Our paper differs from most of these investigations in that we do not restrict ourselves to the case where domain and codomain coincide.…”

Section: Introductionmentioning

confidence: 85%

Classification of polynomial mappings between commutative groups

Schauz

2014

Journal of Number Theory

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Some polynomials P with rational coefficients give rise to well defined maps between cyclic groups, Z q −→ Z r , x + qZ −→ P (x) + rZ . More generally, there are polynomials in several variables with tuples of rational numbers as coefficients that induce maps between commutative groups. We characterize the polynomials with this property, and classify all maps between two given finite commutative groups that arise in this way. We also provide interpolation formulas and a Taylor-type theorem for the calculation of polynomials that describe given maps.

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Section: Introductionmentioning

confidence: 85%

Classification of polynomial mappings between commutative groups

Schauz

2014

Journal of Number Theory

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“…Proof. Again, given f, g ∈ I, we show that there exists h ∈ I with Ad (2). Suppose Q is maximal with respect to not containing any M-unitvalued function.…”

Section: Prime Ideals Corresponding To Ultrafiltersmentioning

confidence: 92%

“…Whether (2) holds or not for a particular D and a particular subring of D E will have to be examined separately. It is, in some sense, a question of density of the subring in the product e∈E D.…”

Section: Introductionmentioning

confidence: 99%

On the spectrum of rings of functions

Frisch

2018

Journal of Pure and Applied Algebra

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For D a domain and E ⊆ D, we investigate the prime spectrum of rings of functions from E to D, that is, of rings contained in e∈E D and containing D. Among other things, we characterize, when M is a maximal ideal of finite index in D, those prime ideals lying above M which contain the kernel of the canonical map to e∈E (D/M ) as being precisely the prime ideals corresponding to ultrafilters on E. We give a sufficient condition for when all primes above M are of this form and thus establish a correspondence to the prime spectra of ultraproducts of residue class rings of D. As a corollary, we obtain a description using ultrafilters, differing from Chabert's original one which uses elements of the M -adic completion, of the prime ideals in the ring of integer-valued polynomials Int(D) lying above a maximal ideal of finite index.2010 Mathematics Subject Classification. Primary 13F20; Secondary 13L05, 13B25, 13A15, 13G05, 12L10.

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“…In particular, it is possible to interpolate a set of points using integervalued polynomials with rational coefficients [17,24]. However, these polynomials can only be used if the verification…”

Section: Appendix A: Proof Of Soundness and Completenessmentioning

confidence: 99%

Automatic equivalence checking of programs with uninterpreted functions and integer arithmetic

Lopes

Monteiro

2015

Int J Softw Tools Technol Transfer

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Proving equivalence of programs has several important applications, including algorithm recognition, regression checking, compiler optimization verification and validation, and information flow checking. Despite being a topic with so many important applications, program equivalence checking has seen little advances over the past decades due to its inherent (high) complexity. In this paper, we propose, to the best of our knowledge, the first semi-algorithm for the automatic verification of partial equivalence of two programs over the combined theory of uninterpreted function symbols and integer arithmetic (UF+IA). The proposed algorithm supports, in particular, programs with nested loops. The crux of the technique is a transformation of uninterpreted functions (UFs) applications into integer polynomials, which enables the precise summarization of loops with UF applications using recurrences. The equivalence checking algorithm then proceeds on loop-free, integer only programs. We implemented the proposed technique in CORK, a tool that automatically verifies the correctness of compiler optimizations, and we show that it can prove more optimizations correct than state-of-the-art techniques.

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Interpolation Domains

Cited by 23 publications

References 5 publications

Classification of polynomial mappings between commutative groups

Classification of polynomial mappings between commutative groups

On the spectrum of rings of functions

Automatic equivalence checking of programs with uninterpreted functions and integer arithmetic

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