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Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.


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TL;DR: In this paper, the integral cohomology of Γ up to p-power torsion for small primes p was derived for the case N = 3, D = − 3, − 4 when N = 4.

32 citations

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TL;DR: In this article, it was shown that Mazur's conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers in the rational numbers.
Abstract: We show that Mazur's conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers in the rational numbers. We also prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over that finite field. Both proofs depend upon Matijasevich's theorem.

32 citations

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TL;DR: In this article, it was shown that for any geometric point f(x) in U(Qbar) we have a Zariski dense open subset U defined over Q in A^d such that for every geometric point F(x), P(f mod P) = HP(A^d), where P is any prime ideal in the ring of integers of Q(f) lying over p. This proves a conjecture of Daqing Wan.
Abstract: Let d>2 and let p be a prime coprime to d. Let Z_pbar be the ring of integers of Q_pbar. Suppose f(x) is a degree-d polynomial over Qbar and Z_pbar. Let P be a prime ideal over p in the ring of integers of Q(f), where Q(f) is the number field generated by coefficients of f in Qbar. Let A^d be the dimension-d affine space over Qbar, identified with the space of coefficients of degree-d monic polynomials. Let NP(f mod P) denote the p-adic Newton polygon of L(f mod P;T). Let HP(A^d) denote the p-adic Hodge polygon of A^d. We prove that there is a Zariski dense open subset U defined over Q in A^d such that for every geometric point f(x) in U(Qbar) we have lim_{p-->oo} NP(f mod P) = HP(A^d), where P is any prime ideal in the ring of integers of Q(f) lying over p. This proves a conjecture of Daqing Wan.

32 citations

Journal ArticleDOI
TL;DR: In this article, Borwein and Erdélyi derived upper and lower estimates for the integer transfinite diameter of small intervals + 3B4, which is a fixed rational and ≥ 0.6.
Abstract: In this paper we build on some recent work of Amoroso, and Borwein and Erdélyi to derive upper and lower estimates for the integer transfinite diameter of small intervals + 03B4], is a fixed rational and 03B4 ~ 0. We also study functions g-, g, g+ associated with transfinite diameters of Farey intervals. Then we consider certain polynomials, which we call critical polynomials, associated to a given interval I. We show how to estimate from below the proportion of roots of an integer polynomial which is sufficiently small on I which must also be roots of the critical polynomial. This generalises now classical work of Aparicio, and extends the techniques of Borwein and Erdélyi from the critical polynomial x for [0,1] to any critical polynomial for an arbitrary interval. As an easy consequence of our results, we obtain an inequality about algebraic integers of independent interest: if 03B1 is totally real, with minimum conjugate 03B11, then, with a small number of explicit exceptions, the mean value of 03B1 and its conjugates is at least 03B11 + 1.6. Manuscrit reçu le 6 septembre 1996 138

31 citations

Journal ArticleDOI
P. M. Cohn1
TL;DR: In this article, it was shown that the Gaussian integers are not quasi-free for GE 2, and that Th. 9.3 of [2] holds under weaker hypotheses which are satisfied by the Gaussians.
Abstract: Let G be any group and G′ its derived, then G/G′ —the group G made abelian—will be denoted by G a . Over any ring R , denote by E 2 ( R ) the group generated by the matrices as x ranges over R ; the structure of E 2 ( R) a has been described in a recent theorem [2; Th. 9.3] for certain rings R , the “quasi-free rings for GE 2 ” ( cf. §2 below). Now over a commutative Euclidean domain, E 2 ( R ) is just the special linear group SL 2 ( R ); this suggests applying the theorem to the ring of integers in a Euclidean number field. However, the only number fields whose rings were shown to be quasi-free for GE 2 in [2] were the non -Euclidean imaginary quadratic fields. In fact that leaves the application of Th. 9.3 of [2] to the ring of Gaussian integers unjustified (I am indebted to J.-P. Serre for drawing this oversight to my attention). In order to justify this application one would have to show either (a) that the Gaussian integers are quasi-free for GE 2 ., or (b) that Th. 9.3 of [2] holds under weaker hypotheses which are satisfied by the Gaussian integers. Our object in this note is to establish (b)–indeed our only course, since the Gaussian integers turn out to be not quasi-free.

31 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202250
2021117
2020121
2019111
201896