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Ring of integers
About: Ring of integers is a(n) research topic. Over the lifetime, 1856 publication(s) have been published within this topic receiving 15882 citation(s).
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TL;DR: The theory of multiple polylogarithm Hopf algebras from an analytic, Hodge and motivic point of view has been studied in this article, where the authors define the category of mixed Tate motives over a ring of integers in a number field.
Abstract: We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe explicitly the multiple polylogarithm Hopf algebra.
416 citations
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TL;DR: In this paper, the fundamental group theory of algebraic geometry from the Kronecker point of view is presented, allowing one to treat on an equal footing the case of an algebraic variety in the usual sense and that of the ring of integers in a number field, for instance.
Abstract: Le texte pr\'esente les fondements d'une th\'eorie du groupe fondamental en G\'eom\'etrie Alg\'ebrique, dans le point de vue ``kroneckerien'' permettant de traiter sur le m\^eme pied le cas d'une vari\'et\'e alg\'ebrique au sens habituel, et celui d'un anneau des entiers d'un corps de nombres, par exemple.
The text presents the foundations of a theory of the fundamental group in Algebraic Geometry from the Kronecker point of view, allowing one to treat on an equal footing the case of an algebraic variety in the usual sense, and that of the ring of integers in a number field, for instance.
408 citations
TL;DR: In this article, the degree and height of polynomials in the integer ring ℤ over the integers have been derived for sparse polynomial systems, and the proof of these results relies heavily on the notion of local height of an affine variety defined over a number field.
Abstract: We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring ℤ. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo. We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application we derive an effective arithmetic Nullstellensatz for sparse polynomial systems. The proof of these results relies heavily on the notion of local height of an affine variety defined over a number field. We introduce this notion and study its basic properties.
181 citations
TL;DR: The method introduces a generalization of the ring of integers, called well-endowed rings, which possesses a very efficient parallel implementation of the basic (+,−,×) ring operations.
Abstract: It is shown that a probabilistic Turing acceptor or transducer running within space bound S can be simulated by a time S2 parallel machine and therefore by a space S2 deterministic machine. (Previous simulations ran in space S6.) In order to achieve these simulations, known algorithms are extended for the computation of determinants in small arithmetic parallel time to computations having small Boolean parallel time, and this development is applied to computing the completion of stochastic matrices. The method introduces a generalization of the ring of integers, called well-endowed rings. Such rings possess a very efficient parallel implementation of the basic (+,−,×) ring operations.
167 citations
01 Jan 1960
TL;DR: In this article, the intrinsic functorial characterizations of the functors Hom and X were obtained for homological algebra, and they were used to account in part for the distinguished role played by then in homology algebra.
Abstract: Our purpose here is to obtain intrinsic functorial characterizations of the functors Hom and X and thus to account in part for the distinguished role played by then in homological algebra. In all that follows, A, r are rings with unit, Z the ring of integers. The category of all F-A-bimodules with r operating on the left, A on the right, is denoted by r91lA, the category of left A-modules (F-modules) by A91(rM1Z), etc. All functors are assumed additive. We use throughout the terminology of [I].
160 citations