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Author: Paul-Jean Cahen Publisher: American Mathematical Soc. ISBN: 0821803883 Category : Mathematics Languages : en Pages : 345
Book Description
Integer-valued polynomials on the ring of integers have been known for a long time and have been used in calculus. Polya and Ostrowski generalized this notion to rings of integers of number fields. More generally still, one may consider a domain $D$ and the polynomials (with coefficients in its quotient field) mapping $D$ into itself. They form a $D$-algebra - that is, a $D$-module with a ring structure. Appearing in a very natural fashion, this ring possesses quite a rich structure, and the very numerous questions it raises allow a thorough exploration of commutative algebra. Here is the first book devoted entirely to this topic. This book features: thorough reviews of many published works; self-contained text with complete proofs; and numerous exercises.
Author: Paul-Jean Cahen Publisher: American Mathematical Soc. ISBN: 0821803883 Category : Mathematics Languages : en Pages : 345
Book Description
Integer-valued polynomials on the ring of integers have been known for a long time and have been used in calculus. Polya and Ostrowski generalized this notion to rings of integers of number fields. More generally still, one may consider a domain $D$ and the polynomials (with coefficients in its quotient field) mapping $D$ into itself. They form a $D$-algebra - that is, a $D$-module with a ring structure. Appearing in a very natural fashion, this ring possesses quite a rich structure, and the very numerous questions it raises allow a thorough exploration of commutative algebra. Here is the first book devoted entirely to this topic. This book features: thorough reviews of many published works; self-contained text with complete proofs; and numerous exercises.
Author: Amitabh Kumer Halder Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
An integer-valued polynomial on a subset, S, of the set of integers, Z, is a polynomial f(x) 2 Q[x] such that f(S) Z. The collection, Int(S;Z), of such integer-valued polynomials forms a ring with many interesting properties. The concept of p-ordering and the associated p-sequence due to Bhargava [2] is used for nding integer-valued polynomials on any subset, S, of Z. In this thesis, we concentrate on extending the work of Keith Johnson and Kira Scheibelhut [14] for the case S = L, the Lucas numbers, where they work on integervalued polynomials on S = F, Fibonacci numbers. We also study integer-valued polynomials on the general 3 term recursion sequence, G, of integers for a given pair of initial values with some interesting properties. The results are well-agreed with those of [14].
Author: Marco Fontana Publisher: Springer ISBN: 3319658743 Category : Mathematics Languages : en Pages : 374
Book Description
This volume presents a collection of articles highlighting recent developments in commutative algebra and related non-commutative generalizations. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and non-Noetherian ring theory, module theory and integer-valued polynomials along with connections to algebraic number theory, algebraic geometry, topology and homological algebra. Most of the eighteen contributions are authored by attendees of the two conferences in commutative algebra that were held in the summer of 2016: “Recent Advances in Commutative Ring and Module Theory,” Bressanone, Italy; “Conference on Rings and Polynomials” Graz, Austria. There is also a small collection of invited articles authored by experts in the area who could not attend either of the conferences. Following the model of the talks given at these conferences, the volume contains a number of comprehensive survey papers along with related research articles featuring recent results that have not yet been published elsewhere.
Author: Nicholas J. Werner Publisher: ISBN: Category : Languages : en Pages : 148
Book Description
Abstract: When D is an integral domain with field of fractions K, the ring Int(D) of integer-valued polynomials over D is defined to be the set of all polynomials f(a) in K[x] such that f(a) is in D for all a in D. The goal of this dissertation is to extend the integer-valued polynomial construction to certain noncommutative rings. Specifically, for any ring R, we define the R-algebra RQ to be the set of elements of the form a + bi + cj + dk, where i, j, and k are the standard quaternion units satisfying the relations i^2 = j^2 = -1 and ij = k = -ji. When this is done with the integers Z, we obtain a noncommutative ring ZQ; when this is done with the rational numbers Q, we get a division ring QQ. Our main focus is on the construction and study of Int(ZQ), the set of integer-valued polynomials over ZQ. We also consider Int(R), where R is an overring of ZQ in QQ. In this treatise, we prove that for such an R, Int(R) has a ring structure and investigate elements, generating sets, and prime ideals of Int(R). The final chapter examines the idea of integer-valued polynomials on subsets of ZQ.
Author: Mario V. Marshall Publisher: ISBN: Category : Ideals (Algebra) Languages : en Pages : 46
Book Description
Let D be an integral domain and f a polynomial that is integer-valued on D. We prove that Int(f(D);D) has the Skolem Property and give a description of its spectrum. For certain discrete valuation domains we give a basis for the ring of integer-valued even polynomials. For these discrete valuation domains, we also give a series expansion of continuous integer-valued functions.
Author: Paul-Jean Cahen
Author: Paul-Jean Cahen
Author: Sanjai K. Gupta
Author: Amitabh Kumer Halder
Author: Todor Petkov Kitchev
Author: Marco Fontana
Author: Hantsje Zantema
Author: Reeve Mac Arthur Garrett
Author: Kira Scheibelhut
Author: Nicholas J. Werner
Author: Mario V. Marshall