Integer Valued Polynomials

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.

This volume presents a multi-dimensional collection of articles highlighting recent developments in commutative algebra. 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 as well as integer-valued polynomials and functions. Specific topics include: · Homological dimensions of Prüfer-like rings · Quasi complete rings · Total graphs of rings · Properties of prime ideals over various rings · Bases for integer-valued polynomials · Boolean subrings · The portable property of domains · Probabilistic topics in Intn(D) · Closure operations in Zariski-Riemann spaces of valuation domains · Stability of domains · Non-Noetherian grade · Homotopy in integer-valued polynomials · Localizations of global properties of rings · Topics in integral closure · Monoids and submonoids of domains The book includes twenty articles written by many of the most prominent researchers in the field. Most contributions are authored by attendees of the conference in commutative algebra held at the Graz University of Technology in December 2012. There is also a small collection of invited articles authored by those who did not attend the conference. Following the model of the Graz conference, the volume contains a number of comprehensive survey articles along with related research articles featuring recent results that have not yet been published elsewhere.

This volume contains selected refereed papers based on lectures presented at the 'Fifth International Fez Conference on Commutative Algebra and Applications' that was held in Fez, Morocco in June 2008. The volume represents new trends and areas of classical research within the field, with contributions from many different countries. In addition, the volume has as a special focus the research and influence of Alain Bouvier on commutative algebra over the past thirty years.

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.

Results in the approximation of functions by polynomials with coefficients which are integers have been appearing since that of Pal in 1914. The body of results has grown to an extent which seems to justify this book. The intention here is to make these results as accessible as possible. The book addresses essentially two questions. The first is the question of what functions can be approximated by polynomials whose coefficients are integers and the second question is how well are they approximated (Jackson type theorems). For example, a continuous function $f$ on the interval $-1,1$ can be uniformly approximated by polynomials with integral coefficients if and only if it takes on integral values at $-1,0$ and $+1$ and the quantity $f(1)+f(0)$ is divisible by $2$. The results regarding the second question are very similar to the corresponding results regarding approximation by polynomials with arbitrary coefficients. In particular, nonuniform estimates in terms of the modules of continuity of the approximated function are obtained. Aside from the intrinsic interest to the pure mathematician, there is the likelihood of important applications to other areas of mathematics; for example, in the simulation of transcendental functions on computers. In most computers, fixed point arithmetic is faster than floating point arithmetic and it may be possible to take advantage of this fact in the evaluation of integral polynomials to create more efficient simulations. Another promising area for applications of this research is in the design of digital filters. A central step in the design procedure is the approximation of a desired system function by a polynomial or rational function. Since only finitely many binary digits of accuracy actually can be realized for the coefficients of these functions in any real filter the problem amounts (to within a scale factor) to approximation by polynomials or rational functions with integral coefficients.

"Presents the proceedings of the recently held Third International Conference on Commutative Ring Theory in Fez, Morocco. Details the latest developments in commutative algebra and related areas-featuring 26 original research articles and six survey articles on fundamental topics of current interest. Examines wide-ranging developments in commutative algebra, together with connections to algebraic number theory and algebraic geometry."

Covers its topic in greater depth than the typical standard books on polynomial algebra