zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Diophantine equations and power integral bases. New computational methods. (English) Zbl 1016.11059
Boston, MA: Birkhäuser. xviii, 184 p. $ 42.95; EUR 59.00; sFr. 79.00 (2002).
This book covers practically all results on index form equations for algebraic number fields obtained until now. The problem under consideration is usually the following: Given an order $O$ of an algebraic number field (usually an equation order, sometimes the maximal order) one is interested in calculating a basis for that order whose elements are successive powers of a single element, a so-called power basis. The index of the order generated by the successive powers of any element of $O$ is the value of a homogeneous multivariate polynomial of high degree, the so-called index form. Hence, the existence of a power basis is tantamount to the index form attaining the value $\pm 1$. Because of the number of variables ($n-1$ if the field degree is $n$) and the size of the degree of the index form $(n(n-1)/2)$ the task of solving index form equations is very difficult in general. The first three chapters of this book introduce to the problem and necessary auxiliary tools like Baker’s method (yielding bounds for potential solutions), the Davenport lemma (reducing those bounds substantially) and enumeration strategies to be employed for checking the remaining potential solutions. Also, other equations like Thue and norm form equations are used in the context of solving index form equations. In chapter 4 the author discusses various aspects of index form equations if the underlying field has special properties. Then chapters 5 to 8 present what is presently known about solving index forms in fields of degree 3 to 6. The last two chapters treat the case of relative index forms. When subfields exist the relative point of view is often superior. This is demonstrated by fields of higher degree, especially of degrees 8 and 9. The final chapter 11 contains more than 20 pages of tables. This book is a nice survey on the subject. The author has spent a considerable part of his mathematical life in studying index forms and this is in large parts a compilation of his achievements. The book is well written and is certainly to be recommended to the audience it addresses: advanced undergraduates, graduates and researchers.

11Y50Computer solution of Diophantine equations
11D57Multiplicative and norm form diophantine equations
11-02Research monographs (number theory)
11D59Thue-Mahler equations
11R33Integral representations related to algebraic numbers
11-04Machine computation, programs (number theory)