By David Hunt (auth.), Jonathan M. Borwein, Igor Shparlinski, Wadim Zudilin (eds.)

ISBN-10: 1461466415

ISBN-13: 9781461466413

ISBN-10: 1461466423

ISBN-13: 9781461466420

“Number concept and similar Fields” collects contributions according to the complaints of the "International quantity conception convention in reminiscence of Alf van der Poorten," hosted by way of CARMA and held March 12-16th 2012 on the college of Newcastle, Australia. the aim of the convention used to be to advertise quantity conception learn in Australia whereas commemorating the legacy of Alf van der Poorten, who had written over one hundred seventy papers regarding quantity idea and collaborated with dozens of researchers. The study articles and surveys offered during this ebook have been written by means of probably the most uncommon mathematicians within the box of quantity conception, and articles will comprise similar issues that target a number of the examine pursuits of Dr. van der Poorten.

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**Example text**

In165 , Alf, Marek Karpinski and Igor Shparlinski obtain new algorithms for testing whether a multivariate polynomial over a p-adic field given by a black box is identical to zero. Zero testing of polynomials in residue rings is also discussed. The results complement known results on zero testing of polynomials over the integers, the rationals and finite fields. ´ In218 , Alf and Akos Pint´er give a lower bound on the number of simple and distinct zeros of elements of a function field over an algebraically closed field, defined by linear recurrence sequences.

3 Quadratic Forms and the Composition of Binary Quadratic Forms The study of binary quadratic forms goes back at least to Gauss, who introduced both equivalence of forms and composition of forms of a given discriminant. With various authors Alf has made significant contributions both to the theory and to exposition of the ideas. In177 Alf begins with the “cute” observation due to Hendrik Lenstra, that 122 + 2 33 = 1233 which satisfies the Diophantine equation a2 + b2 = 10n a + b, which is a particular case of the more general Diophantine equation (*) x2 + y2 = n.

The authors state that very little of the material is original. Much of the material is of recent origin and adequate references are given. The level of the exposition is high and its beauty reflects the care and effort of the authors. A sequence of statements called ‘observations’ replaces the traditional sequence of theorems. However, there seems to be no difference between a theorem and an ‘observation’. Full proofs are often left to the reader, but adequate background is usually given to permit this.

### Number Theory and Related Fields: In Memory of Alf van der Poorten by David Hunt (auth.), Jonathan M. Borwein, Igor Shparlinski, Wadim Zudilin (eds.)

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