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Computational excursions in analysis and number theory. (English) Zbl 1020.12001
CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. 10. New York, NY: Springer. x, 220 p. EUR 69.95/net; sFr 116.00; £49.00; \$ 69.95 (2002).

The author considers a series of problems about topics on the boundary between analysis and number theory. They are all related to the properties of certain polynomials, they are at least 40 years old, and none of them is completely solved. They are also suited for computational experiments. The book contains a variety of techniques from several branches of mathematics to enable the reader to carry out his own research.

In the introduction various sets of polynomials are defined as well as several measures. This is followed by a few results from complex analysis. Then the author gives a list of 17 open problems around which the later chapters are centered. (Those are again listed in Appendix D.)

In Chapter 2 (shortly) and Appendix B (at length) lattice basis reduction techniques and integer relation finding methods are discussed. Then the applicability of these methods for finding polynomials with prescribed properties of their coefficients is developed.

The next chapters contain Pisot and Salem numbers and special series of polynomials. They are followed by a discusion of the location of the zeros and Diophantine approximation of those. Then several open problems are considered:

– The Integer Chebyshev Problem of finding non-zero polynomials of $ℤ\left[x\right]$ of smallest supremum norm in the interval $\left[0,1\right]$ and analyzing the asymptotic behaviour when the degree tends to infinity.

– The Prouhet-Tarry-Escott Problem of finding polynomials of $ℤ\left[x\right]$ divisible by ${\left(x-1\right)}^{n}$ and minimal sum of the absolute values of the coefficients.

– The Erdős-Szekeres Problem of minimizing

${∥\prod _{i=1}^{n}\left(1-{x}^{{a}_{i}}\right)∥}_{\infty }$

for ${a}_{i}\in ℕ$ and proving that these minima grow faster than ${n}^{\beta }$ for any positive constant $\beta$.

– The Littlewood Problem of finding polynomials with coefficients in $\left\{±1\right\}$ with smallest ${L}_{p}$-norm on the unit disk.

Additional chapters deal with a variant of Waring’s problem, Barker polynomials and Golay pairs, and spectra of values of classes of polynomials evaluated at a fixed number $q$. Besides the aforementioned Appendices B and D, Appendix A contains a compendium of inequalities and Appendix C on explicit merit factor formulae. Each chapter ends with lists of computational and research problems and selected additional references.

The treatment of the various topics is very concise. Clearly, the book is written by one of the leading experts in this area of mathematics.

Reviewer: M.Pohst (Berlin)
MSC:
 12-02 Research monographs (field theory) 11-02 Research monographs (number theory) 11R09 Polynomials over global fields 11C08 Polynomials (number theory) 11Y99 Computational number theory 12D05 Factorization of real or complex polynomials 12D10 Algebraic theorems of location of zeros of polynomials over R or C 12E05 Polynomials over general fields 12Y05 Computational aspects of field theory and polynomials 42C05 General theory of orthogonal functions and polynomials