Auxiliary Polynomials in Number Theory

David Masser is Emeritus Professor in the Department of Mathematics and Computer Science at the University of Basel, Switzerland. He started his career with Alan Baker, which gave him a grounding in modern transcendence theory and began his fascination with the method of auxiliary polynomials. His subsequent interest in applying the method to areas outside transcendence, which involved mainly problems of zero estimates, culminated in his works with Gisbert Wüstholz on isogeny and polarization estimates for abelian varieties, for which he was elected a Fellow of the Royal Society in 2005. This expertise proved beneficial in his more recent works with Umberto Zannier on problems of unlikely intersections, where zero estimates make a return appearance.

Introduction; 1. Prologue; 2. Irrationality I; 3. Irrationality II - Mahler's method; 4. Diophantine equations - Runge's method; 5. Irreducibility; 6. Elliptic curves - Stepanov's method; 7. Exponential sums; 8. Irrationality measures I - Mahler; 9. Integer-valued entire functions I - Pólya; 10. Integer-valued entire functions II - Gramain; 11. Transcendence I - Mahler; 12. Irrationality measures II - Thue; 13. Transcendence II - Hermite–Lindemann; 14. Heights; 15. Equidistribution - Bilu; 16. Height lower bounds - Dobrowolski; 17. Height upper bounds; 18. Counting - Bombieri–Pila; 19. Transcendence III - Gelfond–Schneider–Lang; 20. Elliptic functions; 21. Modular functions; 22. Algebraic independence; Appendix: Néron's square root; References; Index.