Auxiliary polynomials in number theory (Q2814160)

Several features of this book are original. First of all: the topic. The author collects a surprisingly large number of different instances, mainly (but not only) in number theory, where one of the main character is an auxiliary polynomial. The existence of such an auxiliary polynomial is most often deduced from an application of Dirichlet's box Principle. It is a sort of miracle that such a simple minded argument is able to lead to so many different deep achievements.NEWLINENEWLINENext, thanks to the unique style of the author, this book offers a pleasant reading; a number of nice jokes enable the reader to have a good time while learning high level serious mathematic. However this book, which is extremely carefully written, is not likely to enable the author to reach his goal when he writes in the introduction: \textit{I was once thanked in print by a non-English author for ``teaching him mistakes''. I hope to be able to pass on these skills to my readers}.NEWLINENEWLINEEach chapter usually starts with questions involving easy explicit numerical examples. The number 1948 (birth year of the author) occurs more often than other values. Often, these questions are down to earth, it is amazing to see how they lead to such far reaching developments. The solutions are given by using what at first sight might seem to be ad hoc arguments. Then, the author explains how to remove some superfluous assumptions, to bypass a few snags and to exploit the underlying ideas in order to reach very general results, by introducing a suitable auxiliary polynomial, as expected. In the prologue (Chapter 1), two questions are raised, which are not of a Diophantine nature, the answer to which involves an auxiliary polynomial. The first one is: given a polynomial \(f\) in one variable \(t\), does there exist a multiple of \(f\) which is a polynomial involving only powers \(t^p\) with \(p\) prime? The second one is: given two polynomials \(f\) and \(g\) in one variable \(t\), does there exist a non zero polynomial in two variables \(P\) such that \(P(f,g)=0\)? Elimination theory with the resultant (explained in Chapter 5) is not the final word.NEWLINENEWLINEThere is no surprise to see chapters devoted to irrationality (Chapters 2, 3), irrationality measures (Chapters 8, 12), transcendence (Chapters 11, 13, 19), algebraic independence (Chapter 22). Even in these classical subjects, in each of these chapters, there is some new material. The author revisits these topics and discusses a number of issues which are not commonly addressed. Just to take one example, when he discusses Siegel's Lemma, he includes in the exercises (8.3 and 8.17) some results on the optimality of the statement. Most often (unless explicitly stated), the results are proved with all necessary details; the author provides carefully written proofs of all results, including complete proofs of the auxiliary lemmas. Two chapters (9: Polya and 10: Gramain) deal with integer valued entire functions.NEWLINENEWLINEThe introduction to the method of Thue in Chapter 12 is specially interesting. Using numerical analysis, the author starts with Newton's method and expands the ideas up to the proof of Thue's result in the general case. He does not pretend that this was the path originally followed by Axel Thue, but this is not the point.NEWLINENEWLINEOne of the first mathematicians to have used auxiliary polynomials is Runge. In Chapter 4 the author starts with some easy Diophantine equations; he proves that there are only finitely many solutions in integers \(x\), \(y\) to the Diophantine equations \(y(y^2-x^2)=x\), \(y(y^3-x^3)=x\) and \(y(y^3-2x^3)=x\). After such special cases, he is ready to prove a general result by means of Runge's method.NEWLINENEWLINEChapter 5, ``Irreducibility'', refers to Hilbert's Irreducibility Theorem (HIT, including a Strong version of it) and expands a method due to V. G. Sprindzuck. The main result of this chapter is new. As an example the author shows that there are only finitely many \(y\in{\mathbb{Z}}\) for which the polynomial \(X(X^3-2y^3)-y\) is reducible. This chapter includes a nice introduction, of independent interest, to elimination theory. The author does not hesitate to write \(13\times 13\) determinants in order to show clearly how the general case looks like. The surprisingly efficient elementary method of Stepanov to obtain results in the direction of Weil's conjectures is explained in Chapter 6. The author shows that for a cubic polynomial \(C\) in \({\mathbb{F}}_p[X]\), the number \(N\) of \((x,y)\in{\mathbb{F}}_p^2\) with \(y^2=C(x)\) satisfies \(|N-p|\leq 8\sqrt{p}\). A classical result of Hasse gives a sharper upper bound; but here, nothing is needed from the theory of elliptic curves.NEWLINENEWLINESeveral elements of Stepanov's method were used by R. Heath Brown when he succeeded to give the first nontrivial upper bounds for exponential sums like NEWLINE\[NEWLINE H=\sum_{k=1}^p \exp\left(\frac{2i\pi k^p}{p^2}\right) NEWLINE\]NEWLINE introduced by Heilbronn. Here, \(p\) is an odd prime. In Chapter 7 the estimate \(|H|\leq 4 p^{11/12}\) is proved.NEWLINENEWLINEHeights play an important role in Diophantine problems; in Chapter 14 is proved the upper bound \({\mathrm{H}}(\alpha)\leq 8\) when \(\alpha\not = 0,1\) is an algebraic number such that \(\alpha\) and \(1-\alpha\) are multiplicatively dependent.NEWLINENEWLINEChapter 15 is devoted to Bilu's equidistribution theorem: the number \(n\) of conjugates in a sector of angle \(\theta\) based at the origin of an algebraic number of degree \(d\) and absolute logarithmic height \(h\) is asymptotically \(\frac{\theta}{2\pi} d\) as \(h\) tends to infinity. An explicit upper bound for \(n-\frac{\theta}{2\pi} d\) in terms of \(d\) and \(h\) is proved, using the Erdős-Turan Theorem and an auxiliary polynomial introduced by Mignotte.NEWLINENEWLINEDobrowolski's important contribution to Lehmer's problem is the topic of Chapter 16. The author uses some ideas of the previous chapter for proving a weak form of Dobrowolski's result, namely for any \(\kappa>1\) there is a constant \(c=c(\kappa)>1\) such that the absolute logarithmic height of a nonzero algebraic number of degree \(d\) which is not a root of unity satisfies \({\mathrm{h}}(\alpha)\geq c/d^\kappa\). The method which is developed in this chapter is suitable for generalizations (F. Amoroso and S. David).NEWLINENEWLINEIn Chapter 17, is proved the upper bound \({\mathrm{H}}(\alpha)\leq 10^{120}\) when \(\alpha\) is an algebraic number such that \(\alpha^n+(1-\alpha)^n=1\) for some \(n\geq 2\). For this example as well as other ones, the author quotes and gives references when stronger estimates are available, he selects the statements in order to give a neat proof where the arguments are crystal clear.NEWLINENEWLINEChapter 18 deals with a method originated by E. Bombieri and J. Pila for counting rational points on analytic curves. This topic is at the origin of a fashionable and fruitful research field which includes a number of deep developments (related to the Zilber-Pink Conjecture, unlikely intersections, o-minimality). Here, the author offers a simple introduction by proving an upper bound for the number of rational numbers with denominator dividing \(n\) such that a given transcendental analytic function on an open set containing \([0,1]\) can take rational values with denominator dividing \(n\). After that, he produces an analog of the Bombieri-Pila result for \({\mathbb{Q}}(i)\).NEWLINENEWLINEChapter 20 on elliptic functions (36 pages) contains a lot of material (this is a subject of which the author is a specialist [Elliptic functions and transcendence. Berlin-Heidelberg-New York: Springer-Verlag (1975; Zbl 0312.10023)]). The main classical transcendence results are of course there, but not only; for instance the last 7 exercises of this chapter give explicit quasi projective embeddings of the algebraic groups which are extensions of an elliptic curve by either the additive or the multiplicative group.NEWLINENEWLINEThe last two chapters are more advanced, but most of the necessary tools have been introduced earlier, so the proofs, which are given with some detail, are not so difficult to digest. Chapter 21, ``Modular functions'', includes the so-called Théorème Stéphanois on the transcendance of the value \(J(q)\) of the modular function \(J\) for \(q\) algebraic with \(0<|q|<1\). Chapter 22, ``Algebraic independence'', deals with the method introduced by A.O. Gel'fond for proving the algebraic independence of \(2^{\root 3 \of{2}}\) and \(2^{\root 3 \of{4}}\) for instance; it also gives the state of the art on this topic.NEWLINENEWLINEAn appendix, ``Néron square root'', deals with specific examples of upper bounds for points on elliptic curves, for which the author offers an interesting and original discussion on their optimality. It arises from a joint work of the author with F. Amoroso and U. Zannier.NEWLINENEWLINEAltogether, there are 709 exercises, including the two last ones: \textit{A.33 Google ``Catalan Numbers''} and \textit{A.34 That's enough auxiliary polynomials. Ed.}NEWLINENEWLINEThese exercises contain a large amount of material. In each chapter they are split into easy ones and difficult ones - several of them among the difficult ones end with: \textit{I don't know} or \textit{no-one knows}. Many open problems and research topics can be found in this book.NEWLINENEWLINEThis book includes a large number of statements, proofs, ideas, problem which will be of great value for the specialists; but it should interest also any mathematician, including students, who wish to expand their knowledge and see a superb example of a topic having a surprisingly large number of different applications in several directions.