Discriminant equations in Diophantine number theory (Q2825402)

The area of \textit{Diophantine equations} is one of the most ancient and interesting branches of number theory. The \textit{discriminant equations} is an important class of Diophantine equations which are closely related to algebraic number theory, Diophantine approximation and Diophantine geometry. Discriminant equations include equations of the form \(D(f) = \delta\) and \(D(F) = \delta\) to be solved in polynomials \(f(X)\) or in binary forms \(F(X,Y)\) with coefficients in an integral domain \(A\), where \(\delta \in A\setminus \{0\}\) and \(D(f)\), \(D(F)\) are the discriminants of \(f(X)\) and \(F(X,Y)\), respectively. This book is the first comprehensive and up-to-date treatment of discriminant equations and their applications. It describes important developments in this topic during the last 40 years. It also contains new results which are not yet published. The book is aimed at anyone with knowledge of basic algebra and algebraic number theory. Its content is divided into three parts and may be summarized as follows.NEWLINENEWLINEPart I contains four chapters and consists of preliminary material. Chapter 1 gives a detailed overview of finite étale algebras over fields which play an important role in this monograph. Chapters 2 and 3 contains the most basic facts about Dedekind domains and algebraic number fields, respectively, used in this monograph. In Chapter 4, an overview without proofs of effective and ineffective results on unit equations is given, since the results on discriminant equations to be presented are based on them.NEWLINENEWLINEPart II discusses the topic of monic polynomials and integral elements of given discriminant, and monogenic orders. It consists of Chapters 5--11. NEWLINENEWLINEChapter 5 starts with some basic theory on discriminant equations for monic polynomials and integral elements of finite étale algebras, and discriminant form and index form equations. It presents the basic finiteness theorems due to the second author [C. R. Math. Acad. Sci., Soc. R. Can. 4, 75--80 (1982; Zbl 0507.12001)] for discriminant equations for monic polynomials over finitely generated domains \(A\) over \(\mathbb Z\), and as a consequence finiteness results for discriminant equations in integral elements, for discriminant form and index form equations, and for monogenic orders are given. NEWLINENEWLINEChapter 6 contains the second author's effective finiteness results [Acta Arith. 23, 419--426 (1973; Zbl 0269.12001); Publ. Math. 21, 125--144 (1974; Zbl 0303.12001); ibid. 23, 141--165 (1976; Zbl 0354.10041)] for discriminant equations of the form \(D(f) = D\) in monic polynomials \(f \in \mathbb Z[x]\) of degree \(n\), where \(D(f)\) is the discriminant of \(f\), and of related discriminant form and index form equations with the best explicit bounds to date for the sizes of solutions. An important application of these results is the design of an algorithm that decides whether an order of an algebraic number field \(K\) is of the form \(\mathbb Z[\alpha]\) and determines all \(\alpha\) with this property. NEWLINENEWLINEChapter 7 deals with the resolution of discriminant form equations of the form \(D_{L/\mathbb Q}(\omega_2x_2+\cdots +\omega_n x_n) = D\), in \(x_2,\ldots,x_n\in \mathbb Z\), where \(D\) is a given non-zero integer, \(L\) an algebraic number field of degree \(n \geq 3\), \(\{1,\omega_2,\ldots,\omega_n\}\) an integral basis of \(L\) and \(D_{L/\mathbb Q}\) the discriminant of \(L\) over \(\mathbb Q\). Furthermore, related index form equations are studied. First, the general approach involving unit equations is presented, and an algorithm for the resolution of such equations over quintic field with some numerical examples. Next, the resolution of the above equations is studied in cubic, quartic and in some other fields via Thue equations. Finally, a survey is given about the solvability of index equations in various special number fields. NEWLINENEWLINEChapter 8 generalizes the results of Chapter 6 to the number field cases when the ground ring is the ring of \(S\)-integers of a number field \(K\), where \(S\) is any finite of places in \(K\) containing all the infinite ones. The main result of the chapter are effective finiteness theorems in explicit form on discriminant equations of monic polynomials, and in integral elements of a finite étale \(K\)-algebra. The results concerning étale \(K\)-algebras are new. Several applications are presented to integral elements in a number field, to algebraic numbers of given degree, to index equations, monogenic orders, and about arithmetical properties of discriminants and indices of integral elements. NEWLINENEWLINEChapter 9 is devoted to the computation of uniform upper bounds for the number of equivalence classes of solutions to discriminant equations, both in monic polynomials with coefficients in the ring of \(S\)-integers of an algebraic number field \(K\), and in integral elements from an étale \(K\)-algebra. Furthermore, it is proved that every finite étale \(\mathbb Q\)-algebra has only finitely many three times monogenic orders. Many results of this chapter are new. NEWLINENEWLINEChapter 10 generalizes some effective finiteness results from Chapter 8. It studies discriminant equations in monic polynomials with coefficients from an arbitrary, effectively given, integrally closed and finitely generated integral domain \(A\) over \(\mathbb Z\) of characteristic 0 and in elements of an \(A\)-order of a finite étale \(K\)-algebra, where \(K\) is the quotient field of \(A\). By the results of Chapter 5, these equations have only finitely many \(A\)-strong equivalent classes of solutions. In Chapter 10, it is proved that a full set of representative can be determined effectively. NEWLINENEWLINEIn Chapter 11, two interesting applications of the results from Chapters 6 and 8 are given. The first one is a method to decide whether an algebraic number field \(K\) has a canonical number system, and if so determines the bases of all canonical number systems of \(K\). The second one is a method that effectively given an \(O_S\)-order \(\mathcal D\) of a finite étale algebra over an algebraic number field computes the minimal number \(r\) of generators of \(\mathcal D\) as an \(O_S\)-algebra, and also elements \(\alpha_1,\ldots,\alpha_r\) such that \({\mathcal D} = O_S[\alpha_1,\ldots,\alpha_r]\).NEWLINENEWLINEPart III investigates discriminant equations of the form \(D(F)= \delta\), in binary forms \(F \in A[X,Y]\), where \(A\) is a given integral domain \(\delta \in A\setminus \{0\}\) and \(D(F)\) is the discriminant of \(F\). It consists of Chapters 12--18. NEWLINENEWLINEChapter 12 gives an overview of basic finiteness theorems for binary forms of given discriminant or given invariant order. The following chapters contains effective and quantitative versions of these results. NEWLINENEWLINEIn Chapter 13, an extension of the reduction theory of Hermite (1851) and Julia (1917) to binary forms with coefficients in the ring of \(S\)-integers of a number field is presented. NEWLINENEWLINEChapter 14 is devoted to the proof of the fact that every binary form \(F \in O_S[X,Y]\) of degree \(n \geq 2\) with non-zero discriminant \(D(F)\) is \(\mathrm{GL}(2,O_S)\)-equivalent to a binary form \(F^*\) with height less than an explicit constant \(C\) depending only on \(n\), \(K\), \(S\) and the norm \(N_S(D(F))\). This result was proved by the authors in [Compos. Math. 79, No. 2, 169--204 (1991; Zbl 0746.11020)]. The proof described here is an alternative proof of the one presented in 1991 where the constants were effective but no explicit. NEWLINENEWLINEIn Chapter 15, the following conjecture is stated: Every binary form \(F\in \mathbb Z[X,Y]\) of degree \( n\geq 4\) with nonzero discriminant \(D(F)\) is \(\mathrm{GL}(2,\mathbb Z)\)-equivalent to a binary form \(F^*\) of height \(H(F^*) \leq C_1(n) |D(F)|^{C_2(n)}\), where \(C_1(n)\) and \(C_2(n)\) depend only on \(n\). Furthermore, the authors prove a result of the above type where the exponent \(C_2\) depends only on \(n\) and is effectively computable, whereas \(C_1\) depends only on \(n\) and the splitting field of \(F\), and cannot be effectively computed by the method of the proof. NEWLINENEWLINEChapter 16 introduces the invariant order associated with a binary form and some basic properties are proved. Further, a proof of the classical result of Delone and Faddeev about the relation between binary cubic forms and orders of rank 3 is given. NEWLINENEWLINEIn Chapter 17, explicit upper bounds are given for the number of \(\mathrm{GL}(2,O_S)\)-equivalent classes of binary forms \(F\in O_S[X,Y]\) with given invariant order and for the number of \(\mathrm{GL}(2,O_S)\)-equivalent classes of binary forms \(F\in O_S[X,Y]\) with given non-zero discriminant and a given splitting field. Furthermore, binary forms defined over an integrally closed integral domain \(A \supset \mathbb Z\) which are finitely generated as \(\mathbb Z\)-algebras are considered, and it is proved that there are only finitely many \(\mathrm{GL}(2,A)\)-equivalence classes of binary forms \(F \in A[X,Y]\) with given invariant order. NEWLINENEWLINEFinally, in Chapter 18 two applications of results of Chapters 8, 14 and 15 are given. The first application deals with the problem of giving good lower bounds for the difference between the zeros of a polynomial and the second is an effective proof of the Shafarevich conjecture in the case of hyperelliptic curves.NEWLINENEWLINEIn conclusion, the book is very interesting and well written. It contains the motivational material necessary for those entering in the field of discriminant equations and succeeds to bring the reader to the forefront of research. Graduates and researchers in the field of number theory will find it a very valuable resource.