Dorin Andrica

An Introduction to Diophantine Equations

In this article we will only touch on a few tiny parts of the fiel d of linear Diophantine equations. Some of the tools introduced, however, will be useful in many other parts of the subject.

Recent Results on the Size of Critical Sets

In the first part of this survey we review some special cases of \({\varphi }_{\mathcal{F}}\)-category of a pair (M, N) of manifolds such as φ-category, Morse-Smale characteristic, and Morse-Smale characteristic for circular functions. Section 2 presents examples of pairs with finite φ, and Sect. 3 provides lower estimates for the size of the critical sets in terms of topological dimension. We employ the cardinality when the manifolds admit maps with finitely many critical points and the topological dimension when no such maps exist.

A review of the book Quadratic Diophantine Equations

and many other articles on this topic, professors Titu Andreescu (School of Natural Sciences and Mathematics, University of Texas at Dallas, USA) and Dorin Andrica (Faculty of Mathematics and Computer Science, ”BabeşBolyai” University, Cluj-Napoca, Romania) delight readers again with Quadratic Diophantine Equations, Springer (2015), a true monograph on this important area of Number Theory. The first chapter of the book (entitled ”Why Quadratic Diophantine Equations?”) is one of preliminaries, a kind background for the next chapters. The authors recall Thue’s theorem, Hilbert’s tenth problem, Hecke operators, Hecke groups, continued fractions, computing self-intersections of closed geodesics, standard homogeneous Eisenstein manifolds and Diophantine equations, etc. The second chapter (entitled ”Continued Fractions, Diophantine Approximation, and Quadratic Rings”) introduces the readers in the theory of continued fractions, theory used in the later chapters, in order to find the minimal solution of a Pell’s equation and not only. The chapter ends with a paragraph about units and norms in quadratic rings. After these two chapters, the readers have all the theoretical support necessary to solve any type of Pell’s equations or Diophantine equations reducible by Pell’s equations.

General Pell’s Equation

This chapter gives the general theory and useful algorithms to find positive integer solutions (x, y) to general Pell’s equation (4.1.1), where D is a nonsquare positive integer, and N a nonzero integer.

Continued Fractions, Diophantine Approximation, and Quadratic Rings

The main goal of this chapter is to lay out basic concepts needed in our study in Diophantine Analysis. The first section contains fundamental results pertaining to continued fractions, some without proofs. The Theory of Continued Fractions is not new but it plays a growing role in contemporary mathematics.

Why Quadratic Diophantine Equations

In order to motivate the study of quadratic type equations, in this chapter we present several problems from various mathematical disciplines leading to such equations. The diversity of the arguments to follow underlines the importance of this subject.

Diophantine Representations of Some Sequences

In 1900, David Hilbert asked for an algorithm to decide whether a given Diophantine equation is solvable or not and put this problem tenth in his famous list of 23.

Pell’s Equation

Euler, after a cursory reading of Wallis’s Opera Mathematica, mistakenly attributed the first serious study of nontrivial solutions to equations of the form \(x^{2} - Dy^{2} = 1\), where x ≠ 1 and y ≠ 0, to John Pell. However, there is no evidence that Pell, who taught at the University of Amsterdam, had ever considered solving such equations. They should be probably called Fermat’s equations, since it was Fermat who first investigated properties of nontrivial solutions of such equations.

Equations Reducible to Pell’s Type Equations

An interesting problem concerning the Pell’s equation \(u^{2} - Dv^{2} = 1\) is to study when the second component of a solution (u, v) is a perfect square.

Aspects of Global Analysis of Circle-Valued Mappings

We deal with the minimum number of critical points of circular functions with respect to two different classes of functions. The first one is the whole class of smooth circular functions and, in this case, the minimum number is the so called circular \(\varphi\) -category of the involved manifold. The second class consists of all smooth circular Morse functions, and the minimum number is the so called circular Morse–Smale characteristic of the manifold. The investigations we perform here for the two circular concepts are being studied in relation with their real counterparts. In this respect, we first evaluate the circular \(\varphi\)-category of several particular manifolds. In Sect. 5, of more survey flavor, we deal with the computation of the circular Morse–Smale characteristic of closed surfaces. Section 6 provides an upper bound for the Morse–Smale characteristic in terms of a new characteristic derived from the family of circular Morse functions having both a critical point of index 0 and a critical point of index n. The minimum number of critical points for real or circle valued Morse functions on a closed orientable surface is the minimum characteristic number of suitable embeddings of the surface in \(\mathbb{R}^{3}\) with respect to some involutive distributions. In the last section we obtain a lower and an upper bound for the minimum characteristic number of the embedded closed surfaces in the first Heisenberg group with respect to its noninvolutive horizontal distribution.