The Euclidean algorithm, lotuses and singularities

Abstract

The anthyphairetic process leads from a pair (a,b) of coprime positive integers to the pair (1,1) by successive subtractions of the smaller number from the bigger one. This process, which is a slow version of Euclid's algorithm applied to the pair (a,b), corresponds naturally to the process of successive blowups leading to the minimal embedded resolution of the plane curve defined by y^a - x^b = 0. This blowup process may be represented graphically by a special two-dimensional simplicial complex called a lotus. This allows to localize the various numbers appearing either during the anthyphairetic process or during the Euclidean algorithm at precise positions inside the lotus. In this introductory article, I recall first the construction of this lotus starting from the sequence of quotients generated by the Euclidean algorithm. I present then an alternative way of constructing it directly from the sequence of pairs of coprime integers generated by the anthyphairetic process, using what I call anthyphairetic rectangles. I conclude by explaining how to reconstruct from a lotus the corresponding sequence of pairs of coprime integers. This is a simple illustration of the way lotuses may serve as computational architectures.