The Greatest Common Divisor: A Case Study for Program Extraction from Classical Proofs

A Jacobi Algorithm and Metric Theory for Greatest Common

Greatest common divisor algorithms are used to provide a natural motivation for considering a class of Jacobi-Perron algorithms which includes the original Jacobi algorithm. This work proves convergence and establishes metric properties for one of these algorithms. The proofs generalize to the larger class of algorithms. Full connections with the calculation of greatest common divisors will be ...

Reened Program Extraction from Classical Proofs: Some Case Studies

Orders at Infinity of Modular Forms with Heegner Divisors

Borcherds described the exponents a(n) in product expansions f = q Q∞ n=1(1−q ) of meromorphic modular forms with a Heegner divisor. His description does not directly give any information about h, the order of vanishing at infinity of f . We give p-adic formulas for h in terms of generalized traces given by sums over the zeroes and poles of f . Specializing to the case of the Hilbert class poly...

Torsion Points And

We study the behavior of the greatest common divisor of a k − 1 and b k − 1, where a, b are fixed integers or polynomials, and k varies. In the integer case, we conjecture that when a and b are multiplicatively independent and in addition a− 1 and b − 1 are coprime, then a k − 1 and b k − 1 are coprime infinitely often. In the polynomial case, we prove a strong version of this conjecture. To do...

On the pointfree counterpart of the local definition of classical continuous maps

The familiar classical result that a continuous map from a space $X$ to a space $Y$ can be defined by giving continuous maps $varphi_U: U to Y$ on each member $U$ of an open cover ${mathfrak C}$ of $X$ such that $varphi_Umid U cap V = varphi_V mid U cap V$ for all $U,V in {mathfrak C}$ was recently shown to have an exact analogue in pointfree topology, and the same was done for the familiar cla...