Math 788M: Computational Number Theory

A computational introduction to number theory and algebra

Explicit resolutions of cubic cusp singularities

Resolutions of cusp singularities are crucial to many techniques in computational number theory, and therefore finding explicit resolutions of these singularities has been the focus of a great deal of research. This paper presents an implementation of a sequence of algorithms leading to explicit resolutions of cusp singularities arising from totally real cubic number fields. As an example, the ...

The hexagonal versus the square lattice

Schmutz Schaller’s conjecture regarding the lengths of the hexagonal versus the lengths of the square lattice is shown to be true. The proof makes use of results from (computational) prime number theory. Using an identity due to Selberg, it is shown that, in principle, the conjecture can be resolved without using computational prime number theory. By our approach, however, this would require a ...

Balancing domain decomposition for problems with large jumps in coefficients

The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introduced recently by the first-named author is used to develop condition number bounds for conforming linear ele...

Parallel integer relation detection: Techniques and applications

Let {x1, x2, · · · , xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 + a2x2 + · · · + anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, d...