Overview

Magma’s built-in types for p-adic computation (FldPad, RngPad, RngPadRes, etc) are all approximate in the sense that their elements are computed immediately to some precision, and therefore an element really corresponds to an equivalence class of genuine p-adic elements.

In this package, we provide a new representation of p-adics which is exact in the sense that a p-adic number is represented to infinite precision. This is achieved by having the representation be lazy in the sense that in addition to each element being represented by a finite-precision approximation, it carries around a function which allows it to update its approximation arbitrarily high.

Advantages of this approach are:

• The user is freed from the burden of choosing a precision at the beginning of a computation; often this precision will be insufficient and the user must increase it manually until no precision errors occur.
• All computations are performed to the minimal precision necessary to compute an output, sometimes giving a gain in speed.

Disadvantages are:

• Dependency tracking overhead: Elements now depend on each other, and tracking these dependencies can be computationally expensive.
• Memory overhead: Every (exact) intermediate value in a computation which the answer depends on will be kept in memory, so more memory is required.

Therefore the main use-cases of this package are:

• Performing computations interactively in Magma; the overheads will be unnoticed.
• Implementing high-level algorithms in Magma; that is, ones where the main operations are themselves expensive enough that the overheads are comparatively small.