Larger Projects

These are large projects that will improve the state of p-adics and local rings in Sage. For each I've tried to estimate the difficulty (which takes into account both the amount of code to be written and any technical challenges I currently forsee), and an opinion on how important the completion of this project is. Note that some of these projects depend on others; these dependencies can be found on the project pages and are NOT included in the estimated difficulty.

Each project page should eventually contain the following information, though many are still in progress:


  1. General extension rings

    • Goal -- the ability to create arbitrary absolute extensions of Qp and Zp, given a monic unramified polynomial f over Zp and an Eisenstein polynomial with coefficients in the extension of Zp defined by f.

    • Priority -- High

    • Difficulty -- Hard

  2. p-Adic polynomial precision

    • Goal -- Finally finish incorporating the changes of #6084, which conceptually splits the precision a polynomial is defined up to from an approximation of the coefficients.

    • Priority -- High

    • Difficulty -- Hard

  3. Hensel Lifting

    • Goal -- Define Hensel lifting for roots and factorizations of polynomials over Henselian rings.

    • Priority -- High

    • Difficulty -- Medium-Easy

  4. Precision for matrices and vectors over local rings

    • Goal -- Separate the precision for matrices and vectors from the approximation of their entries.

    • Priority -- High

    • Difficulty -- Hard

  5. Factoring in Sage for polynomials over local fields

    • Goal -- Implement algorithms for factoring polynomials over local fields. Define extensions of local fields using any polynomial.

    • Priority -- High

    • Difficulty -- Hard

  6. Completions of number fields

    • Goal -- Make the command "K.completion(p, absprec=30)" work when K is a number field and p is a prime of K

    • Priority -- Medium-High

    • Difficulty -- Medium-Easy

  7. More templates

    • Goal -- Implement unramified and Eistenstein extensions of Zp and Qp (the one-step extensions) using templates. Switch the other finite field classes to using templates.

    • Priority -- Medium-Low

    • Difficulty -- Medium

  8. Optimized Zp and Qp elements for the case that p^prec fits in a long

    • Goal -- Write new classes of elements, analogous to IntegerMod_int and IntegerMod_int64 for the case that self.unit or self.value fits into machine precision.

    • Priority -- Medium-Low

    • Difficulty -- Medium

  9. Cyclotomic extensions and Gauss normal basis

    • Goal -- Design a specialized p-adic extension type for cyclotomic extensions of Qp and Zp. Design element classes for unramified extensions that take advantage of a Gauss normal basis for faster arithmetic.

    • Priority -- Medium

    • Difficulty -- Medium-Hard

  10. Univariate power series over exact rings

    • Goal -- revamp univariate power series rings to hew more closely to the syntax of p-adics (relative and absolute precision; capped-rel, capped-abs and fixed-mod rings types; no more elements of infinite precision (except 0 in a capped-relative ring)

    • Priority -- Low

    • Difficulty -- Easy

  11. Univariate power series over p-adics

    • Goal -- Separate precision from approximation for power series. Include precision types that give information about the valuations for ALL coefficients (not just the first n). Write functions for solving p-adic differential equations.

    • Priority -- Medium-Low

    • Difficulty -- Medium-Hard

  12. Multivariate power series

    • Goal -- Continue to improve multivariate power series, building on the work in #1956. Create the ability to work with different precision types for multivariate power series and laurent series.

    • Priority -- Medium-Low

    • Difficulty -- Medium-Hard

  13. New matrix class for Zmod(n)

    • Goal -- Write a specialized class for matrices over Zmod(n) for n > 46341

    • Priority -- High

    • Difficulty -- Medium

  14. Add the database of p-adic fields to Sage

    • Goal -- Add this database of p-adic fields to Sage

    • Priority -- Medium

    • Difficulty -- Easy

  15. Witt vectors over a general ring

    • Goal -- Implement Witt vectors over a general ring

    • Priority -- Low

    • Difficulty -- Low

Smaller Projects

These are smaller projects that are hopefully more accessible to someone new to p-adics in Sage or with less time to devote. The difficulty range for these projects has been rescaled to account for the fact that they're all shorter than the projects above.

  1. Speed up inversion using Hensel lifting

    • Goal -- Optimize inversion in Zp, Qp, and extensions by finding the right choice of precision at which to use extended gcd and then lift with newton's method

    • Priority -- High

    • Difficulty -- Medium-Hard

  2. Teichmuller defining polynomial

    • Goal -- Create an option for Zq and Qq to generate their defining polynomial by lifting from GF(p)[x] to a factor of x^q-1 (as opposed to lifting naively)

    • Priority -- Medium

    • Difficulty -- Medium

  3. Exponentials and logarithms for elements of extension rings

    • Goal -- Fix implementation of p-adic exponential and logarithm maps for extensions.

    • Priority -- Medium-High

    • Difficulty -- Medium

  4. Square and nth roots

    • Goal -- Implementation of nth root for Zp, Qp and extensions

    • Priority -- High

    • Difficulty -- Medium

  5. More functions for Qp and Zp

    • Goal -- implement functions such as Artin-Hasse exponential and p-adic gamma function that exist in Magma.

    • Priority -- Low

    • Difficulty -- Medium

External Projects

These are external projects that will benefit p-adics and local rings in Sage.

  1. FLINT library for computing with p-adics

    • Goal -- write C code on top of Flint (1 or 2) for optimized computations in p-adic extensions.