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== John McKay CHALLENGE system of polynomial equations ==

http://www.cargo.wlu.ca/McKay/

MSRI 2007 Parallel Computation Problem List

  1. [:msri07/threadsafety: Thread Safety of the SAGE Libraries]
  2. [:msri07/pthread_sagex: Add Pthread support to SageX]
  3. [:msri07/anlist: Implementation in SAGE parallel computation of elliptic curve a_p for all p up to some bound]
  4. [:msri07/matrixadd: Implementation in SAGE matrix ADDITION over the rational numbers (say) using a multithreaded approach.]
  5. [:msri07/pointcount: Brute force count points on a variety over a finite field in parallel.]

Parallel Implementations

For each of the following, make remarks about how specific practical implementable parallel algorithms could be used to enhance mathematics software libraries (e.g., SAGE).

  • Arithmetic in Global Commutative Rings
    • The ring {Z} of Integers

    • The ring {Q} of Rational Numbers

    • Arbitrary Precision Real (and Complex) Numbers
    • Univariate Polynomial Rings
    • Number Fields
    • Multivariate Polynomial Rings
  • Arithmetic in Local Commutative Rings
    • Univariate Power series rings
    • p-adic numbers

  • Linear Algebra
    • Arithmetic of Vectors
      • Addition
      • Scalar Multiplication
      • Vector times Matrix
    • Rational reconstruction of a matrix
    • Echelon form
      • Echelon form over Finite Field
      • Echelon form over {Q}

      • Echelon form over Cyclotomic Fields
      • Echelon form (Hermite form) over {Z}

    • Kernel
      • Kernel over Finite Field
      • Kernel over {Q}

      • Kernel over {Z}

    • Matrix multiplication
      • Matrix multiplication over Finite Fields
      • Matrix multiplication over {Z}

      • Matrix multiplication over Extensions of {Z}

  • Noncommutative Rings
  • Group Theory
  • Groebner Basis Computation
  • Elliptic Curves
    • Generic elliptic curve operations
      • Group Law
      • Invariants
      • Division Polynomials
    • Elliptic curves over finite fields
      • Order of the group E({{F}}_{p})

      • Order of the group E({{F}}_{q})

      • Order of a point
    • Elliptic curves over {{Q}} - part I

      • Birch and Swinnerton-Dyer Conjecture
      • Fourier coefficients
      • Canonical height of a point
      • Order of a point
      • Periods
      • Tate's algorithm
      • Conductor and Globally minimal model
      • CPS height bound
      • Torsion subgroup
      • Nagell-Lutz
      • An l-adic algorithm

      • Another l-adic algorithm

      • Mordell-Weil via 2-descent
      • Saturation
      • Heegner points
      • Heegner discriminants
      • Heegner Hypothesis
      • Heegner point index and height
    • Elliptic curves over {{Q}} - part II

      • Root number
      • Special values of L-series
      • Sha bound
      • Isogenies
      • Attributes of primes
      • p-adic height

      • Modular Degree
      • Modular Parameterization
  • Hyperelliptic Curves
  • Modular Forms
    • Presentation of spaces of modular symbols
    • Hecke operators on modular symbols
    • Decomposition of spaces under the Hecke operators
    • Trace formulas
  • Computation of tables
    • Elliptic curves
    • Modular forms
    • Number fields
  • Cryptography
  • Coding Theory
  • Constants, functions and numerical computation

John McKay CHALLENGE system of polynomial equations

http://www.cargo.wlu.ca/McKay/

msri07/problems (last edited 2008-11-14 13:42:04 by localhost)