Lecture 2: Newton's Method and P-adic Lifting.

Date: Apr. 5

Background Material and Further Resources

Reading

Topics

  1. Modular algorithm for univariate polynomial integral resultant computation
    1. res(A,B) mod p = res(A mod p, B mod p) provided p does not divide ldcf(A) or ldcf(B).
    2. L(res(A,B)) is dominated by nL(d), where deg(A), deg(B) <= n and |A|_1, |B|_1 <= d. This bound the number of mod p images, assuming p is a "word-sized" prime.
    3. The time for a single evaluation of A mod p is dominated by n*L(d). The total time spent on evaluation is nL(d)*nL(d).
    4. The modular resultant can be computed in time dominated by n^2, using a modification of the Euclidean algorithm (e.g. subresultant PRS). The total time spent computing modular resultants is nL(d)*n^2 = n^3L(d).
    5. The time for applying the CRT to the resulting modular resultants is (nL(d))^2.
    6. The total time is therefore dominated by n^2L(d)^2 + n^3L(d).
  2. Discrimants and Squarefree polynomials.
  3. Computing rational zeros be exhaustive search
  4. P-adic representation
  5. Computing rational zeros by p-adic lifting.

Programs

  1. Maple worksheet illustrating varous algorithms for computing the rational zeros of integral prolynomials (ratzero.mws).
  2. SACLIB programs:

Assignment

Created: Apr. 6, 2000 by jjohnson@mcs.drexel.edu