FACTORING POLYNOMIALS AND GRÖBNER BASES

Factoring polynomials is a central problem in computational algebra and number theory and is a basic routine in most computer algebra systems (e.g. Maple, Mathematica, Magma, etc). It has been extensively studied in the last few decades by many mathemati-cians and computer scientists. The main approaches include Berlekamp’s method (1967) based on the kernel of Frobenius map, Niederreiter’s method (1993) via an ordinary dif-ferential equation, Zassenhaus’s modular approach (1969), Lenstra, Lenstra and Lovasz’s lattice reduction (1982), and Gao’s method via a partial differential equation (2003). These methods and their recent improvements due to van Hoeij (2002) and Lecerf et al (2006– 2007) provide efficient algorithms that are widely used in practice today. This thesis studies two issues on polynomial factorization. One is to improve the efficiency of modular approach for factoring bivariate polynomials over finite fields. The usual modular approach first solves a modular linear equation (from Berlekamp’s equation or Niederreiter’s differential equation), then performs Hensel lifting of modular factors, and finally finds right combinations. An alternative method is presented in this thesis tha