Factorization of polynomials

Abstract

In 1975-1976, A. Ostrowski published a deep, general and extensive research on multiplication and factorization of polynomials based principally on the generalized notions of highest and lowest terms of a polynomial, called general mappings, lambda, of a polynomial into an extreme aggregate of its terms. The first part of Ostrowski's works discusses all possible orderings, omega, in the set of products of powers of independent variables under a very general set of postulates, which contains the usual lexicographical principle as a special case. A one-to-one correspondence between omega and lambda is established and all realizations of postulates defining omega are investigated using the ideas of weight functions and the baric polyhedron of a polynomial. The second part of Ostrowski's works contains applications of the results in the first part to the problem of irreducibility of polynomials. In particular, the cases of 2 and 3 term polynomials are completely determined, while that of 4 term polynomials a complete discussion is only carried out for the case of the baria polygon being a triangle. In this thesis, we carry out a comprehensive study on the above-mentioned works of Ostrowski by analyzing, clarifying, proving and supplying relevant examples to all his results. In addition, the irreducibility of 4 term polynomials whose baric polygon is a quadrangle is investigated and complete results for some large classes of polynomials are obtained.