ON IRREDUCIBLE FACTORS OF THE POLYNOMIAL f(x) - g(y)

Abstract

In this paper, all irreducible factors of bivariate polynomials of the form f(x) - g(y) over an arbitrary field are described. It also proves that the number of irreducible factors of f(x) - g(y) (counting multiplicities) does not exceed the greatest common divisor of the degrees of f(x) and g(y), yielding a well known result of Tverberg regarding the irreducibility of f(x) - g(y). It proves that if f(x) and g(y) are non-constant polynomials with coefficients in the field ℚ of rational numbers and deg f(x) is a prime number, then f(x) - g(y) is a product of at most two irreducible polynomials over ℚ. This contributes to a problem raised by Cassels which asks for the polynomials f, such that the polynomial [Formula: see text] is reducible.

On Difference Quotients of Chebyshev Polynomials

“…Cassels et al [2,3] studied factorizations of f (x) − g(y) as the polynomial in the pair of variables x and y. They also considered a trivial case when f and g are the same polynomial since f (x) − f (y) is divisible by x − y.…”

Roots of Difference Quotient Forms of Chebyshev Polynomials

“…Let and be polynomials in the single independent variables and with coefficients in the field of complex numbers. Cassels et al [1,2] studied factorizations of as the polynomial in the pair of variables and . They also considered a trivial case when and are the same polynomial since is divisible by .…”