An Extension of the Irreducibility Criteria of Ehrenfeucht and Tverberg

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

“…In this paper we show that there are best possible decompositions f = f (1) • f (2) , g = g (1) • g (2) over K, which are unique up to linear equivalence, such that a factor P (x, y) of f (x) − g(y) is irreducible over K if and only if P (x, y) can be written as P (1) (f (2) (x), g (2) (y)) for some irreducible factor P (1) (1) (y); moreover f (1) (x) and g (1) (y) are of the same degree, provided these degrees are not divisible by the characteristic of K. Indeed, we prove:…”

“…. , x n ] where K is an algebraically closed field of characteristic p > 0 -by proving that for n ≥ 3, F is reducible over K if and only if F = g 1 (x 1 ) p + · · · + g n (x n ) p + a[g 1 (x 1 ) + · · · + g n (x n )] for some a in K and g i (x i ) in K[x i ] (see [1;13,Cor. 2]).…”

“…In what follows, when a polynomial f (x) with coefficients in a field K is written as a composition f (1) (2) is called a decomposition of f over K and the polynomial f (1) (respectively f (2) ) will be referred to as a left (respectively right) decomposition factor of f .…”

“…Let f (x) and g(y) be polynomials of positive degrees in different variables x, y such that f (x) − g(y) is reducible over K. If f = f (1) • f (2) and g = g (1) • g (2) are any decompositions of f, g over K, then it is clear that a proper factor P (1) (1) (y) over K gives rise to a proper factor P (x, y) = P (1) (f (2) (x), g (2) (y)) of f (x) − g(y). In this paper we show that there are best possible decompositions f = f (1) • f (2) , g = g (1) • g (2) over K, which are unique up to linear equivalence, such that a factor P (x, y) of f (x) − g(y) is irreducible over K if and only if P (x, y) can be written as P (1) (f (2) (x), g (2) (y)) for some irreducible factor P (1) (1) (y); moreover f (1) (x) and g (1) (y) are of the same degree, provided these degrees are not divisible by the characteristic of K. Indeed, we prove:…”

“…Then there exist decompositions f = f (1) • f (2) , g = g (1) • g (2) of f, g respectively over K, satisfying the following properties: (2) are any decompositions of f, g over K for which properties (i) and (ii) are satisfied with f (i) , g (i) replaced by F (i) , G (i) , i = 1, 2, then F (2) and G (2) are right decomposition factors of f (2) and g (2) respectively. (iv) The decompositions f = f (1) • f (2) , g = g (1) • g (2) satisfying (i), (ii), (iii) are unique up to linear equivalence over K. (v) The polynomials f (1) (x) − y and g (1) (x) − y have the same splitting field over K(y). (vi) In case degree of f (1) and g (1) are not divisible by the characteristic of K, then deg f (1) (x) = deg g (1) (y); in particular the degrees of f (x) and g(y) are not coprime.…”