On polynomials with integer coefficients

Abstract

Let f( x 1, x 2,…, x n ) be a polynomial with rational integral coefficients. Let d( f) be the greatest common divisor of all integers of the form f( z 1, z 2,…, z n ) where z 1, z 2,…, z n ∈ Z. More generally if A i = { sa i + b i } s∈ Z , a i , b i ∈ Z, is an arithmetic progression for i = 1, 2,…, n we set A=A 1×A 2×…×A n and call d( A, f) the greatest common divisor of all integers of the form f( z 1, z 2,…, z n ) where z i ∈ A i , i = 1, 2,…, n. A rather simple expression for d( A, f) is derived in terms of the coefficients of f, and we show that d( A, f) = d( f) when a 1 a 2… a n is relatively prime to f( b 1, b 2,…, b n ). When f is primitive it is shown that d( A, f) divides II i=t n a i m i ·m i! where m i is the degree of f in x i , i = 1, 2,…, n.