Arithmetic properties of polynomials

Abstract

In this paper, first, we prove that the Diophantine system \[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)\] has infinitely many integer solutions for $f(X)=X(X+a)$ with nonzero integers $a\equiv 0,1,4\pmod{5}$. Second, we show that the above Diophantine system has an integer parametric solution for $f(X)=X(X+a)$ with nonzero integers $a$, if there are integers $m,n,k$ such that \[\begin{cases} \begin{split} (n^2-m^2) (4mnk(k+a+1) + a(m^2+2mn-n^2)) &\equiv0\pmod{(m^2+n^2)^2},\\ (m^2+2mn-n^2) ((m^2-2mn-n^2)k(k+a+1) - 2amn) &\equiv0 \pmod{(m^2+n^2)^2}, \end{split} \end{cases}\] where $k\equiv0\pmod{4}$ when $a$ is even, and $k\equiv2\pmod{4}$ when $a$ is odd. Third, we get that the Diophantine system \[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)=\frac{f(r)}{f(s)}\] has a five-parameter rational solution for $f(X)=X(X+a)$ with nonzero rational number $a$ and infinitely many nontrivial rational parametric solutions for $f(X)=X(X+a)(X+b)$ with nonzero integers $a,b$ and $a\neq b$. At last, we raise some related questions.