Arithmetic properties of polynomials

TL;DR

This paper investigates the arithmetic properties of quadratic and cubic polynomials, proving the existence of infinitely many solutions to certain Diophantine systems and providing parametric solutions for specific polynomial forms.

Contribution

It establishes new results on the solvability of complex Diophantine systems involving quadratic polynomials and introduces parametric solutions for related polynomial equations.

Findings

Infinitely many integer solutions for specific quadratic polynomial systems.

Existence of integer parametric solutions under certain modular conditions.

Infinite nontrivial rational solutions for polynomial systems involving cubic 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.