TITLE: On $\{P(x)+Q(y):\ x,y=0,\ldots,p-1\}$ with $p$ prime
QUESTION [7 upvotes]: QUESTION: Is my following conjecture (formulated in 2016) true? How to solve it?
Conjecture. For any non-constant polynomials $P(x)，Q(x)\in\mathbb Z[x]$, there is a positive integer $N(P,Q)$ depending on $P(x)$ and $Q(x)$ such that for any prime $p>N(P,Q)$ the set
$$\{P(x)+Q(y):\ x,y=0,1,\ldots,p-1\}$$
contains a complete system of residues modulo $p$.
For example, it seems that we may take 
$$N(x^4,x^2)=5,\ N(x^5,x^2)=11,\ N(x^3,x^3)=7,\ N(x^6,x^3)=31.$$

REPLY [6 votes]: The conjecture is true when the degrees of $P(x)$ and $Q(y)$ are relatively prime. For general polyomials $F(x,y)$ in two variables, a variant of the conjecture is true.
Theorem 1. Let $P(x)\in\mathbb{Z}[x]$ and $Q(y)\in\mathbb{Z}[y]$ be two polynomials of relatively prime degrees. For any prime $p$, and for all residues $r$ modulo $p$, the congruence $P(x)+Q(y)\equiv r\pmod p$ has $p+O(p^{1/2})$ solutions. The implied constant depends only on $P(x)$ and $Q(y)$.
Sketch of proof. Without loss of generality, the prime $p$ exceeds the leading coefficients of $P(x)$ and $Q(y)$. Then, for all residues $r$ modulo $p$, the polynomial $P(x)+Q(y)-r$ is absolutely irreducible over $\mathbb{F}_p$, hence the congruence $P(x)+Q(y)\equiv r\pmod p$ has $p+O(p^{1/2})$ solutions by the celebrated theorem of Weil (1940). For more details on the quoted results, see Theorems 1A and 1B in Section III.1 of Schmidt: Equations over finite fields (Springer, 1976).
Theorem 2. Let $F(x,y)\in\mathbb{Q}[x,y]$ be an absolutely irreducible polynomial. There exists a positive integer $N(F)$ depending on $F(x,y)$ such that for any prime $p>N(F)$, and for all but $O(1)$ residues $r$ modulo $p$, the congruence $F(x,y)\equiv r\pmod p$ has $p+O(p^{1/2})$ solutions. The implied constants depend only on the degree of $F(x,y)$.
Sketch of proof. It follows from a theorem of Noether (1922) that there exists a positive integer $N(F)$ depending on $F(x,y)$ such that for any prime $p>N(F)$, and for all but $O(1)$ residues $r$ modulo $p$, the polinomial $F(x,y)-r$ is absolutely irreducible over $\mathbb{F}_p$. From here we finish as in the previous proof. For more details on Noether's theorem, see Theorem 2A in Section V.2 of Schmidt: Equations over finite fields (Springer, 1976).