A diophantine related query

Question

Supposing I give you a multivariate equation $$F\in\Bbb Z[x_1,\dots,x_n]$$

Following is undecidable:

'Is there an $(a_1,\dots,a_n)\in\Bbb N^n$ such that $F(a)=0$?'

However is the following always true?

'$\forall F$ that is non-trivial there are infinite $(a_1,\dots,a_n)\in\Bbb N^n$ such that $F(a)\neq 0$?'

Answer 1

It is true that there are infinitely many $n$-tuples $(a_1,\dots,a_n)$ of positive integers such that $P(a_1,\dots,a_n)\ne 0$. We outline a proof by induction on the number $n$ of variables.

For the case $n=1$, if the polynomial is a non-zero constant, there is nothing to prove. Otherwise, let the polynomial have degree $d\ge 1$. Then the term $a_dx^d$ is long-term dominant, so our polynomial is non-zero at any large enough natural number.

Now suppose that the result is true for polynomials in any number of variables less than $n$. We show it is true for polynomials in $n$ variables.

Let $P$ be a polynomial in the variables $x_1$ to $x_n$. Without loss of generality we may assume that $P$ mentions the variable $x_1$. Let $P$ be of degree $d\ge 1$ in $x_1$, and let coefficient of $x^d$ be $Q(x_2,\dots,x_n)$.

By the induction assumption, there exist natural numbers $a_2,\dots,a_n$ such that $Q(a_2,\dots, a_n)\ne 0$. Fix such an $(n-1)$-tuple.

Then $P(x_1,a_2,\dots,a_n)$ is a non-trivial polynomial in $1$ variable, so is nonzero for infinitely many positive integer values of $x_1$.

Answer 2

The request would mean that

$$\lim\limits_{x_{i} \to \infty, |x_{i}| \in N } |F(x_{1},x_{2},...,x_{m})|=0$$ otherwise there would be an infinite number of solutions $F(a) \neq 0$ as soon as we reach sufficiently large values.

Now take any $x_{i}$ and fix the rest at any value where $F(a) \neq 0$. (The situation $F(a) = 0$ for any $a$ is one of the trivial cases.)

Having done this, we are now looking at a linear or higher degree polynomial of $x_{i}$ with at least one value $x_{i}=a_{i},F(a_{i}) \neq 0$. Since there is one value different from 0 for this particular variable, $x_{i}$, we can construct the entire polynomial over this variable that behaves as any other polynomial over one variable.

Any polynomial over one variable does not have $\lim\limits_{x \to \infty, |x| \in N } |P(x)|=0$ , regardless of the condition $|x| \in N$. It is simply unbounded when $x$ tends to $\infty$ since the highest degree factor dominates.

Since we cannot have the limit over this particular variable that will lead $|F(x_{i})|$ to $0$, we cannot have it at all. Obviously we have an infinite number of solutions $F(a) \neq 0$ just by changing $x_{i}$, let alone the remaining variables.