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You are asking about a venerable and active area in Diophantine equations lying at the border of arithmetic geometry and analytic number theory. Some keywords are forms in many variables and circle method.

Let $P(x_1,\ldots,x_n) \in \mathbb{Z}[x_1,\ldots,x_n]$ be a form (i.e., homogeneous polynomial) of degree $d$. As you say, if $n \leq d^2$, then there are very natural examples of forms without nontrivial integral solutions: i.e.e.g., for each $d \geq 2$, take the reduced norm form on a central division algebra of degree $d^2$. (One might express this in shorthand by saying that $\mathbb{Q}$ is not $C_2(d)$ for any $d \geq 2$.)

Note that when $d$ is odd, solutions over $\mathbb{R}$ are guaranteed, so one may simply ask whether taking $n \gg d$ is enough to guarantee the existence of nontrivial $\mathbb{Z}$-solutions. The answer is a resounding yes: this is a celebrated $1957$ theorem of B.J. Birch. Analogous results are known with $\mathbb{Q}$ replaced by any number field $K$: if $K$ has no real places, then the condition that $d$ is odd may be omitted.

So the problem becomes a quantitative one: for (say) odd $d$, just how large must $n$ be compared to $d$ to ensure that a form $P$ necessarily has a nontrivial $\mathbb{Z}$-solution? For instance, taking $d = 3$, we know that $n$ must be at least $10$ and in fact it is conjectured that this is the sharp answer. But we are very far from being able to prove this: in celebrated work of the 1960's, Davenport showed that one may take $n \geq 16$. In a 2007 Inventiones paper, Heath-Brown improved this to $n \geq 14$.

The classical reference for this material is M.J. Greenberg's little book Lectures on Forms in Many Variables. But this was written in the late 1960's and a lot of work has been done since then. A bit of googling finds some nice survey papers, e.g. this one by Trevor Wooley, who is one the current leaders in the field. Note in particular that on page $4$ of this document Wooley relates an example of Cassels and Guy which shows that when $d = 6$, there are forms with arbitrarily large $n$ and points over $\mathbb{R}$ (and also over $\mathbb{Q}_p$ for all $p$) but no nontrivial $\mathbb{Q}$-points. So the case of even $d$ is really different.

1

You are asking about a venerable and active area in Diophantine equations lying at the border of arithmetic geometry and analytic number theory. Some keywords are forms in many variables and circle method.

Let $P(x_1,\ldots,x_n) \in \mathbb{Z}[x_1,\ldots,x_n]$ be a form (i.e., homogeneous polynomial) of degree $d$. As you say, if $n \leq d^2$, then there are very natural examples of forms without nontrivial integral solutions: i.e., for each $d \geq 2$, take the norm form on a division algebra of degree $d^2$. (One might express this in shorthand by saying that $\mathbb{Q}$ is not $C_2(d)$ for any $d \geq 2$.)

Note that when $d$ is odd, solutions over $\mathbb{R}$ are guaranteed, so one may simply ask whether taking $n \gg d$ is enough to guarantee the existence of nontrivial $\mathbb{Z}$-solutions. The answer is a resounding yes: this is a celebrated $1957$ theorem of B.J. Birch. Analogous results are known with $\mathbb{Q}$ replaced by any number field $K$: if $K$ has no real places, then the condition that $d$ is odd may be omitted.

So the problem becomes a quantitative one: for (say) odd $d$, just how large must $n$ be compared to $d$ to ensure that a form $P$ necessarily has a nontrivial $\mathbb{Z}$-solution? For instance, taking $d = 3$, we know that $n$ must be at least $10$ and in fact it is conjectured that this is the sharp answer. But we are very far from being able to prove this: in celebrated work of the 1960's, Davenport showed that one may take $n \geq 16$. In a 2007 Inventiones paper, Heath-Brown improved this to $n \geq 14$.

The classical reference for this material is M.J. Greenberg's little book Lectures on Forms in Many Variables. But this was written in the late 1960's and a lot of work has been done since then. A bit of googling finds some nice survey papers, e.g. this one by Trevor Wooley, who is one the current leaders in the field.