Maximum number of algebraically independent homogeneous polynomials of given degree

Question

Given degree $d\geq2$ how many algebraically independent homogeneous polynomials of degree $d$ can we have in $\mathbb Z[x_1,\dots,x_n]$? Is it possible to have $n^2$ or $n^3$ of them or are we limited to $n$?

Answer 1

There are $n$ of them in every degree. For example, $x_1^d,\dots,x_n^d$.

If you believe that $x_1,\dots,x_n$ are algebraically independent, then it follows that $x_1^d,\dots,x_n^d$ are too. (An algebraic relation between them would also be an algebraic relation between the $x_i$'s.) Meanwhile, there can never be more than $n$, because the transcendence degree of $\mathbb{Q}(x_1,\dots,x_n)$ over $\mathbb{Q}$ is $n$, and $\mathbb{Q}(x_1,\dots,x_n)$ contains $\mathbb{Z}[x_1,\dots,x_n]$.

(Transcendence degree is a lot like vector space dimension: there can be up to that many independent elements, but not more. The reasons are very similar. With vector spaces, we know that if $A=v_1,\dots,v_m$ is a linearly independent set of vectors and $B=w_1,\dots,w_n$ is another linearly independent set that is bigger, i.e. $n>m$, then there is some $w_j$ that can be added to $A$ without disrupting its linear independence. From this, we can see that all maximal linearly independent sets are the same size: if $A$ is maximal, it means nothing can be added to $A$ without losing independence, so $B$ can't be bigger. Algebraic independence is exactly the same: if $A=f_1,\dots,f_m$ are algebraically independent polynomials, and $B=g_1,\dots,g_n$ are a bigger set of algebraically independent polynomials, then we can add one of the $g_j$'s to $A$ without disrupting independence. Keyword: "algebraic matroid".)