Let $d>0$ be even. Consider ${\mathbb{R}}[x_1,\ldots, x_n]_d$, i.e. polynomials of degree $d$.

Call a homogeneous polynomial $f$ of degree $d$ a *polynomial in quadratic variables* if it is of the form $f=p(y_1^2,\ldots, y_n^2)$ for some polynomial $p$. Here the $y_i=y_i(x_1,\ldots, x_n)$ are linear forms such that $\{y_1,\ldots, y_n\}$ is a basis of the degree 1 polynomials ${\mathbb{R}}[x_1,\ldots,x_n]_1$.

For example if we take $p(x,y)=5x^3y^4+(x-3y)^3$, and $x-y, x+y$ to be the linear polynomials, then this gives the example of $5(x-y)^6(x+y)^8+((x-y)^2-3(x+y)^2)^3$ $=5(x-y)^6(x+y)^8-(2x^2+8xy+2y^2)^3$ as an example of a polynomial in quadratic variables.

Is it true that polynomials in quadratic variables are dense in the vector space ${\mathbb{R}}[x_1,\ldots, x_n]_d$? In other words, it every polynomial of degree $d$ a limit of polynomials of degree $d$ in quadratic variables?

**Edit:** In response to Pete, Ewan's and Darij's comments, I will rephrase this question in terms of ring automorphisms. Firstly, let's ignore odd degree polynomials. In the language of linear change of variables, let $V$ be a real vector space space of dimension $n$. Consider the ring $\mathbb{R}[x_1^2,\ldots, x_n^2]$. Each choice of basis $\mathcal{B}=\{v_1,\ldots, v_n\}$ of the vector space $V$ induces a ring monomorphism $\mathcal{B}_*:\mathbb{R}[x_1^2,\ldots, x_n^2]\to Sym(V)$ to the symmetric algebra on $V$. This monomorphism is given by extending the map $x_i^2\mapsto v_i$.

Is every even degree element of $Sym(V)$ in the image of some induced monomorphism $\mathcal{B}_*$?

and$(x+y)^2$. Thus then they are not positive definite. $\endgroup$of your variables$y_1,\ldots,y_n$, so the answer to your question is "no". (He's also saying that the general case can be reduced to the special case $(y_1,\ldots,y_n) = (x_1,\ldots,x_n)$ via an automorphism of the ring, but why don't you address his first point first.) $\endgroup$6more comments