Is every polynomial a limit of polynomials in quadratic variables? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:53:52Z http://mathoverflow.net/feeds/question/49620 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49620/is-every-polynomial-a-limit-of-polynomials-in-quadratic-variables Is every polynomial a limit of polynomials in quadratic variables? Colin Tan 2010-12-16T09:20:49Z 2010-12-16T13:23:51Z <p>Let $d>0$ be even. Consider ${\mathbb{R}}[x_1,\ldots, x_n]_d$, i.e. polynomials of degree $d$. </p> <p>Call a homogeneous polynomial $f$ of degree $d$ a <i>polynomial in quadratic variables</i> 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 ${\mathbb{R}}[x_1,\ldots,x_n]_1$. </p> <p>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.</p> <p>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?</p> <p><b>Edit:</b> 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 $x_i^2\mapsto v_i$.</p> <p>Is every even degree element of $Sym(V)$ in the image of some induced monomorphism $\mathcal{B}_*$? </p> http://mathoverflow.net/questions/49620/is-every-polynomial-a-limit-of-polynomials-in-quadratic-variables/49645#49645 Answer by Andreas Thom for Is every polynomial a limit of polynomials in quadratic variables? Andreas Thom 2010-12-16T13:23:51Z 2010-12-16T13:23:51Z <p>The dimension of the linear space of homogenous polynomials of degree $d$ is $\binom{n+d-1}{d}$. The dimension of the space of homogenous polynomials of degree $d$ in squared linear variables is $$\binom{n+d/2-1}{d/2} \cdot (n^2-1),$$ where $n^2-1 = \dim GL_n(\mathbb R)$. One easily see that if $d$ and $n$ are sufficiently large, then the dimension of the set of squared polynomials is too small to exhaust all homogenous polynomials of degree $d$. Indeed,</p> <p>$$\binom{n+d/2-1}{d/2} \cdot (n^2-1) \leq \frac{(n-1)\cdot n \cdots (n+d/2-1)}{1 \cdot 2 \cdots d/2} \cdot n^2$$ $$&lt; \frac{(n-1)\cdot n \cdots (n+d-1)}{1 \cdot 2 \cdots d} = \binom{n+d-1}{d}$$ if</p> <p>$$n^2 &lt; \frac{(n+d/2)\cdots (n+d-1)}{(d/2+1) \cdots d} = \prod_{k=1}^{d/2} \frac{n-1+d/2+k}{d/2+k}.$$</p> <p>But the RHS equals $$\prod_{k=1}^{d/2} \left( 1+\frac{n-1}{d/2+k} \right) \geq \left( 1+\frac{n-1}{d} \right)^{d/2} \sim \exp\left(\frac{n-1}2 \right)$$ for large $d$. So, if $n$ is large, we get $n^2 &lt; \exp\left(\frac{n-1}2 \right) \leq \frac{(n+d/2)\cdots (n+d-1)}{(d/2+1) \cdots d}$ and this implies the claim.</p>