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edited Dec 16 2010 at 12:53 |
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 $\mathcal{B}_*$?
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edited Dec 16 2010 at 12:45 |
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 $V$.
Is every even degree element in of $Sym(V)$ in the image of some inudced monomorphisms induced monomorphism $\mathcal{B}_*$?
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edited Dec 16 2010 at 12:44 |
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 $p(x,y)=5x^3y^8+(x-3y)^3$, 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 $V$.
Is every element in $Sym(V)$ in the image of some inudced monomorphisms $\mathcal{B}_*$?
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edited Dec 16 2010 at 12:43 |
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 $p(x,y)=5x^3y^8+(x-3y)^3$, 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 $V$.
Is every element in $Sym(V)$ in the image of some inudced monomorphisms $\mathcal{B}_*$?
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edited Dec 16 2010 at 11:40 |
Work in the polynomial ring Let ${\mathbb{R}}[x_1,\ldots, x_n]$d>0$ be even. Let Consider $y_1=y_1(x_1,\ldots, x_n)${\mathbb{R}}[x_1,\ldots, x_n]_d$, i.e. ..,$y_n=y_n(x_1,\ldots, x_n)$ be homogeneous polynomials linear in the variables $x_1,\ldots, x_n$. Suppose that $y_1,\ldots, y_n$ span the space of linear polynomials in degree ${\mathbb{R}}[x_1,\ldots, x_n]$. d$.
Call a homogeneous polynomial $f\in{\mathbb{R}}[x_1,\ldots,x_n]$ f$ of degree $d$ a polynomial in quadratic variables if it is of the form $p(y_1^2,\ldots, f=p(y_1^2,\ldots, y_n^2)$ 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 $p(x,y)=5x^3y^8+(x-3y)^3$, 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 polynomial ring vector space ${\mathbb{R}}[x_1,\ldots, x_n]$x_n]_d$? In other words, it every polynomial of degree $d$ a limit of polynomials of degree $d$ in quadratic variables?
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edited Dec 16 2010 at 9:57 |
Work in the polynomial ring ${\mathbb{R}}^n$. {\mathbb{R}}[x_1,\ldots, x_n]$. Let $y_1^2+\cdots+y_n^2$ be a positive-definite quadratic formy_1=y_1(x_1,\ldots, x_n)$, where ...,$y_n=y_n(x_1,\ldots, x_n)$ be homogeneous polynomials linear in the variables $y_i=y_i(x_1,\ldots, x_n)$ are x_1,\ldots, x_n$. Suppose that $y_1,\ldots, y_n$ span the space of linear polynomials in ${\mathbb{R}}[x_1,\ldots, x_n]$.
Call a polynomial $f\in{\mathbb{R}}[x_1,\ldots,x_n]$ a polynomial in quadratic variables if it is of the form $p(y_1,\ldots, y_n)$ p(y_1^2,\ldots, y_n^2)$ for some polynomial $p$.
For example if we take $p(x,y)=5x^3y^8+(x-3y)^3$, and the quadratic form $x-y, x+y$ to be $(x-y)^2+(x+y)^2$, the linear polynomials, then this gives the example of $5(x-y)^6(x+y)^8+((x-y)^2-3(x+y)^2)^3=5(x-y)^6(x+y)^8-(2x^2+8xy+2y^2)^3$ 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 polynomials in quadratic variables are dense in the polynomial ring ${\mathbb{R}}[x_1,\ldots, x_n]$? In other words, it every polynomial a limit of polynomials in quadratic variables?
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asked Dec 16 2010 at 9:20 |
Is every polynomial a limit of polynomials in quadratic variables?
Work in ${\mathbb{R}}^n$. Let $y_1^2+\cdots+y_n^2$ be a positive-definite quadratic form, where the $y_i=y_i(x_1,\ldots, x_n)$ are linear polynomials. Call a polynomial $f\in{\mathbb{R}}[x_1,\ldots,x_n]$ a polynomial in quadratic variables if it is of the form $p(y_1,\ldots, y_n)$ for some polynomial $p$.
For example if we take $p(x,y)=5x^3y^8+(x-3y)^3$, and the quadratic form to be $(x-y)^2+(x+y)^2$, 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 polynomials in quadratic variables are dense in the polynomial ring ${\mathbb{R}}[x_1,\ldots, x_n]$? In other words, it every polynomial a limit of polynomials in quadratic variables?
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