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##The general question:##

Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order $d$ tensors on $V$ and the space of order $d$ homogeneous polynomials in $n$ indeterminates over $k$.

I am wondering whether this identification is ever of much use in the study of tensors and tensor fields. In particular, I am interested in ways in which this identification might simplify problems in differential geometry, but uses in other fields would be interesting also.

##A potential example of what I'm asking about:##

This question is inspired by the following observation I made, which I would also like to confirm is valid:

Let $\omega \in \mathrm{Sym}^1(V)$ be some unknown. Suppose we have a map $$\phi \colon \mathrm{Sym}^1(V) \longrightarrow k$$ along with a few known elements $\eta, \theta, \nu_1, \nu_2 \in \mathrm{Sym}^1(V)$. Additionally, we have the following system of equations: $$ \begin{align} \phi(\omega) \cdot \eta + \phi(\eta) \cdot \omega &= \nu_1 \\ \phi(\omega) \cdot \theta + \phi(\theta) \cdot \omega &= \nu_2 \end{align} $$ Our goal is to solve for $\omega$. In light of the above, we can identify $\mathrm{Sym}^1(V)$ with $k[x_1, \dots, x_n]_1 \subset k[x_1, \dots, x_n]$, form the field of fractions $k(x_1, \dots, x_n)$ and solve the above system for $\omega$ using linear algebra. Doing so, we will arrive at an expression of the form: $$ \left(\phi(\theta)\eta - \phi(\eta)\theta\right)\omega = \theta \cdot \nu_1 + \eta \cdot \nu_2 $$ where all the elements are now considered to be in $k(x_1, \dots, x_n)$, so all the products make sense. Dividing, we can obtain an expression for $\omega$ expressed entirely in terms of objects we know. We can now evaluate this expression on the appropriate vectors to obtain an expression for $\omega$ in terms of a basis for $\mathrm{Sym}^1(V)$.

Note: I originally posted this question herehere on math.se. At the time I was unsure whether it would better to post it on mathoverflow. Given that it has remained unanswered over there for some time, I figured I would try posting it here. I am not as familiar with mathoverflow as some other stack exchange sites, so please feel free to edit my post in order to make things fit better with this site if necessary.

##The general question:##

Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order $d$ tensors on $V$ and the space of order $d$ homogeneous polynomials in $n$ indeterminates over $k$.

I am wondering whether this identification is ever of much use in the study of tensors and tensor fields. In particular, I am interested in ways in which this identification might simplify problems in differential geometry, but uses in other fields would be interesting also.

##A potential example of what I'm asking about:##

This question is inspired by the following observation I made, which I would also like to confirm is valid:

Let $\omega \in \mathrm{Sym}^1(V)$ be some unknown. Suppose we have a map $$\phi \colon \mathrm{Sym}^1(V) \longrightarrow k$$ along with a few known elements $\eta, \theta, \nu_1, \nu_2 \in \mathrm{Sym}^1(V)$. Additionally, we have the following system of equations: $$ \begin{align} \phi(\omega) \cdot \eta + \phi(\eta) \cdot \omega &= \nu_1 \\ \phi(\omega) \cdot \theta + \phi(\theta) \cdot \omega &= \nu_2 \end{align} $$ Our goal is to solve for $\omega$. In light of the above, we can identify $\mathrm{Sym}^1(V)$ with $k[x_1, \dots, x_n]_1 \subset k[x_1, \dots, x_n]$, form the field of fractions $k(x_1, \dots, x_n)$ and solve the above system for $\omega$ using linear algebra. Doing so, we will arrive at an expression of the form: $$ \left(\phi(\theta)\eta - \phi(\eta)\theta\right)\omega = \theta \cdot \nu_1 + \eta \cdot \nu_2 $$ where all the elements are now considered to be in $k(x_1, \dots, x_n)$, so all the products make sense. Dividing, we can obtain an expression for $\omega$ expressed entirely in terms of objects we know. We can now evaluate this expression on the appropriate vectors to obtain an expression for $\omega$ in terms of a basis for $\mathrm{Sym}^1(V)$.

Note: I originally posted this question here on math.se. At the time I was unsure whether it would better to post it on mathoverflow. Given that it has remained unanswered over there for some time, I figured I would try posting it here. I am not as familiar with mathoverflow as some other stack exchange sites, so please feel free to edit my post in order to make things fit better with this site if necessary.

##The general question:##

Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order $d$ tensors on $V$ and the space of order $d$ homogeneous polynomials in $n$ indeterminates over $k$.

I am wondering whether this identification is ever of much use in the study of tensors and tensor fields. In particular, I am interested in ways in which this identification might simplify problems in differential geometry, but uses in other fields would be interesting also.

##A potential example of what I'm asking about:##

This question is inspired by the following observation I made, which I would also like to confirm is valid:

Let $\omega \in \mathrm{Sym}^1(V)$ be some unknown. Suppose we have a map $$\phi \colon \mathrm{Sym}^1(V) \longrightarrow k$$ along with a few known elements $\eta, \theta, \nu_1, \nu_2 \in \mathrm{Sym}^1(V)$. Additionally, we have the following system of equations: $$ \begin{align} \phi(\omega) \cdot \eta + \phi(\eta) \cdot \omega &= \nu_1 \\ \phi(\omega) \cdot \theta + \phi(\theta) \cdot \omega &= \nu_2 \end{align} $$ Our goal is to solve for $\omega$. In light of the above, we can identify $\mathrm{Sym}^1(V)$ with $k[x_1, \dots, x_n]_1 \subset k[x_1, \dots, x_n]$, form the field of fractions $k(x_1, \dots, x_n)$ and solve the above system for $\omega$ using linear algebra. Doing so, we will arrive at an expression of the form: $$ \left(\phi(\theta)\eta - \phi(\eta)\theta\right)\omega = \theta \cdot \nu_1 + \eta \cdot \nu_2 $$ where all the elements are now considered to be in $k(x_1, \dots, x_n)$, so all the products make sense. Dividing, we can obtain an expression for $\omega$ expressed entirely in terms of objects we know. We can now evaluate this expression on the appropriate vectors to obtain an expression for $\omega$ in terms of a basis for $\mathrm{Sym}^1(V)$.

Note: I originally posted this question here on math.se. At the time I was unsure whether it would better to post it on mathoverflow. Given that it has remained unanswered over there for some time, I figured I would try posting it here. I am not as familiar with mathoverflow as some other stack exchange sites, so please feel free to edit my post in order to make things fit better with this site if necessary.

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Is the identification between symmetric tensors and homogeneous polynomials useful?

##The general question:##

Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order $d$ tensors on $V$ and the space of order $d$ homogeneous polynomials in $n$ indeterminates over $k$.

I am wondering whether this identification is ever of much use in the study of tensors and tensor fields. In particular, I am interested in ways in which this identification might simplify problems in differential geometry, but uses in other fields would be interesting also.

##A potential example of what I'm asking about:##

This question is inspired by the following observation I made, which I would also like to confirm is valid:

Let $\omega \in \mathrm{Sym}^1(V)$ be some unknown. Suppose we have a map $$\phi \colon \mathrm{Sym}^1(V) \longrightarrow k$$ along with a few known elements $\eta, \theta, \nu_1, \nu_2 \in \mathrm{Sym}^1(V)$. Additionally, we have the following system of equations: $$ \begin{align} \phi(\omega) \cdot \eta + \phi(\eta) \cdot \omega &= \nu_1 \\ \phi(\omega) \cdot \theta + \phi(\theta) \cdot \omega &= \nu_2 \end{align} $$ Our goal is to solve for $\omega$. In light of the above, we can identify $\mathrm{Sym}^1(V)$ with $k[x_1, \dots, x_n]_1 \subset k[x_1, \dots, x_n]$, form the field of fractions $k(x_1, \dots, x_n)$ and solve the above system for $\omega$ using linear algebra. Doing so, we will arrive at an expression of the form: $$ \left(\phi(\theta)\eta - \phi(\eta)\theta\right)\omega = \theta \cdot \nu_1 + \eta \cdot \nu_2 $$ where all the elements are now considered to be in $k(x_1, \dots, x_n)$, so all the products make sense. Dividing, we can obtain an expression for $\omega$ expressed entirely in terms of objects we know. We can now evaluate this expression on the appropriate vectors to obtain an expression for $\omega$ in terms of a basis for $\mathrm{Sym}^1(V)$.

Note: I originally posted this question here on math.se. At the time I was unsure whether it would better to post it on mathoverflow. Given that it has remained unanswered over there for some time, I figured I would try posting it here. I am not as familiar with mathoverflow as some other stack exchange sites, so please feel free to edit my post in order to make things fit better with this site if necessary.