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Given any two symmetric and homogenous polynomials with complex coefficients, I'm trying to determine if a unitary change of basis relates them. Specifically, assuming the polynomials are of degree $n$ in $m$ variables, I'm looking for a complete list of invariants to ascertain their equivalence under a SU($m$) change of basis.

Let the general form of these polynomials be represented by $f(\vec{x}) = \sum\limits_{i_{1}+...+i_{m}=n} \alpha_{i_{1}...i_{m}}x_{1}^{i_{1}}x_{2}^{i_{2}}...x_{m}^{i_{m}} $ $= \sum\limits_{|\vec{i}|=n} \alpha_{\vec{i}}\vec{x}^{\vec{i}}$ and the action of $\text{U} \in \text{SU}(m)$ on these polynomials is given by the following change of basis: $\vec{x'} = \text{U}\vec{x'}$. I'm then looking to determine: given any two polynomials $f(\vec{x})$ and $g(\vec{x})$, does there exist any $\text{U} \in \text{SU}(m)$ such that $f(\vec{x}) = g(U\vec{x})$.

I understand that the polynomial ring of invariants $\mathbf{C}[x_{1},x_{2},...,x_{m}]^{\text{SU}(m)}$ is finitely generated and the generators can be computed using various methods including the Cayley's $\Omega$-process [Algorithms in Invariant Theory - Bernd Sturmfels]. I am of the understanding that these invariants are polynomials in $\vec{\alpha}$ and not $\vec{\alpha^{*}}$, where $^{*}$ represents the complex conjugate. But are these polynomial invariants sufficient to guarantee the equivalence of the given polynomials or are they only necessary?

The reason I'm asking about the sufficiency of these invariants is that I don't know how to derive other 'non-polynomial' invariants - which are polynomials in not only $\vec{\alpha}$ but $\vec{\alpha^{*}}$ as well - like the

1. $l^{2}$-vector norm (reference),
2. Fisher inner product/Bombieri norm (reference), and
3. degree $n$ of these polynomials.

Can I somehow derive the polynomial invariants in both $\vec{\alpha}$ and $\vec{\alpha^{*}}$ just from the polynomial invariants in $\vec{\alpha}$ given by the classical invariant theory? Maybe there is a trivial relation between the polynomial invariants and these 'non-polynomial' invariants - is there something that I'm missing here?

Given any two symmetric and homogenous polynomials with complex coefficients, I'm trying to determine if a unitary change of basis relates them. Specifically, assuming the polynomials are of degree $n$ in $m$ variables, I'm looking for a complete list of invariants to ascertain their equivalence under a SU($m$) change of basis.

Let the general form of these polynomials be represented by $f(\vec{x}) = \sum\limits_{i_{1}+...+i_{m}=n} \alpha_{i_{1}...i_{m}}x_{1}^{i_{1}}x_{2}^{i_{2}}...x_{m}^{i_{m}} $ $= \sum\limits_{|\vec{i}|=n} \alpha_{\vec{i}}\vec{x}^{\vec{i}}$ and the action of $\text{U} \in \text{SU}(m)$ on these polynomials is given by the following change of basis: $\vec{x'} = \text{U}\vec{x}$. I'm then looking to determine: given any two polynomials $f(\vec{x})$ and $g(\vec{x})$, does there exist any $\text{U} \in \text{SU}(m)$ such that $f(\vec{x}) = g(U\vec{x})$.

I understand that the polynomial ring of invariants $\mathbf{C}[x_{1},x_{2},...,x_{m}]^{\text{SU}(m)}$ is finitely generated and the generators can be computed using various methods including the Cayley's $\Omega$-process [Algorithms in Invariant Theory - Bernd Sturmfels]. I am of the understanding that these invariants are polynomials in $\vec{\alpha}$ and not $\vec{\alpha^{*}}$, where $^{*}$ represents the complex conjugate. But are these polynomial invariants sufficient to guarantee the equivalence of the given polynomials or are they only necessary?

The reason I'm asking about the sufficiency of these invariants is that I don't know how to derive other 'non-polynomial' invariants - which are polynomials in not only $\vec{\alpha}$ but $\vec{\alpha^{*}}$ as well - like the

1. $l^{2}$-vector norm (reference),
2. Fisher inner product/Bombieri norm (reference), and
3. degree $n$ of these polynomials.

Can I somehow derive the polynomial invariants in both $\vec{\alpha}$ and $\vec{\alpha^{*}}$ just from the polynomial invariants in $\vec{\alpha}$ given by the classical invariant theory? Maybe there is a trivial relation between the polynomial invariants and these 'non-polynomial' invariants - is there something that I'm missing here?

Given any two symmetric and homogenous polynomials with complex coefficients, I'm trying to determine if a unitary change of basis relates them. Specifically, assuming the polynomials are of degree $n$ in $m$ variables, I'm looking for a complete list of invariants to ascertain their equivalence under a SU($m$) change of basis.

I understand that the polynomial ring of invariants is finitely generated and the generators can be computed using various methods including the Cayley's $\Omega$-process. But are these polynomial invariants sufficient to guarantee the equivalence of the given polynomials or are they only necessary?

The reason I'm asking about the sufficiency of these invariants is that I don't know how to derive other 'non-polynomial' invariants, like the

1. $l^{2}$-vector norm (reference)
2. Fisher inner product/Bombieri norm (reference)
3. degree $n$ of these polynomials

which are also invariants under unitary action on the variables, from the generators of the ring of polynomial invariants. Maybe there is a trivial relation between the polynomial invariants and these 'non-polynomial' invariants - is there something that I'm missing here?

Given any two symmetric and homogenous polynomials with complex coefficients, I'm trying to determine if a unitary change of basis relates them. Specifically, assuming the polynomials are of degree $n$ in $m$ variables, I'm looking for a complete list of invariants to ascertain their equivalence under a SU($m$) change of basis.

Let the general form of these polynomials be represented by $f(\vec{x}) = \sum\limits_{i_{1}+...+i_{m}=n} \alpha_{i_{1}...i_{m}}x_{1}^{i_{1}}x_{2}^{i_{2}}...x_{m}^{i_{m}} $ $= \sum\limits_{|\vec{i}|=n} \alpha_{\vec{i}}\vec{x}^{\vec{i}}$ and the action of $\text{U} \in \text{SU}(m)$ on these polynomials is given by the following change of basis: $\vec{x'} = \text{U}\vec{x'}$. I'm then looking to determine: given any two polynomials $f(\vec{x})$ and $g(\vec{x})$, does there exist any $\text{U} \in \text{SU}(m)$ such that $f(\vec{x}) = g(U\vec{x})$.

I understand that the polynomial ring of invariants $\mathbf{C}[x_{1},x_{2},...,x_{m}]^{\text{SU}(m)}$ is finitely generated and the generators can be computed using various methods including the Cayley's $\Omega$-process [Algorithms in Invariant Theory - Bernd Sturmfels]. I am of the understanding that these invariants are polynomials in $\vec{\alpha}$ and not $\vec{\alpha^{*}}$, where $^{*}$ represents the complex conjugate. But are these polynomial invariants sufficient to guarantee the equivalence of the given polynomials or are they only necessary?

The reason I'm asking about the sufficiency of these invariants is that I don't know how to derive other 'non-polynomial' invariants - which are polynomials in not only $\vec{\alpha}$ but $\vec{\alpha^{*}}$ as well - like the

1. $l^{2}$-vector norm (reference),
2. Fisher inner product/Bombieri norm (reference), and
3. degree $n$ of these polynomials.

Can I somehow derive the polynomial invariants in both $\vec{\alpha}$ and $\vec{\alpha^{*}}$ just from the polynomial invariants in $\vec{\alpha}$ given by the classical invariant theory? Maybe there is a trivial relation between the polynomial invariants and these 'non-polynomial' invariants - is there something that I'm missing here?

Given any two symmetric and homogenous polynomials with complex coefficients, I'm trying to determine if a unitary change of basis relates them. Specifically, assuming the polynomials are of degree $n$ in $m$ variables, I'm looking for a complete list of invariants to ascertain their equivalence under a SU($m$) change of basis.

I understand that the polynomial ring of invariants is finitely generated and the generators can be computed using various methods including the Cayley's $\Omega$-process. But are these polynomial invariants sufficient to guarantee the equivalence of the given polynomials or are they only necessary?

The reason I'm asking about the sufficiency of these invariants is that I don't know how to derive other 'non-polynomial' invariants, like the

1. $l^{2}$-vector norm (reference)
2. Fisher inner product/Bombieri norm (reference)
3. degree $d$ of these polynomials

which are also invariants under unitary action on the variables, from the generators of the ring of polynomial invariants. Maybe there is a trivial relation between the polynomial invariants and these 'non-polynomial' invariants - is there something that I'm missing here?

Given any two symmetric and homogenous polynomials with complex coefficients, I'm trying to determine if a unitary change of basis relates them. Specifically, assuming the polynomials are of degree $n$ in $m$ variables, I'm looking for a complete list of invariants to ascertain their equivalence under a SU($m$) change of basis.

I understand that the polynomial ring of invariants is finitely generated and the generators can be computed using various methods including the Cayley's $\Omega$-process. But are these polynomial invariants sufficient to guarantee the equivalence of the given polynomials or are they only necessary?

The reason I'm asking about the sufficiency of these invariants is that I don't know how to derive other 'non-polynomial' invariants, like the

1. $l^{2}$-vector norm (reference)
2. Fisher inner product/Bombieri norm (reference)
3. degree $n$ of these polynomials

which are also invariants under unitary action on the variables, from the generators of the ring of polynomial invariants. Maybe there is a trivial relation between the polynomial invariants and these 'non-polynomial' invariants - is there something that I'm missing here?