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Ben Webster ♦
Ben Webster ♦
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Actually, if $N>2$, it's impossible to find $N$ "cyclic polynomials" as you desire. That's the same as asking if the cyclic invariants are a polynomial ring, which is impossible by the Chevalley-Shepard-Todd theorem

I suspect in general, the right thing to look at is the sum of the $z_i$'s and all monomials in $p_h=\sum_{i=1}^N \zeta^{ih}z_i$ where the indices $h$ add to a multiple of $N$ (here $\zeta$ is a primitive $N$th root of unity).

So, that means we look at monomials $p_0^{k_0}p_1^{k_1}\cdots p_{N-1}^{k_{N-1}}$ where $\mathbf{k}\cdot (0,1,2,\dots, N-1)\equiv 0\pmod{N}$. It can't be that hard to come up with vectors that span the cone of such integer vectors as a semi-group, though I'll admit, I can't see what they might be at the moment.

Actually, if $N>2$, it's impossible to find $N$ "cyclic polynomials" as you desire. That's the same as asking if the cyclic invariants are a polynomial ring, which is impossible by the Chevalley-Shepard-Todd theorem

I suspect in general, the right thing to look at is the sum of the $z_i$'s and all monomials in $p_h=\sum_{i=1}^N \zeta^{ih}z_i$ where the indices $h$ add to a multiple of $N$ (here $\zeta$ is a primitive $N$th root of unity).

Actually, if $N>2$, it's impossible to find $N$ "cyclic polynomials" as you desire. That's the same as asking if the cyclic invariants are a polynomial ring, which is impossible by the Chevalley-Shepard-Todd theorem

I suspect in general, the right thing to look at is the sum of the $z_i$'s and all monomials in $p_h=\sum_{i=1}^N \zeta^{ih}z_i$ where the indices $h$ add to a multiple of $N$ (here $\zeta$ is a primitive $N$th root of unity).

So, that means we look at monomials $p_0^{k_0}p_1^{k_1}\cdots p_{N-1}^{k_{N-1}}$ where $\mathbf{k}\cdot (0,1,2,\dots, N-1)\equiv 0\pmod{N}$. It can't be that hard to come up with vectors that span the cone of such integer vectors as a semi-group, though I'll admit, I can't see what they might be at the moment.

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Ben Webster ♦
Ben Webster ♦
  • 44.7k
  • 12
  • 126
  • 260

Actually, if $N>2$, it's impossible to find $N$ "cyclic polynomials" as you desire. That's the same as asking if the cyclic invariants are a polynomial ring, which is impossible by the Chevalley-Shepard-Todd theorem

I suspect in general, the right thing to look at is the sum of the $z_i$'s and all monomials in $p_h=\sum \zeta^{ih}z_i$$p_h=\sum_{i=1}^N \zeta^{ih}z_i$ where the indices $h$ add to a multiple of $N$ (here $\zeta$ is a primitive $N$th root of unity).

Actually, if $N>2$, it's impossible to find $N$ "cyclic polynomials" as you desire. That's the same as asking if the cyclic invariants are a polynomial ring, which is impossible by the Chevalley-Shepard-Todd theorem

I suspect in general, the right thing to look at is the sum of the $z_i$'s and all monomials in $p_h=\sum \zeta^{ih}z_i$ where the indices add to a multiple of $N$ (here $\zeta$ is a primitive $N$th root of unity).

Actually, if $N>2$, it's impossible to find $N$ "cyclic polynomials" as you desire. That's the same as asking if the cyclic invariants are a polynomial ring, which is impossible by the Chevalley-Shepard-Todd theorem

I suspect in general, the right thing to look at is the sum of the $z_i$'s and all monomials in $p_h=\sum_{i=1}^N \zeta^{ih}z_i$ where the indices $h$ add to a multiple of $N$ (here $\zeta$ is a primitive $N$th root of unity).

Source Link
Ben Webster ♦
Ben Webster ♦
  • 44.7k
  • 12
  • 126
  • 260

Actually, if $N>2$, it's impossible to find $N$ "cyclic polynomials" as you desire. That's the same as asking if the cyclic invariants are a polynomial ring, which is impossible by the Chevalley-Shepard-Todd theorem

I suspect in general, the right thing to look at is the sum of the $z_i$'s and all monomials in $p_h=\sum \zeta^{ih}z_i$ where the indices add to a multiple of $N$ (here $\zeta$ is a primitive $N$th root of unity).