Cavender Felsenstein Neyman model with root distribution $\pi = \left(\frac{1}{2}, \frac{1}{2} \right)$.
The transition matrix associated with edge $i$ is of the form
$$M_j = \begin{pmatrix}
a_i & b_i\\
b_i & a_i
\end{pmatrix}$$ and the Fourier parameters are $\left(x^{(i)}_i, x^{(i)}_2\right)$ for all edges $i$.
Phylogenetic tree 4-0-0
Evolutionary model
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Summary
Dimension: 5
Degree: 8
Probability coordinates: 8
Fourier coordinates: 8
dim of Singular locus: 24
deg of Singular locus: 2
MLdeg: 92
EDdeg: 92
Degree: 8
Probability coordinates: 8
Fourier coordinates: 8
dim of Singular locus: 24
deg of Singular locus: 2
MLdeg: 92
EDdeg: 92
Probability parametrization
$\begin{align}
p_{\texttt{0101}} &= \frac{1}{2}(a_1a_3b_2b_4 + a_2a_4b_1b_3) \\[0.5em]
p_{\texttt{0001}} &= \frac{1}{2}(a_1a_2a_3b_4 + a_4b_1b_2b_3) \\[0.5em]
p_{\texttt{0100}} &= \frac{1}{2}(a_1a_3a_4b_2 + a_2b_1b_3b_4) \\[0.5em]
p_{\texttt{0000}} &= \frac{1}{2}(a_1a_2a_3a_4 + b_1b_2b_3b_4) \\[0.5em]
p_{\texttt{0111}} &= \frac{1}{2}(a_1b_2b_3b_4 + a_2a_3a_4b_1) \\[0.5em]
p_{\texttt{0011}} &= \frac{1}{2}(a_1a_2b_3b_4 + a_3a_4b_1b_2) \\[0.5em]
p_{\texttt{0110}} &= \frac{1}{2}(a_1a_4b_2b_3 + a_2a_3b_1b_4) \\[0.5em]
p_{\texttt{0010}} &= \frac{1}{2}(a_1a_2a_4b_3 + a_3b_1b_2b_4)
\end{align}$
Fourier parametrization
$\begin{align}
q_{\texttt{1111}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{2}x^{(4)}_{2} \\[0.5em]
q_{\texttt{0101}} &= x^{(1)}_{1}x^{(2)}_{2}x^{(3)}_{1}x^{(4)}_{2} \\[0.5em]
q_{\texttt{1010}} &= x^{(1)}_{2}x^{(2)}_{1}x^{(3)}_{2}x^{(4)}_{1} \\[0.5em]
q_{\texttt{0000}} &= x^{(1)}_{1}x^{(2)}_{1}x^{(3)}_{1}x^{(4)}_{1} \\[0.5em]
q_{\texttt{1100}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{1}x^{(4)}_{1} \\[0.5em]
q_{\texttt{0011}} &= x^{(1)}_{1}x^{(2)}_{1}x^{(3)}_{2}x^{(4)}_{2} \\[0.5em]
q_{\texttt{0110}} &= x^{(1)}_{1}x^{(2)}_{2}x^{(3)}_{2}x^{(4)}_{1} \\[0.5em]
q_{\texttt{1001}} &= x^{(1)}_{2}x^{(2)}_{1}x^{(3)}_{1}x^{(4)}_{2}
\end{align}$
Equivalent classes of probability parametrization
$\begin{align}
\text{Class } &p_{\texttt{0000}}:\ p_{\texttt{0000}},\ p_{\texttt{1111}} \\[0.5em]
\text{Class } &p_{\texttt{0001}}:\ p_{\texttt{0001}},\ p_{\texttt{1110}} \\[0.5em]
\text{Class } &p_{\texttt{0010}}:\ p_{\texttt{0010}},\ p_{\texttt{1101}} \\[0.5em]
\text{Class } &p_{\texttt{0011}}:\ p_{\texttt{0011}},\ p_{\texttt{1100}} \\[0.5em]
\text{Class } &p_{\texttt{0100}}:\ p_{\texttt{0100}},\ p_{\texttt{1011}} \\[0.5em]
\text{Class } &p_{\texttt{0101}}:\ p_{\texttt{0101}},\ p_{\texttt{1010}} \\[0.5em]
\text{Class } &p_{\texttt{0110}}:\ p_{\texttt{0110}},\ p_{\texttt{1001}} \\[0.5em]
\text{Class } &p_{\texttt{0111}}:\ p_{\texttt{0111}},\ p_{\texttt{1000}}
\end{align}$
Equivalent classes of Fourier parametrization
$\begin{align}
\text{Class } &0:\ q_{\texttt{1000}}, q_{\texttt{0100}}, q_{\texttt{0010}}, q_{\texttt{1110}}, q_{\texttt{0001}}, q_{\texttt{1101}}, q_{\texttt{1011}}, q_{\texttt{0111}} \\[0.5em]
\text{Class } &q_{\texttt{0000}}:\ q_{\texttt{0000}} \\[0.5em]
\text{Class } &q_{\texttt{0011}}:\ q_{\texttt{0011}} \\[0.5em]
\text{Class } &q_{\texttt{0101}}:\ q_{\texttt{0101}} \\[0.5em]
\text{Class } &q_{\texttt{0110}}:\ q_{\texttt{0110}} \\[0.5em]
\text{Class } &q_{\texttt{1001}}:\ q_{\texttt{1001}} \\[0.5em]
\text{Class } &q_{\texttt{1010}}:\ q_{\texttt{1010}} \\[0.5em]
\text{Class } &q_{\texttt{1100}}:\ q_{\texttt{1100}} \\[0.5em]
\text{Class } &q_{\texttt{1111}}:\ q_{\texttt{1111}}
\end{align}$
Invariants in Fourier coordinates
Ideal generated by $$ \begin{align}
q_{1100}q_{0011} - q_{0110}q_{1001} \\
q_{1111}q_{0000} - q_{0110}q_{1001} \\
q_{0101}q_{1010} - q_{0110}q_{1001}
\end{align} $$
Gröbner basis of ideal of invariants
Gröbner basis with elements $$ \begin{align}
q_{1100}q_{0011} - q_{0110}q_{1001} \\
q_{1111}q_{0000} - q_{0110}q_{1001} \\
q_{0101}q_{1010} - q_{0110}q_{1001}
\end{align}$$
with respect to the ordering
degrevlex $ q_{\texttt{1111}}, q_{\texttt{0101}}, q_{\texttt{1010}}, q_{\texttt{0000}}, q_{\texttt{1100}}, q_{\texttt{0011}}, q_{\texttt{0110}}, q_{\texttt{1001}}$
Additional information
The cardinality of the smallest set of generators that define the ideal of phylogenetic invariants: 3
The largest degree of a generator in the degree reverse lexicographic reduced Gröbner basis: 2
The largest degree of an element in a minimal generating set for the ideal of phylogenetic invariants: 2
The cardinality of the degree reverse lexicographic reduced Gröbner basis of the ideal of phylogenetic invariants: 3
The largest degree of a generator in the degree reverse lexicographic reduced Gröbner basis: 2
The largest degree of an element in a minimal generating set for the ideal of phylogenetic invariants: 2
The cardinality of the degree reverse lexicographic reduced Gröbner basis of the ideal of phylogenetic invariants: 3
Specialized Fourier transform
$\frac{1}{1} \begin{pmatrix}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\
1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1\\
1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\
1 & 1 & -1 & -1 & 1 & 1 & -1 & -1\\
1 & 1 & 1 & 1 & -1 & -1 & -1 & -1\\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1
\end{pmatrix} ] $
Inverse specialized Fourier transform
$\frac{1}{8} \begin{pmatrix}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1\\
1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1\\
1 & 1 & -1 & -1 & 1 & 1 & -1 & -1\\
1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\\
1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\
1 & 1 & 1 & 1 & -1 & -1 & -1 & -1
\end{pmatrix} $