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 $\left(x^{(i)}_1, x^{(i)}_i\right)$ for all edges $i$.
Phylogenetic tree 5-0-1
Evolutionary model
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Summary
Dimension: 7
Degree: 44
Probability coordinates: 16
Fourier coordinates: 16
dim of Singular locus: -
deg of Singular locus: -
MLdeg: 2098
EDdeg: 334
Degree: 44
Probability coordinates: 16
Fourier coordinates: 16
dim of Singular locus: -
deg of Singular locus: -
MLdeg: 2098
EDdeg: 334
Probability parametrization
$\begin{align}
p_{\texttt{00111}} &= \frac{1}{2}(a_1a_2a_3a_4a_5b_6 + a_1a_2a_6b_3b_4b_5 + a_3a_4a_5a_6b_1b_2 + b_1b_2b_3b_4b_5b_6) \\[0.5em]
p_{\texttt{00101}} &= \frac{1}{2}(a_1a_2a_3a_5b_4b_6 + a_1a_2a_4a_6b_3b_5 + a_3a_5a_6b_1b_2b_4 + a_4b_1b_2b_3b_5b_6) \\[0.5em]
p_{\texttt{00010}} &= \frac{1}{2}(a_1a_2a_3a_5a_6b_4 + a_1a_2a_4b_3b_5b_6 + a_3a_5b_1b_2b_4b_6 + a_4a_6b_1b_2b_3b_5) \\[0.5em]
p_{\texttt{00000}} &= \frac{1}{2}(a_1a_2a_3a_4a_5a_6 + a_1a_2b_3b_4b_5b_6 + a_3a_4a_5b_1b_2b_6 + a_6b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{01110}} &= \frac{1}{2}(a_1a_3a_4b_2b_5b_6 + a_1a_5a_6b_2b_3b_4 + a_2a_3a_4a_6b_1b_5 + a_2a_5b_1b_3b_4b_6) \\[0.5em]
p_{\texttt{01100}} &= \frac{1}{2}(a_1a_3b_2b_4b_5b_6 + a_1a_4a_5a_6b_2b_3 + a_2a_3a_6b_1b_4b_5 + a_2a_4a_5b_1b_3b_6) \\[0.5em]
p_{\texttt{00011}} &= \frac{1}{2}(a_1a_2a_3a_6b_4b_5 + a_1a_2a_4a_5b_3b_6 + a_3b_1b_2b_4b_5b_6 + a_4a_5a_6b_1b_2b_3) \\[0.5em]
p_{\texttt{00001}} &= \frac{1}{2}(a_1a_2a_3a_4a_6b_5 + a_1a_2a_5b_3b_4b_6 + a_3a_4b_1b_2b_5b_6 + a_5a_6b_1b_2b_3b_4) \\[0.5em]
p_{\texttt{01111}} &= \frac{1}{2}(a_1a_3a_4a_5b_2b_6 + a_1a_6b_2b_3b_4b_5 + a_2a_3a_4a_5a_6b_1 + a_2b_1b_3b_4b_5b_6) \\[0.5em]
p_{\texttt{01101}} &= \frac{1}{2}(a_1a_3a_5b_2b_4b_6 + a_1a_4a_6b_2b_3b_5 + a_2a_3a_5a_6b_1b_4 + a_2a_4b_1b_3b_5b_6) \\[0.5em]
p_{\texttt{01010}} &= \frac{1}{2}(a_1a_3a_5a_6b_2b_4 + a_1a_4b_2b_3b_5b_6 + a_2a_3a_5b_1b_4b_6 + a_2a_4a_6b_1b_3b_5) \\[0.5em]
p_{\texttt{01000}} &= \frac{1}{2}(a_1a_3a_4a_5a_6b_2 + a_1b_2b_3b_4b_5b_6 + a_2a_3a_4a_5b_1b_6 + a_2a_6b_1b_3b_4b_5) \\[0.5em]
p_{\texttt{01011}} &= \frac{1}{2}(a_1a_3a_6b_2b_4b_5 + a_1a_4a_5b_2b_3b_6 + a_2a_3b_1b_4b_5b_6 + a_2a_4a_5a_6b_1b_3) \\[0.5em]
p_{\texttt{01001}} &= \frac{1}{2}(a_1a_3a_4a_6b_2b_5 + a_1a_5b_2b_3b_4b_6 + a_2a_3a_4b_1b_5b_6 + a_2a_5a_6b_1b_3b_4) \\[0.5em]
p_{\texttt{00110}} &= \frac{1}{2}(a_1a_2a_3a_4b_5b_6 + a_1a_2a_5a_6b_3b_4 + a_3a_4a_6b_1b_2b_5 + a_5b_1b_2b_3b_4b_6) \\[0.5em]
p_{\texttt{00100}} &= \frac{1}{2}(a_1a_2a_3b_4b_5b_6 + a_1a_2a_4a_5a_6b_3 + a_3a_6b_1b_2b_4b_5 + a_4a_5b_1b_2b_3b_6)
\end{align}$
Fourier parametrization
$\begin{align}
q_{\texttt{00101}} &= x^{(1)}_{1}x^{(2)}_{1}x^{(3)}_{2}x^{(4)}_{1}x^{(5)}_{2}x^{(6)}_{1} \\[0.5em]
q_{\texttt{10100}} &= x^{(1)}_{2}x^{(2)}_{1}x^{(3)}_{2}x^{(4)}_{1}x^{(5)}_{1}x^{(6)}_{2} \\[0.5em]
q_{\texttt{00000}} &= x^{(1)}_{1}x^{(2)}_{1}x^{(3)}_{1}x^{(4)}_{1}x^{(5)}_{1}x^{(6)}_{1} \\[0.5em]
q_{\texttt{10111}} &= x^{(1)}_{2}x^{(2)}_{1}x^{(3)}_{2}x^{(4)}_{2}x^{(5)}_{2}x^{(6)}_{2} \\[0.5em]
q_{\texttt{01100}} &= x^{(1)}_{1}x^{(2)}_{2}x^{(3)}_{2}x^{(4)}_{1}x^{(5)}_{1}x^{(6)}_{2} \\[0.5em]
q_{\texttt{10010}} &= x^{(1)}_{2}x^{(2)}_{1}x^{(3)}_{1}x^{(4)}_{2}x^{(5)}_{1}x^{(6)}_{2} \\[0.5em]
q_{\texttt{00011}} &= x^{(1)}_{1}x^{(2)}_{1}x^{(3)}_{1}x^{(4)}_{2}x^{(5)}_{2}x^{(6)}_{1} \\[0.5em]
q_{\texttt{01111}} &= x^{(1)}_{1}x^{(2)}_{2}x^{(3)}_{2}x^{(4)}_{2}x^{(5)}_{2}x^{(6)}_{2} \\[0.5em]
q_{\texttt{11110}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{2}x^{(4)}_{2}x^{(5)}_{1}x^{(6)}_{1} \\[0.5em]
q_{\texttt{10001}} &= x^{(1)}_{2}x^{(2)}_{1}x^{(3)}_{1}x^{(4)}_{1}x^{(5)}_{2}x^{(6)}_{2} \\[0.5em]
q_{\texttt{01010}} &= x^{(1)}_{1}x^{(2)}_{2}x^{(3)}_{1}x^{(4)}_{2}x^{(5)}_{1}x^{(6)}_{2} \\[0.5em]
q_{\texttt{11101}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{2}x^{(4)}_{1}x^{(5)}_{2}x^{(6)}_{1} \\[0.5em]
q_{\texttt{11000}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{1}x^{(4)}_{1}x^{(5)}_{1}x^{(6)}_{1} \\[0.5em]
q_{\texttt{00110}} &= x^{(1)}_{1}x^{(2)}_{1}x^{(3)}_{2}x^{(4)}_{2}x^{(5)}_{1}x^{(6)}_{1} \\[0.5em]
q_{\texttt{01001}} &= x^{(1)}_{1}x^{(2)}_{2}x^{(3)}_{1}x^{(4)}_{1}x^{(5)}_{2}x^{(6)}_{2} \\[0.5em]
q_{\texttt{11011}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{1}x^{(4)}_{2}x^{(5)}_{2}x^{(6)}_{1}
\end{align}$
Equivalent classes of probability parametrization
$\begin{align}
\text{Class } &p_{\texttt{00000}}:\ p_{\texttt{00000}},\ p_{\texttt{11111}} \\[0.5em]
\text{Class } &p_{\texttt{00001}}:\ p_{\texttt{00001}},\ p_{\texttt{11110}} \\[0.5em]
\text{Class } &p_{\texttt{00010}}:\ p_{\texttt{00010}},\ p_{\texttt{11101}} \\[0.5em]
\text{Class } &p_{\texttt{00011}}:\ p_{\texttt{00011}},\ p_{\texttt{11100}} \\[0.5em]
\text{Class } &p_{\texttt{00100}}:\ p_{\texttt{00100}},\ p_{\texttt{11011}} \\[0.5em]
\text{Class } &p_{\texttt{00101}}:\ p_{\texttt{00101}},\ p_{\texttt{11010}} \\[0.5em]
\text{Class } &p_{\texttt{00110}}:\ p_{\texttt{00110}},\ p_{\texttt{11001}} \\[0.5em]
\text{Class } &p_{\texttt{00111}}:\ p_{\texttt{00111}},\ p_{\texttt{11000}} \\[0.5em]
\text{Class } &p_{\texttt{01000}}:\ p_{\texttt{01000}},\ p_{\texttt{10111}} \\[0.5em]
\text{Class } &p_{\texttt{01001}}:\ p_{\texttt{01001}},\ p_{\texttt{10110}} \\[0.5em]
\text{Class } &p_{\texttt{01010}}:\ p_{\texttt{01010}},\ p_{\texttt{10101}} \\[0.5em]
\text{Class } &p_{\texttt{01011}}:\ p_{\texttt{01011}},\ p_{\texttt{10100}} \\[0.5em]
\text{Class } &p_{\texttt{01100}}:\ p_{\texttt{01100}},\ p_{\texttt{10011}} \\[0.5em]
\text{Class } &p_{\texttt{01101}}:\ p_{\texttt{01101}},\ p_{\texttt{10010}} \\[0.5em]
\text{Class } &p_{\texttt{01110}}:\ p_{\texttt{01110}},\ p_{\texttt{10001}} \\[0.5em]
\text{Class } &p_{\texttt{01111}}:\ p_{\texttt{01111}},\ p_{\texttt{10000}}
\end{align}$
Equivalent classes of Fourier parametrization
$\begin{align}
\text{Class } &0:\ q_{\texttt{10000}}, q_{\texttt{01000}}, q_{\texttt{00100}}, q_{\texttt{11100}}, q_{\texttt{00010}}, q_{\texttt{11010}}, q_{\texttt{10110}}, q_{\texttt{01110}}, q_{\texttt{00001}}, q_{\texttt{11001}}, q_{\texttt{10101}}, q_{\texttt{01101}}, q_{\texttt{10011}}, q_{\texttt{01011}}, q_{\texttt{00111}}, q_{\texttt{11111}} \\[0.5em]
\text{Class } &q_{\texttt{00000}}:\ q_{\texttt{00000}} \\[0.5em]
\text{Class } &q_{\texttt{00011}}:\ q_{\texttt{00011}} \\[0.5em]
\text{Class } &q_{\texttt{00101}}:\ q_{\texttt{00101}} \\[0.5em]
\text{Class } &q_{\texttt{00110}}:\ q_{\texttt{00110}} \\[0.5em]
\text{Class } &q_{\texttt{01001}}:\ q_{\texttt{01001}} \\[0.5em]
\text{Class } &q_{\texttt{01010}}:\ q_{\texttt{01010}} \\[0.5em]
\text{Class } &q_{\texttt{01100}}:\ q_{\texttt{01100}} \\[0.5em]
\text{Class } &q_{\texttt{01111}}:\ q_{\texttt{01111}} \\[0.5em]
\text{Class } &q_{\texttt{10001}}:\ q_{\texttt{10001}} \\[0.5em]
\text{Class } &q_{\texttt{10010}}:\ q_{\texttt{10010}} \\[0.5em]
\text{Class } &q_{\texttt{10100}}:\ q_{\texttt{10100}} \\[0.5em]
\text{Class } &q_{\texttt{10111}}:\ q_{\texttt{10111}} \\[0.5em]
\text{Class } &q_{\texttt{11000}}:\ q_{\texttt{11000}} \\[0.5em]
\text{Class } &q_{\texttt{11011}}:\ q_{\texttt{11011}} \\[0.5em]
\text{Class } &q_{\texttt{11101}}:\ q_{\texttt{11101}} \\[0.5em]
\text{Class } &q_{\texttt{11110}}:\ q_{\texttt{11110}}
\end{align}$
Invariants in Fourier coordinates
Ideal generated by $$-$$
Gröbner basis of ideal of invariants
Gröbner basis with $0$ elements
Additional information
The cardinality of the smallest set of generators that define the ideal of phylogenetic invariants: -
The largest degree of a generator in the degree reverse lexicographic reduced Gröbner basis: -
The largest degree of an element in a minimal generating set for the ideal of phylogenetic invariants: -
The cardinality of the degree reverse lexicographic reduced Gröbner basis of the ideal of phylogenetic invariants: -
The largest degree of a generator in the degree reverse lexicographic reduced Gröbner basis: -
The largest degree of an element in a minimal generating set for the ideal of phylogenetic invariants: -
The cardinality of the degree reverse lexicographic reduced Gröbner basis of the ideal of phylogenetic invariants: -
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\\
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\\
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\\
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}{16} \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\\
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\\
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\\
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} $