Discovered Conservative Matrix Fields

This page contains some of the conservative matrix fields we discovered. They can all be generated through \( f, \bar{f} \) representation.

The conservative matrix field of \(e\):

\(
M_X = \begin{pmatrix} 0 & x+1 \\\\ 1 & -(x+y+1) \end{pmatrix},
M_Y = \begin{pmatrix} -1 & x+1 \\\\ 1 & -(x+y+2) \end{pmatrix}
\)

It can also be constructed by using \(a=-1, c=0, \bar{a}=0, \bar{c}=-1\)

The conservative matrix field of \(\pi\)

\(
M_X = \begin{pmatrix} 0 & -(2x+1)x \\\\ 1 & y+3x+2 \end{pmatrix},
M_Y = \begin{pmatrix} y-x & -(2x+1)x \\\\ 1 & 2x+2y+1 \end{pmatrix}
\)

It can also be constructed by using \(a=2, c=1, \bar{a}=1, \bar{c}=0\)

The conservative matrix field of \(\zeta(2)\)

\(
M_X = \begin{pmatrix} 0 & -x^2 \\\\ (x+1)^2 & x^2 + (x+1)^2 +y(y-1) \end{pmatrix}
M_Y = \begin{pmatrix} -x^2 + xy -y^2/2 & -x^2 \\\\ x^2 & x^2 +xy+y^2/2\end{pmatrix}
\)

It can be constructed using the degree-2 structure with the coefficients \(\vec{c}=(0,0,0,1)\)

The conservative matrix field of \(\zeta(3)\)

\(
M_X(x,y)=\begin{pmatrix} 0 & -x^6 \\\\ 1 & x^3 + (x+1)^3 + 2y(y-1)(2x+1) \end{pmatrix}
M_Y(x,y) = \begin{pmatrix}-(x-y)(x^2-xy+y^2) & -x^6 \\\\ 1 & (x+y)(x^2+xy+y^2)\end{pmatrix}
\)

It can be constructed using the first degree-3 structure with the coefficients \(\vec{c}=(0,1)\)