A new algorithm for computing regular representations for radicals of parametric differential ideals

Figures & data

Table 1. Discussion of the system $P_1$ (Lorenz system)

Case	Regular ideals
$a\ne 0$	$[au-av+u_x,-abu+auw+au+a{u}_x+u_x+u_{x,x},$ $-abc{u}^2+a{u}^4+ac{u}^2+acu{u}_x+u^3{u}_x+au{u}_{x,x}-a{u}_x^2+cu{u}_x$ $+cu{u}_{x,x}+u{u}_{x,x}+u{u}_{x,x,x}-u_x^2-u_x{u}_{x,x}]:\{u{\}}^{\mathrm{\infty }},$ $[u,v,cw+w_x]$
$a=0$	$[u_x,-bu+uw+v+v_x,bcu-u^{2}v-cv-c{v}_x-v_x-v_{x,x}]:$ $\{u{\}}^{\mathrm{\infty }},[u,v+v_x,cw+w_x]$

Table 2. Discussion of system $P_3$

Case	Regular ideals
$a,b,c,a+1\ne 0$	$[4ac{u}_x-b^{2}v+4c{u}_x,-b{v}_x-2c,v_y,a{u}_{y,y}-1],$ $[av{u}_{x,x}^2+bv{u}_{x,x}+v{u}_{x,x}^2+c{u}_x,u_{x,y},v_y,a{u}_{y,y}-1]:\{2av{u}_{x,x}+bv$ $+2v{u}_{x,x}{\}}^{\mathrm{\infty }},[u_x,a{u}_{y,y}-1]$
$a,c,a+1\ne 0$, $b=0$	$[u_x,a{u}_{y,y}-1],[av{u}_{x,x}^2+bv{u}_{x,x}+v{u}_{x,x}^2+c{u}_x,u_{x,y},v_y,$ $a{u}_{y,y}-1]:\{2av{u}_{x,x}+bv+2v{u}_{x,x}{\}}^{\mathrm{\infty }}$
$a,a+1\ne 0$, $b=c=0$	$[u_{x,x},u_{x,y},a{u}_{y,y}-1],[v,u_{x,y},a{u}_{y,y}-1]$
$a,a+1,b\ne 0$, $c=0$	$[u_{x,y},a{u}_{x,x}+b+u_{x,x},a{u}_{y,y}-1],[u_{x,y},u_{x,x},a{u}_{y,y}-1],$ $[v,u_{x,y},a{u}_{y,y}-1]$
$a=-1,b,c\ne 0$	$[v,u_x,-u_{y,y}-1],[u_x,-u_{y,y}-1]:\{v{\}}^{\mathrm{\infty }},$ $[bv{u}_{x,x}+2c{u}_x,u_{x,y},v_y,-u_{y,y}-1]:\{v{\}}^{\mathrm{\infty }}$
$a=-1,c=0$, $b\ne 0$	$[v,u_{x,y},-u_{y,y}-1],[u_{x,y},u_{x,x},-u_{y,y}-1]$
$a=-1,b=0$, $c\ne 0$	$[u_x,-u_{y,y}-1]$
$a=-1,b,c=0$	$[u_{x,y},-u_{y,y}-1]$