Recovery from Power Sums

Figures & data

Table

${\mathcal{A}}_2$ and ${\mathcal{A}}_3$	Number of solutions to (6)	Possibilities for $P\prime$ in Conjecture 12
$0\in {\mathcal{A}}_2,0\in {\mathcal{A}}_3$	0	–
$0\notin {\mathcal{A}}_2,0\in {\mathcal{A}}_3$	2	$(1:-1:0),(1:0:-1)$
$0\in {\mathcal{A}}_2,0\notin {\mathcal{A}}_3$	2	$(1:\zeta :{\zeta }^2),(1:{\zeta }^2:\zeta )$
$0\notin {\mathcal{A}}_2,0\notin {\mathcal{A}}_3$	4	$(1\mathrm{}:\mathrm{}-1\mathrm{}:\mathrm{}0),(1\mathrm{}:\mathrm{}0\mathrm{}:\mathrm{}-1)$, $(1\mathrm{}:\mathrm{}\zeta \mathrm{}:\mathrm{}{\zeta }^2),(1\mathrm{}:\mathrm{}{\zeta }^2\mathrm{}:\mathrm{}\zeta )$

Fig. 1 The image of the norm map ${\psi }_{\mathcal{A}}$ for $n=6,m=3$ is a curvy hexagon in a triangle. The color coding on the left shows the progression of images for $n=3,4,5,6$. The color coding on the right shows the algebraic degrees 12 (red) and 24 (blue) of the curvy segments.