  For maps from $S^3$ and $\RP^3$ into $S^2$ and $\RP^2$, we study the problem
of minimizing the root set by deforming the maps through homotopies. After
presenting the classification of the homotopy classes of such maps, we prove
that the minimal root set for a non null-homotopic map is either a circle or
the disjoint union of two circle, according its range is $S^2$ or $\RP^2$,
respectively.
