Open problems, finite certificates

Construct a counterexample $F:\mathbb{C}^3\to\mathbb{C}^3$. No degree or sparsity restriction is imposed.

Let $u=1+xy$. A known three-dimensional construction is
$F_1=u^3z+y^2u(4+3xy),\qquad F_2=y+3xu^2z+3xy^2(4+3xy),\qquad F_3=2x-3x^2y-x^3z.$
It satisfies $\det J_F=-2$, and
$(0,0,-\tfrac14),\ (1,-\tfrac32,\tfrac{13}{2}),\ (-1,\tfrac32,\tfrac{13}{2})$
all map to $(-\tfrac14,0,0)$. The submission must be a genuinely new counterexample algebraically inequivalent to this reference. Reformulations, variable or component permutations, reparametrizations, and constructions obtainable by invertible algebraic coordinate changes in the source or target are inadmissible.

Construct a counterexample $F:\mathbb{C}^3\to\mathbb{C}^3$ with $\max_i\deg F_i\le 7$. Supply at least two distinct algebraic points $p_1,p_2$ in one common fiber, so that $F(p_1)=F(p_2)$.

Let $u=1+xy$. A known construction is
$F_1=u^3z+y^2u(4+3xy),\qquad F_2=y+3xu^2z+3xy^2(4+3xy),\qquad F_3=2x-3x^2y-x^3z.$
Its component degrees are $(7,6,4)$, $\det J_F=-2$, and the three points $(0,0,-\tfrac14)$, $(1,-\tfrac32,\tfrac{13}{2})$, and $(-1,\tfrac32,\tfrac{13}{2})$ share the image $(-\tfrac14,0,0)$. The submission must be a genuinely new counterexample algebraically inequivalent to this reference. Reformulations and constructions obtainable by invertible algebraic coordinate changes are inadmissible.

Construct a counterexample $F:\mathbb{C}^3\to\mathbb{C}^3$ whose polynomial degree $\deg F:=\max_i\deg F_i$ is strictly smaller than the currently known minimum $7$. Supply at least two distinct algebraic points $p_1,p_2$ in one common fiber, so that $F(p_1)=F(p_2)$.

For reference only, let $u=1+xy$ and consider
$F_1=u^3z+y^2u(4+3xy),\qquad F_2=y+3xu^2z+3xy^2(4+3xy),\qquad F_3=2x-3x^2y-x^3z.$
This map has $\det J_F=-2$, three known collision points, and $\deg F=7$. The task is to improve the currently known minimum, so the reference construction itself does not qualify.

Construct a counterexample $F:\mathbb{C}^3\to\mathbb{C}^3$ with generic fiber degree $\deg_{\mathrm{gen}}F=4$ and polynomial degree $\deg F:=\max_i\deg F_i\le 11$. The currently known construction with generic fiber degree $4$ has polynomial degree $12$; improve this bound and supply four pairwise-distinct algebraic points $p_1,p_2,p_3,p_4$ in one common fiber.

For structural reference only, the first known three-dimensional map
$F_1=(1+xy)^3z+y^2(1+xy)(4+3xy),\quad F_2=y+3xz(1+xy)^2+3xy^2(4+3xy),\quad F_3=2x-3x^2y-x^3z$
satisfies $\det J_F=-2$ and has three supplied collision points mapping to $(-\tfrac14,0,0)$. It is not a solution to this task: the target is a four-sheeted generic fiber with $\deg F\le 11$, improving the currently known degree-$12$ construction.

Construct a counterexample $F:\mathbb{C}^2\to\mathbb{C}^2$. No degree or sparsity restriction is imposed.

For reference only, the known three-dimensional map
$F_1=(1+xy)^3z+y^2(1+xy)(4+3xy),\quad F_2=y+3xz(1+xy)^2+3xy^2(4+3xy),\quad F_3=2x-3x^2y-x^3z$
has $\det J_F=-2$ and three collision points. Because it is a map on $\mathbb{C}^3$, it is only a structural reference for this two-dimensional task.