Let $X$ be a compact, convex subset of $\mathbb R^n$ and $P:X\to X$ denote a system of $n$ polynomial equations with integer coefficients, where the polynomial of each equation $P_i$ has finite degree. The number of equations equals the number of variables.
Is it true that $P$ has either uncountably many or finite solutions?
A solution of $P$ is $x \in X$ such that $P_i(x)=0$ for all $i=1,...,n$, where $0 \in X$.
No. For instance, let $a_n=(1/n,1/n^2)$ for each $n\in\mathbb{Z}_+$, let $X\subset\mathbb{R}^2$ be the closed convex hull of the $a_n$ together with $(0,1)$ (concretely, $X$ is the infinite-sided "polygon" formed by connecting the points $a_n$ in sequence, and then connecting their limit $(0,0)$ to $(0,1)$, and then connecting $(0,1)$ back to $a_1=(1,1)$). Let $P:X\to X$ be defined by $P(x,y)=(0,y-x^2)$. Then the solutions to $P$ in $X$ are exactly the countably infinitely many points $a_n$ together with $(0,0)$, because $x\mapsto x^2$ is strictly convex so all the "sides" of $X$ are above the parabola where $y=x^2$ except at their endpoints.