Say that a compact convex polytope is *rational* if is the intersection of half-spaces whose bounding hyperplanes are the zero-sets of affine functions of the coordinates with rational coefficients. Say that a continuous function $f$ from a rational compact convex polytope $K$ into itself is *rational continuous piecewise affine* if there exists a dissection of $K$ into finitely many rational polytopes $K\_1,\dots,K\_n$ (whose interiors are disjoint and whose union is $K$) such that for all $1 \leq i \leq n$ the restriction of $f$ to $K\_i$ is an affine map with rational coefficients. (People sometimes write "piecewise linear" where I write "piecewise affine" and abbreviate "continuous piecewise linear" as "cpl".)


The Brouwer fixed point theorem assures us that such a map $f$ must have a fixed point. But must it have a fixed point with rational coordinates?



Fixed points of rational continuous piecewise affine maps
A fixed-point is obtained as the intersection of two affine subspaces (a piece of the graph of $f$ and the identity graph) whose equations are defined over $\mathbb{Q}$. Therefore Cramer's formula (applied to a maximal rank subsystem) ensures that if there exists a solution then there exists a rational one.


