On the decidability of Diophantine problems in combinatorial geometry

Bernd Sturmfels
1987 Bulletin of the American Mathematical Society  
In spite of Matiyasevic's solution to Hubert's 10th problem some fifteen years ago it is still unknown whether there exists an algorithm to decide the solvability of diophantine equations within the field of rational numbers. In this note we show the equivalence of this problem with a conjecture of B. Grünbaum [6] on rational coordinatizability in combinatorial geometry. Such an algorithm exists if and only if the rational realizability problems for matroids, oriented matroids, and convex
more » ... s, and convex polytopes (Steinitz problem) are decidable.
doi:10.1090/s0273-0979-1987-15532-7 fatcat:q3lav4g6yjc4fcnkzzv7eflufm