Linear network coding and the model theory of linear rank inequalities

Let n ≥ 4. Can the entropic region of order n be defined by a finite list of polynomial inequalities? This question was first asked by Chan and Grant. We showed, in a companion paper, that if it were the case one could solve many algorithmic problems coming from network coding, index coding and secret sharing. Unfortunately, it seems that the entropic regions are not semialgebraic. Are the Ingleton regions semialgebraic sets? We provide some evidence showing that the Ingleton regions are semialgebraic. Furthermore, we show that if the Ingleton regions are semialgebraic, then one can solve many algorithmic problems coming from Linear Network Coding.