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 show 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 of order larger than four are not semialgebraic. Actually, we guess that it is the case and we provide strong evidence supporting our conjecture.
We study the definability of the almost-entropic regions by finite lists of algebraic inequalities. First, we study linear information inequalities and polyhedrality, we present a proof of a theorem of Matus, which claims that the almost-entropic regions are not polyhedral. Then, we study polynomial inequalities and semilagebraicity, we show that the semialgebracity of the almost-entropic regions is something that depends on the essentially conditionality of a certain class of conditional information inequalities. Those results suggest that the almost-entropic regions are not semialgebraic. We conjecture that those regions are not decidable.
We study the definability of the almost-entropic regions by finite lists of algebraic inequalities. First, we study linear information inequalities and polyhedrality, we present a proof of a theorem of Matus, which claims that the almost-entropic regions are not polyhedral. Then, we study polynomial inequalities and semilagebraicity, we show that the semialgebracity of the almost-entropic regions is something that depends on the essentially conditionality of a certain class of conditional information inequalities. Those results suggest that the almost-entropic regions are not semialgebraic. We conjecture that those regions are not decidable.
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