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This is related to another question of mine

Let $V=\Bbb R^n$. Morelli defined the (commutative unital) ring $L(V)$ to be the additive group generated by the indicator functions of convex polytopes in $V$ with multiplication induced by Minkowski sum.

Question: Are there any positive-dimensional polytopes $P$ satisfy the following condition (*)? If there are, is there a “good” (take that to mean what you will) characterization of them?

(*) Viewed as elements of $L(V)$, the set of faces of $P$ (including $P$ itself) is algebraically independent (over $\Bbb Z$).

Example: (*) is always false for zonotopes essentially by definition.

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  • $\begingroup$ what is the addition operation in the ring $L(V)$? $\endgroup$ – Yossi Lonke Nov 21 '18 at 8:05
  • $\begingroup$ @YossiLonke Minkowski sum $\endgroup$ – Avi Steiner Nov 21 '18 at 13:46
  • $\begingroup$ You write that "multiplication is induced by Minkowski sum." In a ring, there are two operations: multiplication and addition. So I repeat my question: what is the addition operation in this ring? $\endgroup$ – Yossi Lonke Nov 21 '18 at 22:06
  • $\begingroup$ @YossiLonke The multiplication operation is minkowski sum, while the addition operation is function addition $\endgroup$ – Avi Steiner Nov 22 '18 at 2:01

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