# When is the set of faces of a convex polytope algebraically independent?

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.

• what is the addition operation in the ring $L(V)$? – Yossi Lonke Nov 21 '18 at 8:05
• @YossiLonke Minkowski sum – Avi Steiner Nov 21 '18 at 13:46
• 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? – Yossi Lonke Nov 21 '18 at 22:06
• @YossiLonke The multiplication operation is minkowski sum, while the addition operation is function addition – Avi Steiner Nov 22 '18 at 2:01