**Background**: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan.

$X_P$ is always projective, because the collection of characters corresponding to the points $\mathbb{Z}^n \cap P$ together give an embedding of $X_P$ into projective space.

However, the dimension of this embedding is the number of integer points, which is generally exponentially large in a reasonable description of $P$.

**Questions**:

- Suppose that $P$ is given by $Ax \leq b$ in $\mathbb{Q}^n$, with $A \in M_{n \times m} (\mathbb{Q})$ and $b \in \mathbb{Q}^n$, and that we are promised that $P$ is integral. Is there a projective embedding of $X_P$ which only requires $POLY( |A|, |b|)$ bits to specify?
- Is there a family $A_n, b_n$ such that the minimal dimension of an embedding grows exponentially in $|A|, |b|$?
- Are there some parameters of the polytope (in the sense of parametrized complexity) that control the size of a minimal (and efficiently computable) embedding?

I am being purposefully vague about whether the encoding of $A$ is in binary or unary; both polynomial or pseudopolynomial size embeddings would be interesting to me.

**Motivation**: I am curious about whether there are polytope parameters that become apparent through simple embeddings of the corresponding toric variety, and which could help with computational problems on the polytope side.

For example, if we know that $X_P$ is a smooth complete intersection and we have the equations cutting it out, we can compute its Euler characteristic using the formula on page 146 of "On the Chern Classes and the Euler Characteristic for Nonsingular Complete Intersections" by Vicente Navarro Aznar. This would count the number of vertices of $P$, which is generally a $\#P $ hard problem. Of course, most polytopes won't give a smooth toric variety or a complete intersection, and very likely computing the embedding is hard, so this observation is of limited use.

Alternatively, one could use this formula relating $F_p$ point counts to the fan in order to count the number of vertices, provided that the variety was such that counting $F_p$ points at sufficiently many (or sufficiently) large Galois fields was computationally tractable.

Anyhow, I'm curious about whether we can measure the complexity of the polytoe by the complexity of the toric variety as a projective variety. The basic question is whether or not we can efficiently find small embeddings in general, hence this question.