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Lev Borisov
Lev Borisov
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In general, $g(n)$ is strictly less than $f(n)$, although they have the same leading term. One sufficient condition to get $g(n)=f(n)$ is to require smoothness of the corresponding toric variety. Combinatorially, this means that at every vertex $q$ of $P$ there are exactly $\dim P$ edges coming out of it; moreover if you take $q_i$ to be the closest points to $q$ on these edges, the simplex with vertices $q,q_1,...,q_{\dim P}$ has minimum possible volume. I believe the statement is pretty much local near all the vertices: if you can generate all points in $kP$ that are close to any given vertex, then you will be able to eventually generate all points.

Edit: I made a mistake in claiming that $g$ and $f$ have the same leading term. This only happens if the differences of points in $P$ generate the whole lattice. A simple counterexample in dimension three is the convex hull of $(0,0,0),(1,1,0),(1,0,N),(0,1,N)$ for some $N$.

In general, $g(n)$ is strictly less than $f(n)$, although they have the same leading term. One sufficient condition to get $g(n)=f(n)$ is to require smoothness of the corresponding toric variety. Combinatorially, this means that at every vertex $q$ of $P$ there are exactly $\dim P$ edges coming out of it; moreover if you take $q_i$ to be the closest points to $q$ on these edges, the simplex with vertices $q,q_1,...,q_{\dim P}$ has minimum possible volume. I believe the statement is pretty much local near all the vertices: if you can generate all points in $kP$ that are close to any given vertex, then you will be able to eventually generate all points.

In general, $g(n)$ is strictly less than $f(n)$, although they have the same leading term. One sufficient condition to get $g(n)=f(n)$ is to require smoothness of the corresponding toric variety. Combinatorially, this means that at every vertex $q$ of $P$ there are exactly $\dim P$ edges coming out of it; moreover if you take $q_i$ to be the closest points to $q$ on these edges, the simplex with vertices $q,q_1,...,q_{\dim P}$ has minimum possible volume. I believe the statement is pretty much local near all the vertices: if you can generate all points in $kP$ that are close to any given vertex, then you will be able to eventually generate all points.

Edit: I made a mistake in claiming that $g$ and $f$ have the same leading term. This only happens if the differences of points in $P$ generate the whole lattice. A simple counterexample in dimension three is the convex hull of $(0,0,0),(1,1,0),(1,0,N),(0,1,N)$ for some $N$.

the word "minimum" is clarified to "minimum possible volume"
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Lev Borisov
Lev Borisov
  • 5.2k
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  • 22
  • 38

In general, $g(n)$ is strictly less than $f(n)$, although they have the same leading term. One sufficient condition to get $g(n)=f(n)$ is to require smoothness of the corresponding toric variety. Combinatorially, this means that at every vertex $q$ of $P$ there are exactly $\dim P$ edges coming out of it; moreover if you take $q_i$ to be the closest points to $q$ on these edges, the simplex with vertices $q,q_1,...,q_{\dim P}$ ishas minimum possible volume. I believe the statement is pretty much local near all the vertices: if you can generate all points in $kP$ that are close to any given vertex, then you will be able to eventually generate all points.

In general, $g(n)$ is strictly less than $f(n)$, although they have the same leading term. One sufficient condition to get $g(n)=f(n)$ is to require smoothness of the corresponding toric variety. Combinatorially, this means that at every vertex $q$ of $P$ there are exactly $\dim P$ edges coming out of it; moreover if you take $q_i$ to be the closest points to $q$ on these edges, the simplex with vertices $q,q_1,...,q_{\dim P}$ is minimum. I believe the statement is pretty much local near all the vertices: if you can generate all points in $kP$ that are close to any given vertex, then you will be able to eventually generate all points.

In general, $g(n)$ is strictly less than $f(n)$, although they have the same leading term. One sufficient condition to get $g(n)=f(n)$ is to require smoothness of the corresponding toric variety. Combinatorially, this means that at every vertex $q$ of $P$ there are exactly $\dim P$ edges coming out of it; moreover if you take $q_i$ to be the closest points to $q$ on these edges, the simplex with vertices $q,q_1,...,q_{\dim P}$ has minimum possible volume. I believe the statement is pretty much local near all the vertices: if you can generate all points in $kP$ that are close to any given vertex, then you will be able to eventually generate all points.

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Lev Borisov
Lev Borisov
  • 5.2k
  • 1
  • 22
  • 38

In general, $g(n)$ is strictly less than $f(n)$, although they have the same leading term. One sufficient condition to get $g(n)=f(n)$ is to require smoothness of the corresponding toric variety. Combinatorially, this means that at every vertex $q$ of $P$ there are exactly $\dim P$ edges coming out of it; moreover if you take $q_i$ to be the closest points to $q$ on these edges, the simplex with vertices $q,q_1,...,q_{\dim P}$ is minimum. I believe the statement is pretty much local near all the vertices: if you can generate all points in $kP$ that are close to any given vertex, then you will be able to eventually generate all points.