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Are there any results stating that a given family of convex polytopes have Ehrhart polynomials with non-negative coefficients?

What methods are available for proving such a property for some family of polytopes?

Remember, the Ehrhart polynomial $p(k)$ for a convex polytope $P$ with integer vertices is given by the property that $p(k)$ counts the number of integer lattice points in the $k$-dilation of $P$, where $k$ is a positive integer.

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I have converted a comment (slightly modified) to the following answer as requested by Per Alexandersson. If $Ω_P(k)$ is the order polynomial of a poset $P$, then $Ω_P(k+1)$ is the Ehrhart polynomial of the order polytope $\mathcal{O}(P)$. For any n≥1, the Ehrhart polynomial of the order polytope of the poset $P_n$ with one minimal element covered by $n$ other elements is $\sum_{i=1}^{k+1}i^n$. For $n=20$ the coefficient of $k$ is $−168011/330$, so Ehrhart polynomials of 0/1 polytopes (or even order polytopes) need not have nonnegative coefficients. Incidentally, it was easy to check using Stembridge's posets package for Maple that the Ehrhart polynomial of the order polytope of any poset with at most eight elements has nonnegative coefficients.

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  • $\begingroup$ Richard Stanley's example of a non-Ehrhart positive order polytope is discussed in more detail here: arxiv.org/abs/1806.08403 $\endgroup$ – Sam Hopkins Jun 25 '18 at 23:52
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There seems to be a lot of information (including the conjecture that this holds for the Birkhoff polytope) in Richard Stanley's slides. Of course, since Richard is a frequent participant, he can (and surely will) say more.

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    $\begingroup$ All the known explicit Ehrhart polynomials for the Birkhoff polytope are at math.binghamton.edu/dennis/Birkhoff $\endgroup$ – Brendan McKay Oct 29 '14 at 22:57
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    $\begingroup$ The Ehrhart polynomial of an order polytope can have negative coefficients. See EC1, 2nd ed., Exericse 3.164. Ehrhart polynomials of integer zonotopes do have nonnegative coefficients (e.g., Theorem 2.2 of math.mit.edu/~rstan/pubs/pubfiles/83.pdf). I am not sure if it's known whether or not the generalized permutohedra of Postnikov (math.mit.edu/~apost/papers/permutohedron_full.pdf) have this property. $\endgroup$ – Richard Stanley Mar 18 '15 at 0:49
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    $\begingroup$ @RichardStanley: I might be mistaken, but the first poset Figure 3.87 in EC1 2nd ed, gives the inequalities $z \leq x_1 \leq y$, $z\leq x_2$ and $z\leq x_3$ together with the condition that all values are between $0$ and $k$. Asking Mathematica for counting solutions gives the polynomial $\frac{1}{120} (k+1) (k+2) (k+3) \left(12 k^2+33 k+20\right)$. The code I used for counting lattice points: Table[Length@List@ToRules@Reduce[ And[z <= x1 <= y, z <= x2, z <= x3] && And[0<=z<=k,0<=y<=k,0<=x1<=k,0<=x2<=k,0<=x3<= k], {z, y, x1, x2, x3}, Integers],{k, 0, 5}] $\endgroup$ – Per Alexandersson Mar 19 '15 at 14:44
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    $\begingroup$ @PerAlexandersson: yow, you are right! I have made this mistake before. If $\Omega_P(k)$ is the order polynomial of a poset $P$, then $\Omega_P(k+1)$ (not $\Omega_P(k)$) is the Ehrhart polynomial of the order polytope. However, for any $n\geq 1$, the order polytope of the poset $P_n$ with one minimal element covered by $n$ other elements is $\sum_{i=1}^{k+1}i^n$. For $n=20$ the coefficient of $k$ is $-168011/330$, so Ehrhart polynomials of 0/1 polytopes need not have nonnegative coefficients. $\endgroup$ – Richard Stanley Mar 20 '15 at 18:17
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    $\begingroup$ The zeros of the Ehrhart polynomial $H_n(k)$ of the Birkhoff polytope of $n\times n$ doubly stochastic matrices look very interesting. For $9\times 9$ matrices see math.mit.edu/~rstan/zeros/magic9.pdf. Assuming that this behavior generalizes to all $n$, we would have the following. Let $c(n,i)$ be the coefficient of $k^{(n-1)^2-i}$ in $H_n(k)$. Then for fixed $k$, $c(n,i)/c(n,0)\sim n^{3i}/2^ii!$. See EC1, 2nd ed., Exercise~4.54. $\endgroup$ – Richard Stanley Mar 21 '15 at 0:11
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Fu Liu recently put on the arXiv a nice survey about this question: https://arxiv.org/abs/1711.09962v1

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Proposition 4 of Morelli's paper "Pick's Theorem and the Todd class of a toric variety" gives a sufficient condition: it describes a setting in which there is a positive formula for coefficient of $x^k$ as a sum over $k$-dimensional faces.

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