# Roots of Hilbert polynomials for Fano toric varieties

Half a year ago Sergey Galkin, Swarnava Mukhopadhyay and I put a preprint on the arXiv regarding the *canonical strip hypothesis*, where we gave the first counterexamples violating this hypothesis. Let me recall the setup.

If $X$ is a smooth projective Fano variety (i.e. $\omega_X$ is ample) we can look at its canonical polarisation $\bigoplus_{n\geq 0}\mathrm{H}^0(X,\omega_X^{\otimes n})$. As such we obtain a Hilbert series, and therefore a *Hilbert polynomial*. The canonical strip hypothesis concerns the locations of the roots of this polynomial. Because Serre duality involves the canonical bundle $\omega_X$, we obtain a symmetry around the axis (or *critical line*) $\Re=-\frac{1}{2}$ for these roots.

Based on inequalities involving Chern numbers Golyshev hypothesised that these roots are not only symmetric around this critical line, but are actually contained within the "canonical strip", where $-1\leq\Re\leq 0$. This is the case in low dimension, and for all complete intersections in partial flag varieties.

In the paper you can find the details for a family of Fano varieties violating this hypothesis, and a picture of these roots suggesting that something interesting happening with the positions of these roots. This blogpost is not about this picture.

I want to discuss a different picture. Whilst writing the paper we also computed Hilbert polynomials for all smooth projective toric Fano varieties up to dimension 7 (there are 80891 of these), and observed that the hypothesis holds. The results of this computation can be found in the GitHub repository `pbelmans/ehrhart-polynomials-toric-fanos`

As a fun exercise in using D3 I decided to plot these pictures, and share them with you. Below you can find the picture up to dimension 5, and a standalone version also includes dimension 6. There's no picture up to dimension 7, as this would involve too many roots.

I do not know what one should conclude from these pictures. It seems that there might be a pattern, but it is unclear to me what it should be.

Maybe I'll continue my D3 exercise at some point. It'd be nice to have a toggling option, zooming and dragging. But for now this will be it.