Symplectic embeddings II revisited

As the title suggests, this is not so much a new instalment of the symplectic embeddings saga; more a development of the last one. To set the scene again, I have been exploring the role lattice point counts play in symplectic embedding problems – specifically, those involving ellipsoids – and have some new comments to make, which I found interesting at the time.

While the coefficients of Ehrhart polynomials of lattice polygons have a clear intepretation, which is more or less Pick’s theorem, the coefficients appearing in Ehrhart quasipolynomials for rational polygons are less well understood. One can do some manipulations to make a bit of progress – for instance, the leading term of each polynomial constituting the quasipolynomial is always the volume of the \ell\text{th} dilate of the polygon, where \ell is its index – although the linear coefficients especially seem to be quite mysterious. The periodic nature of quasipolynomials mean that certain inequalities are satisfied between coefficients. These give rise to the claims made above and allow one to more efficiently test whether or not an ellipsoid can be symplectically embedded in another.

As discussed previously, McDuff’s result states that, to test whether or not one can embed E(a,b) into E(c,d), one must establish that the sequences

\mathcal{N}(a,b)\leq\mathcal{N}(c,d).

Ideally one would have some result saying that this holds iff the first blah entries of each are bounded in this way, but such a claim seems hard to come by, if possible with a uniform bound on blah. Anyway, at least one can test it via a finite check, which I improved to 2\ell-1 over 3\ell

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