Gevrey Expansions of Hypergeometric Integrals II

We study integral representations of the Gevrey series solutions of irregular hypergeometric systems under certain assumptions. We prove that, for such systems, any Gevrey series solution, along a coordinate hyperplane of its singular support, is the asymptotic expansion of a holomorphic solution given by a carefully chosen integral representation.

Fano schemes for generic sums of products of linear forms

We study the Fano scheme of [Formula: see text]-planes contained in the hypersurface cut out by a generic sum of products of linear forms. In particular, we show that under certain hypotheses, linear subspaces of sufficiently high dimension must be contained in a coordinate hyperplane. We use our results on these Fano schemes to obtain a lower bound for the product rank of a linear form. This provides a new lower bound for the product ranks of the [Formula: see text] Pfaffian and [Formula: see text] permanent, as well as giving a new proof that the product and tensor ranks of the [Formula: see text] determinant equal five. Based on our results, we formulate several conjectures.

Stretching Convex Domains to Capture Many Lattice Points

We consider an optimal stretching problem for strictly convex domains in $\mathbb{R}^d$ that are symmetric with respect to each coordinate hyperplane, where stretching refers to transformation by a diagonal matrix of determinant 1. Specifically, we prove that the stretched convex domain which captures the most positive lattice points in the large volume limit is balanced: the (d − 1)-dimensional measures of the intersections of the domain with each coordinate hyperplane are equal. Our results extend those of Antunes and Freitas, van den Berg, Bucur and Gittins, Ariturk and Laugesen, van den Berg and Gittins, and Gittins and Larson. The approach is motivated by the Fourier analysis techniques used to prove the classical $\#\{(i,j) \in \mathbb{Z}^2 : i^2 +j^2 \le r^2 \} =\pi r^2 + \mathcal{O}(r^{2/3})$ result for the Gauss circle problem.