Title: The zeros of the integer point transform
Data: 15 de julho de 2024 – segunda-feira – 14h - https://meet.google.com/zfm-qcsk-yqk
Student: Ana Carolina Laurini Malara

Abstract: Given a convex polytope P ⊂ Rd, its integer point transform σP encodes the infor- mation contained in P, and it is in fact a complete invariant of P [13]. One motivation for the study of σP is that it is an extension of the number of integer points in P, called the integer point enumerator of P. Although Minkowski initiated the study of integer point enumerators of convex bodies, their integer point transforms have only been studied in more recent times. Here we continue the study the integer point transform σP by considering its null set NullR(σP) := {x ∈ R d | σP (x) = 0}.

We use the software Desmos to generate many conjectures, some of which we prove here. For many integer polygons P, we give a complete list of the rational zeroes of σP , and it sometimes turns out that these comprise all of their zeroes. Such zeros have been studied before and have been called cyclotomic zeroes of algebraic curves, such as in the work of Aliev and Smyth [1]. We study symmetric properties of integer point transforms, and prove that their null sets are equivariant under the hyperoctahedral group. Along the way, we discover that the integer point transforms of unimodular simplices are intimately related to Vandermonde determinants. Finally, we consider the local minima of σP, for some centrally-symmetric polytopes P, a problem that is related to the classical problem of minima of cosine polynomials.