| Resumo: |
Given a convex polytope \\PP\\subset \\R^d, its integer point transform \\sigma_{\\PP} encodes the information contained in \\PP and is in fact a complete invariant of \\PP, as recently shown in \\cite. One motivation for the study of \\sigma_{\\PP} is that it is an extension of the number of integer points in \\PP, called the integer point enumerator of \\PP. 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 \\sigma_{\\PP} by considering its null set \\Nullreal(\\sigma_{\\PP}):= \\{ x \\in \\R^d \\mid \\sigma_{\\PP}(x) = 0 \\}. We computed many examples using the Desmos software package \\cite, which is an advanced graphing calculator implemented as a web application. With these examples we would be able to create new conjectures, some of which we prove here. For many integer polygons \\PP, we give a complete list of the rational zeroes of \\sigma_{\\PP}, and it sometimes turns out that these comprise all of their zeroes. Such zeros have been studied before and have been called cyclotomic points on algebraic curves, such as in the work of Aliev and Smyth \\cite. We also study symmetric properties of integer point transforms, and prove that their null sets are equivariant under all rigid motions over the integers. 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 \\sigma_{\\PP}, for some centrally-symmetric polytopes \\PP, a problem that is related to the classical problem of minima of cosine polynomials. |
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