Linear Recursions for Integer Point Transforms

Jochemko, Katharina

doi:10.1007/978-3-030-98327-7_10

Cited by 1 publication

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“…The work of Fink, Mészáros, and Dizier [4] gives applications of integer point transforms to Schubert polynomials. The recent work of Katharina Jochemko [6] shows that the sequence of integer point transforms σ kP +Q k≥0 satisfies a multivariate linear recursion, where P is an integer polytope, and Q is any polytope. Here we do not assume knowledge of Ehrhart theory, and rather proceed from first principles.…”

Section: Introductionmentioning

confidence: 99%

The integer point transform as a complete invariant

Robins¹

2023

Communications in Mathematics

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The integer point transform $\sigma_\PP$ is an important invariant of a rational polytope $\PP$, and here we show that it is a complete invariant. We prove that it is only necessary to evaluate $\sigma_\PP$ at one algebraic point in order to uniquely determine $\PP$, by employing the Lindemann-Weierstrass theorem. Similarly, we prove that it is only necessary to evaluate the Fourier transform of a rational polytope $\PP$ at a single algebraic point, in order to uniquely determine $\PP$. We prove that identical uniqueness results also hold for integer cones. In addition, by relating the integer point transform to finite Fourier transforms, we show that a finite number of \emph{integer point evaluations} of $\sigma_\PP$ suffice in order to uniquely determine $\PP$. We also give an equivalent condition for central symmetry of a finite point set, in terms of the integer point transform, and prove some facts about its local maxima. Most of the results are proven for arbitrary finite sets of integer points in $\R^d$.

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Section: Introductionmentioning

confidence: 99%

The integer point transform as a complete invariant

Robins¹

2023

Communications in Mathematics

View full text Add to dashboard Cite

show abstract

Linear Recursions for Integer Point Transforms

Cited by 1 publication

References 14 publications

The integer point transform as a complete invariant

The integer point transform as a complete invariant

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