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The integer point transform as a complete invariant (Sinai Robins, University of São Paulo, Brazil)
October 7, 2025 @ 12:15 pm - 1:10 pm
Given any finite set of integer points S, there is an associated function f_S that encodes S, which we call its integer point transform. One can think of this integer point transform f_S algebraically or analytically. Here we focus on its analytic properties, showing that it is a complete invariant. In fact, we prove that it is only necessary to evaluate f_S at one algebraic point in order to uniquely determine the finite set S, by employing the Lindemann-Weierstrass theorem. Similarly, we prove that it’s only necessary to evaluate the Fourier transform of a rational polytope P (as well as rational cones) at a single algebraic point, in order to uniquely determine S. Finally, by relating the integer point transform to finite Fourier transforms, we show that a finite number of integer point evaluations of f_S suffice in order to uniquely determine S.
