On moments of a polytope

Gravin, Nick; Ṗasechnik, Dmitrii V.; Shapiro, Boris; Shapiro, Michael

doi:10.1007/s13324-018-0226-8

Cited by 10 publications

(17 citation statements)

References 23 publications

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“…These classical equations characterize polynomials that are products of linear factors, among all polynomials of degree If we are given numerical values in Q for the moments m I then the factorization (18) is found in exact arithmetic by the built-in factorization methods in any computer algebra system, provided the vertex coordinates x kl of our simplex are rational numbers. If the moments m I are rational but the x kl are not rational then they are algebraic over Q, and one can use algorithms for absolute factorization to obtain the right-hand side of (18). If the moments are floating point numbers then one uses tools from numerical algebraic geometry (e.g.…”

Section: Simplicesmentioning

confidence: 99%

“…We summarize our derivation of the affine invariants of ternary cubics as follows: modulo one homogeneous relation of degree (24,18,18). Hence its Hilbert series equals…”

Section: Symmetry and Invariantsmentioning

confidence: 99%

“…Let Q be a quadrilateral in the plane. The moment variety M [3] (Q) is a hypersurface in P 9 , whose defining polynomial has 5100 terms of degree (18,12,12 where the affine invariants in (33) are normalized to have content one and leading monomials s = m 00 m 02 m 12 m 30 + · · ·, t = m 2 00 m 2 03 m 2 30 + · · ·, h = m 00 m 02 m 20 + · · ·, g = m 3 00 m 3 02 m 2 30 + · · · .…”

Section: Quadrilaterals and Beyondmentioning

confidence: 99%

“…The above formula was found as follows. By Table 1 below, the Z 3degree of the hypersurface is (18,12,12). We used Proposition 5.9 to generate affine invariants of this degree with indeterminate coefficients.…”

Section: Quadrilaterals and Beyondmentioning

confidence: 99%

“…It has appeared in logarithmic potential theory [6,29], and in connection with the mother body problem [22]. Algorithms for reconstructing P from its axial moments can be found in [17,18]. A practical application for moments of planar polygons was suggested by Sharon and Mumford [31] as a tool to reconstruct arbitrary planar shapes from their fingerprints.…”

Section: Introductionmentioning

confidence: 99%

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Moment varieties of measures on polytopes

Kohn¹,

Shapiro²,

Sturmfels³

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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The uniform probability measure on a convex polytope induces piecewise polynomial densities on its projections. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex coordinates. We study projective varieties that are parametrized by finite collections of such rational functions. Our focus lies on determining the prime ideals of these moment varieties. Special cases include Hankel determinantal ideals for polytopal splines on line segments, and the relations among multisymmetric functions given by the cumulants of a simplex. In general, our moment varieties are more complicated than in these two special cases. They offer challenges for both numerical and symbolic computing in algebraic geometry.

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Section: Simplicesmentioning

confidence: 99%

“…We summarize our derivation of the affine invariants of ternary cubics as follows: modulo one homogeneous relation of degree (24,18,18). Hence its Hilbert series equals…”

Section: Symmetry and Invariantsmentioning

confidence: 99%

Section: Quadrilaterals and Beyondmentioning

confidence: 99%

Section: Quadrilaterals and Beyondmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Moment varieties of measures on polytopes

Kohn¹,

Shapiro²,

Sturmfels³

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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An Identity Theorem for the Fourier–Laplace Transform of Polytopes on Nonzero Complex Multiples of Rationally Parameterizable Hypersurfaces

Engel

2022

Discrete Comput Geom

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A set $$\mathcal {S}$$ S of points in $$\mathbb {R}^n$$ R n is called a rationally parameterizable hypersurface if there is vector function $$\varvec{\sigma }:\mathbb {R}^{n-1}\rightarrow \mathbb {R}^n$$ σ : R n - 1 → R n having as components rational functions defined on some common domain D such that $$\mathcal {S}=\{\varvec{\sigma }(\textbf{t}):\textbf{t}\in D\}$$ S = { σ ( t ) : t ∈ D } . A generalized n-dimensional polytope in $$\mathbb {R}^n$$ R n is a union of a finite number of convex n-dimensional polytopes in $$\mathbb {R}^n$$ R n . The Fourier–Laplace transform of such a generalized polytope $$\mathcal {P}$$ P in $$\mathbb {R}^n$$ R n is defined by $$F_{\mathcal {P}}(\textbf{z})=\int _{\mathcal {P}}e^{\textbf{z}\cdot \textbf{x}}\,\textbf{dx}$$ F P ( z ) = ∫ P e z · x dx . Let $$\gamma $$ γ be a fixed nonzero complex number. We prove that $$F_{\mathcal {P}_1}(\gamma \varvec{\sigma }(\textbf{t}))=F_{\mathcal {P}_2}(\gamma \varvec{\sigma }(\textbf{t}))$$ F P 1 ( γ σ ( t ) ) = F P 2 ( γ σ ( t ) ) for all $$\textbf{t} \in O$$ t ∈ O implies $$\mathcal {P}_1=\mathcal {P}_2$$ P 1 = P 2 if O is an open subset of D satisfying some well-defined conditions and we present similar results for the null set of the Fourier–Laplace transform of $$\mathcal {P}$$ P . Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres.

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Algebraic vertices of non-convex polyhedra

Akopyan

Bárány

Robins

2017

Advances in Mathematics

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In this article we define an algebraic vertex of a generalized polyhedron and show that it is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope $P$ is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of $P$. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier--Laplace transform. We show that a point $\mathbf{v}$ is an algebraic vertex of a generalized polyhedron $P$ if and only if the tangent cone of $P$, at $\mathbf{v}$, has non-zero Fourier--Laplace transform.Comment: 13 pages, 3 figure

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On moments of a polytope

Cited by 10 publications

References 23 publications

Moment varieties of measures on polytopes

Moment varieties of measures on polytopes

An Identity Theorem for the Fourier–Laplace Transform of Polytopes on Nonzero Complex Multiples of Rationally Parameterizable Hypersurfaces

Algebraic vertices of non-convex polyhedra

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