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Preserving Hodge Vectors of Lattice Polytopes

Vadym Kurylenko, Benjamin Nill

Abstract

Given lattice polytopes $P_1, \ldots, P_k$ contained in a $k$-dimensional subspace $U \subseteq \mathbb{R}^d$ and a $d$-dimensional lattice polytope $Q \subset \mathbb{R}^d$, we compute the Hodge vector of the Cayley polytope $P_1 * \cdots * P_k * Q$, and show that it equals the mixed volume of $P_1, \ldots, P_k$ times the Hodge vector of the projection of $Q$ along $U$. Here, the Hodge vector of a lattice polytope is its local $h^*$-vector with leading and trailing zeroes removed. This result allows finding infinitely many high-dimensional lattice polytopes with the same Hodge vector that are not free joins. The proof relies on a closed formula for the Hodge-Deligne polynomial of generic complete intersections in the torus in terms of the bivariate/mixed $h^*$-polynomial. A special case of our construction is what we call Lawrence twists: extending the Gale transform by centrally-symmetric pairs of vectors. As applications, we can produce many new thin polytopes answering a question by Borger, Kretschmer and the second author, and we provide an alternative explanation of the thinness of $B_k$-polytopes answering a question of Selyanin.

Preserving Hodge Vectors of Lattice Polytopes

Abstract

Given lattice polytopes contained in a -dimensional subspace and a -dimensional lattice polytope , we compute the Hodge vector of the Cayley polytope , and show that it equals the mixed volume of times the Hodge vector of the projection of along . Here, the Hodge vector of a lattice polytope is its local -vector with leading and trailing zeroes removed. This result allows finding infinitely many high-dimensional lattice polytopes with the same Hodge vector that are not free joins. The proof relies on a closed formula for the Hodge-Deligne polynomial of generic complete intersections in the torus in terms of the bivariate/mixed -polynomial. A special case of our construction is what we call Lawrence twists: extending the Gale transform by centrally-symmetric pairs of vectors. As applications, we can produce many new thin polytopes answering a question by Borger, Kretschmer and the second author, and we provide an alternative explanation of the thinness of -polytopes answering a question of Selyanin.
Paper Structure (19 sections, 13 theorems, 51 equations)

This paper contains 19 sections, 13 theorems, 51 equations.

Table of Contents

  1. Introduction and main result
  2. Hodge-Deligne polynomials and bivariate Ehrhart theory
  3. Hodge-Deligne polynomials of affine hypersurfaces in the torus
  4. Bivariate Ehrhart theory
  5. Isomorphisms, Cayley polytopes, free joins and lattice pyramids
  6. A closed formula for the Hodge-Deligne polynomial of an affine complete intersection in the torus
  7. Generalized Lawrence twists
  8. Lawrence twists
  9. The original motivation
  10. Lawrence twists via Gale duality
  11. Lawrence twists are generalized Lawrence twists
  12. A direct relation between h*-polynomials of lattice projections and Cayley polytopes
  13. Applications
  14. Infinitely many non-free-joins lattice polytopes with same Hodge vector
  15. Many more thin polytopes
  16. ...and 4 more sections

Key Result

Theorem 1

Let $P_1,\ldots,P_k, P_{k+1} \subset \mathbb{R}^d$ be lattice polytopes, where the first $k$ polytopes are contained in a $k$-dimensional rational subspace $U$ and $\dim P_{k+1}=d \geq 1$. Let $V$ denote the mixed volume ${\rm MV}(P_1, \ldots, P_k)$ of $P_1, \ldots, P_k$. Then the Hodge vector of $P

Theorems & Definitions (31)

  • Definition 1
  • Theorem 1
  • Proposition 1: danilov_newton_1987, Proposition 3.9
  • Definition 2
  • Example 1
  • Definition 3
  • Definition 4
  • Proposition 2
  • Proposition 3
  • proof
  • ...and 21 more