Semi-algebraic sets of f-vectors

TL;DR

This paper investigates the geometric and algebraic structure of f-vector sets of polytopes, revealing that many such sets are not semi-algebraic, thus challenging previous assumptions and connecting to deep problems like Hilbert's Tenth.

Contribution

The paper develops proof techniques to show that certain f-vector sets of polytopes are not semi-algebraic sets of lattice points, expanding understanding of their algebraic complexity.

Findings

Certain f-vector sets are proven not to be semi-algebraic sets of lattice points.

The set of all f-vectors of 4-polytopes remains an open problem.

Connections are drawn between f-vector sets and Hilbert's Tenth problem.

Abstract

Polytope theory has produced a great number of remarkably simple and complete characterization results for face-number sets or f-vector sets of classes of polytopes. We observe that in most cases these sets can be described as the intersection of a semi-algebraic set with an integer lattice. Such "semi-algebraic sets of lattice points" have not received much attention, which is surprising in view of a close connection to Hilbert's Tenth problem, which deals with their projections. We develop proof techniques in order to show that, despite the observations above, some f-vector sets are NOT semi-algebraic sets of lattice points. This is then proved for the set of all pairs $(f_1,f_2)$ of 4-dimensional polytopes, the set of all f-vectors of simplicial $d$-polytopes for $d\ge6$, and the set of all f-vectors of general $d$-polytopes for $d\ge6$. For the f-vector set of all 4-polytopes this remains open.