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Unbounded convex polyhedra as polynomial images of Euclidean spaces
Algebraic Geometry · Submitted 23 Jan 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Namely, \em ${\mathcal K}$ is a polynomial image of ${\mathbb R}^n$ if and only if $\vec{{\mathcal C}}({\mathcal K})$ has dimension $n$\em.
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In a previous work we proved that each $n$-dimensional convex polyhedron ${\mathcal K}subset{\mathbb R}^n$ and its relative interior are regular images of ${\mathbb R}^n$. As the image of a non-constant polynomial map is an unbounded semialgebraic set, it is not possible to substitute regular maps by polynomial maps in the previous statement. In this work we determine constructively all unbounded $n$-dimensional convex polyhedra ${\mathcal K}\subset{\mathbb R}^n$ that are polynomial images of ${\mathbb R}^n$. We also analyze for which of them the interior ${\rm Int}({\mathcal K})$ is a polynomial image of ${\mathbb R}^n$. A discriminating object is the recession cone $\vec{\mathcal C}({\mathcal K})$ of ${\mathcal K}$. Namely, \em ${\mathcal K}$ is a polynomial image of ${\mathbb R}^n$ if and only if $\vec{{\mathcal C}}({\mathcal K})$ has dimension $n$\em. In addition, \em ${\rm Int}({\mathcal K})$ is a polynomial image of ${\mathbb R}^n$ if and only if $\vec{{\mathcal C}}({\mathcal K})$ has dimension $n$ and ${\mathcal K}$ has no bounded faces of dimension $n-1$\em. A key result is an improvement of Pecker's elimination of inequalities to represent semialgebraic sets as projections of algebraic sets. Empirical approaches suggest us that there are `few' polynomial maps that have a concrete convex polyhedron as a polynomial image and that there are even fewer for which it is affordable to show that their images actually correspond to our given convex polyhedron. This search of a `needle in the haystack' justifies somehow the technicalities involved in our constructive proofs.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In a previous work we proved that each $n$-dimensional convex polyhedron ${\mathcal K}subset{\mathbb R}^n$ and its relative interior are regular images of ${\mathbb R}^n$. | background 1.00 | 0.36 | 0.03 | asserted 1.00 |
| s1 | As the image of a non-constant polynomial map is an unbounded semialgebraic set, it is not possible to substitute regular maps by polynomial maps in the previous statement. | background 0.52 | 0.23 | 0.03 | asserted 1.00 |
| s2 | In this work we determine constructively all unbounded $n$-dimensional convex polyhedra ${\mathcal K}\subset{\mathbb R}^n$ that are polynomial images of ${\mathbb R}^n$. | contribution 0.99 | 0.96 | 0.34 | asserted 1.00 |
| s3 | We also analyze for which of them the interior ${\rm Int}({\mathcal K})$ is a polynomial image of ${\mathbb R}^n$. | contribution 1.00 | 0.93 | 0.06 | asserted 0.90 |
| s4 | A discriminating object is the recession cone $\vec{\mathcal C}({\mathcal K})$ of ${\mathcal K}$. | contribution 0.54 | 0.63 | 0.05 | asserted 1.00 |
| s5 | Namely, \em ${\mathcal K}$ is a polynomial image of ${\mathbb R}^n$ if and only if $\vec{{\mathcal C}}({\mathcal K})$ has dimension $n$\em. | result 1.00 | 0.90 | 0.66 | asserted 1.00 |
| s6 | In addition, \em ${\rm Int}({\mathcal K})$ is a polynomial image of ${\mathbb R}^n$ if and only if $\vec{{\mathcal C}}({\mathcal K})$ has dimension $n$ and ${\mathcal K}$ has no bounded faces of dimension $n-1$\em. | result 1.00 | 0.92 | 0.26 | asserted 1.00 |
| s7 | A key result is an improvement of Pecker's elimination of inequalities to represent semialgebraic sets as projections of algebraic sets. | contribution 0.56 | 0.77 | 0.05 | asserted 1.00 |
| s8 | Empirical approaches suggest us that there are `few' polynomial maps that have a concrete convex polyhedron as a polynomial image and that there are even fewer for which it is affordable to show that their images actually correspond to our given convex polyhedron. | interpretation 0.61 | 0.27 | 0.02 | hedged possibility 0.99 |
| s9 | This search of a `needle in the haystack' justifies somehow the technicalities involved in our constructive proofs. | interpretation 0.81 | 0.34 | 0.02 | hedged possibility 0.86 |
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Original abstract
In a previous work we proved that each $n$-dimensional convex polyhedron ${\mathcal K}subset{\mathbb R}^n$ and its relative interior are regular images of ${\mathbb R}^n$. As the image of a non-constant polynomial map is an unbounded semialgebraic set, it is not possible to substitute regular maps by polynomial maps in the previous statement. In this work we determine constructively all unbounded $n$-dimensional convex polyhedra ${\mathcal K}\subset{\mathbb R}^n$ that are polynomial images of ${\mathbb R}^n$. We also analyze for which of them the interior ${\rm Int}({\mathcal K})$ is a polynomial image of ${\mathbb R}^n$. A discriminating object is the recession cone $\vec{\mathcal C}({\mathcal K})$ of ${\mathcal K}$. Namely, \em ${\mathcal K}$ is a polynomial image of ${\mathbb R}^n$ if and only if $\vec{{\mathcal C}}({\mathcal K})$ has dimension $n$\em. In addition, \em ${\rm Int}({\mathcal K})$ is a polynomial image of ${\mathbb R}^n$ if and only if $\vec{{\mathcal C}}({\mathcal K})$ has dimension $n$ and ${\mathcal K}$ has no bounded faces of dimension $n-1$\em. A key result is an improvement of Pecker's elimination of inequalities to represent semialgebraic sets as projections of algebraic sets. Empirical approaches suggest us that there are `few' polynomial maps that have a concrete convex polyhedron as a polynomial image and that there are even fewer for which it is affordable to show that their images actually correspond to our given convex polyhedron. This search of a `needle in the haystack' justifies somehow the technicalities involved in our constructive proofs.
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To ease the presentation of the full picture of what is known [ref] about the representation of semialgebraic sets with piecewise linear boundary as either polynomial or regular images of some euclidean space $^m$ we introduce the following two invariants.
On complements of convex polyhedra as polynomial images of ${\mathbb R}^n$ · Introduction · 0.99
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