[Paper Review] Unbounded convex polygons as polynomial images of the plane

Carlos Ueno|arXiv (Cornell University)|Jul 31, 2013

Point processes and geometric inequalities7 references3 citations

TL;DR

This paper proves that unbounded convex polygons with nonparallel unbounded edges—known as V-polygons—are polynomial images of the real plane ℝ², while their interiors are polynomial images of ℝ³. The key contribution is a constructive proof using affine transformations and polynomial maps that systematically cover missing boundary points, completing the characterization of such sets as polynomial images in algebraic geometry.

ABSTRACT

In this note we show that unbounded convex polygons with nonparallel unbounded edges are polynomial images of ${\mathbb R}^2$, whereas their interiors are polynomial images of ${\mathbb R}^3$

Motivation & Objective

• To characterize which unbounded convex polygons can be represented as polynomial images of ℝ².
• To determine whether the interior of such polygons can be realized as a polynomial image of ℝ³.
• To complete the classification of convex polygons that are polynomial images, distinguishing between bounded, parallel-edge, and V-polygon cases.
• To address the open question of whether V-polygons can be realized via proper polynomial maps.

Proposed method

• Transform any V-polygon into a curtain in the upper half-plane using affine coordinates, ensuring unbounded edges align with coordinate axes.
• Use a sequence of polynomial maps to successively cover missing vertices, starting from a punctured polygon.
• Apply Lemma 2.3 to construct a polynomial map that covers a missing vertex by modifying the image via a product of linear forms.
• Utilize the fact that ℝ³ can be decomposed as ℝ² × ℝ to lift a 2D polynomial map to 3D, enabling image expansion into the interior.
• Construct a map f₁: ℝ³ → ℝ² by translating the image of a 2D polynomial map along a direction (1,1), parameterized by the third variable.
• Prove that the image of this 3D map restricted to ℝ² × (0,∞) equals the interior of the V-polygon, using the openness of the half-plane image.

Experimental results

Research questions

• RQ1 Can every unbounded convex polygon with nonparallel unbounded edges be realized as the image of a polynomial map from ℝ² to ℝ²?
• RQ2 Is the interior of such a polygon a polynomial image of ℝ³, and if so, how can this be constructed explicitly?
• RQ3 Can a V-polygon be represented as the image of a proper polynomial map f: ℝ² → ℝ², or must such maps necessarily be non-proper?
• RQ4 What is the role of affine transformations and curtain structures in simplifying the construction of polynomial images of semialgebraic sets?
• RQ5 How do the boundary structure and edge directions of a convex polygon constrain its realizability as a polynomial image?

Key findings

• Any unbounded convex polygon with nonparallel unbounded edges (a V-polygon) is the image of a polynomial map f: ℝ² → ℝ².
• The interior of such a V-polygon is the image of a polynomial map f: ℝ³ → ℝ², which cannot be reduced to ℝ² in general.
• The construction relies on successive polynomial maps that cover missing vertices by exploiting the product of linear forms defining the polygon’s edges.
• The proof uses affine transformations to place the polygon in a canonical position (e.g., vertex at origin, edges on axes), simplifying the mapping process.
• The method ensures that the image of the final composition covers the entire polygon, including all boundary points, even when intermediate maps miss vertices.
• The result completes the classification: only V-polygons and their interiors (under specific dimension constraints) can be polynomial images among unbounded convex polygons.