# Compactness criteria for real algebraic sets and newton polyhedra

## TL;DR

This paper establishes new criteria based on Newton polyhedra to determine the compactness and stable compactness of real algebraic sets defined by polynomial equations.

## Contribution

It introduces necessary and sufficient conditions for the compactness of real algebraic sets using Newton polyhedra, advancing understanding in real algebraic geometry.

## Key findings

- Necessary criterion for compactness of zero sets.
- Sufficient condition for compactness of zero sets.
- Criteria for stable compactness of algebraic sets.

## Abstract

Let $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be a polynomial and $\mathcal{Z}(f)$ its zero set. In this paper, in terms of the so-called Newton polyhedron of $f,$ we present a necessary criterion and a sufficient condition for the compactness of $\mathcal{Z}(f).$ From this we derive necessary and sufficient criteria for the stable compactness of $\mathcal{Z}(f).$