H\"older-Type Global Error Bounds for Non-degenerate Polynomial Systems

TL;DR

This paper establishes H"older-type global error bounds for a broad class of polynomial systems that are non-degenerate at infinity, providing explicit bounds based on polynomial degrees and combinatorial data.

Contribution

It proves the existence of explicit H"older-type error bounds for non-degenerate polynomial systems, expanding understanding of their stability and error estimates.

Findings

Global error bounds hold for non-degenerate polynomial systems.

Bounds are explicitly expressed in terms of polynomial degrees and combinatorial data.

The class of non-degenerate polynomial maps is open, dense, and semi-algebraic.

Abstract

Let $F := (f_1, \ldots, f_p) \colon {\Bbb R}^n \to {\Bbb R}^p$ be a polynomial map, and suppose that $S := \{x \in {\Bbb R}^n \ : \ f_i(x) \le 0, i = 1, \ldots, p\} \ne \emptyset.$ Let $d := \max_{i = 1, \ldots, p} \deg f_i$ and $\mathcal{H}(d, n, p) := d(6d - 3)^{n + p - 1}.$ Under the assumption that the map $F \colon {\Bbb R}^n \rightarrow {\Bbb R}^p$ is convenient and non-degenerate at infinity, we show that there exists a constant $c > 0$ such that the following so-called {\em H\"older-type global error bound result} holds $$c d(x,S) \le [f(x)]_+^{\frac{2}{\mathcal{H}(2d, n, p)}} + [f(x)]_+ \quad \textrm{ for all } \quad x \in \mathbb{R}^n,$$ where $d(x, S)$ denotes the Euclidean distance between $x$ and $S,$ $f(x) := \max_{i = 1, \ldots, p} f_i(x),$ and $[f(x)]_+ := \max \{f(x), 0 \}.$ The class of polynomial maps (with fixed Newton polyhedra), which are non-degenerate at infinity, is generic in the sense that it is an open and dense semi-algebraic set. Therefore, H\"older-type global error bounds hold for a large class of polynomial maps, which can be recognized relatively easily from their combinatoric data.