# Complementary problems with polynomial data

## TL;DR

This paper studies the polynomial complementary problem involving polynomial maps, analyzing solution set properties such as existence, uniqueness, and error bounds, thereby extending known results in nonlinear complementarity problems.

## Contribution

It provides new insights into the solution set properties of polynomial complementarity problems, including genericity, nonemptiness, compactness, and explicit error bounds, generalizing previous results.

## Key findings

- Solution set properties characterized, including conditions for nonemptiness and compactness.
- Explicit error bounds with determined exponents established.
- Generalizations of known results in nonlinear complementarity problems achieved.

## Abstract

Given polynomial maps $f, g \colon \mathbb{R}^n \to \mathbb{R}^n,$ we consider the {\em polynomial complementary problem} of finding a vector $x \in \mathbb{R}^n$ such that \begin{equation*} f(x) \ \ge \ 0, \quad g(x) \ \ge \ 0, \quad \textrm{ and } \quad \langle f(x), g(x) \rangle \ = \ 0. \end{equation*} In this paper, we present various properties on the solution set of the problem, including genericity, nonemptiness, compactness, uniqueness as well as error bounds with exponents explicitly determined. These strengthen and generalize some previously known results, and hence broaden the boundary knowledge of nonlinear complementarity problems as well.