# Real monodromy action

## TL;DR

This paper introduces a new concept called real monodromy structure to analyze how real solutions of polynomial systems change over real parameter spaces, especially in kinematic applications, extending beyond traditional complex monodromy groups.

## Contribution

It defines a novel real monodromy structure that captures solution behavior over real parameters, overcoming limitations of the naive complex extension.

## Key findings

- Naive real monodromy extension is highly restrictive.
- Real monodromy structure provides detailed solution change information.
- Application to kinematics demonstrates practical utility.

## Abstract

The monodromy group is an invariant for parameterized systems of polynomial equations that encodes structure of the solutions over the parameter space. Since the structure of real solutions over real parameter spaces are of interest in many applications, real monodromy action is investigated here. A naive extension of monodromy action from the complex numbers to the real numbers is shown to be very restrictive. Therefore, we define a real monodromy structure which need not be a group but contains tiered characteristics about the real solutions. This real monodromy structure is applied to an example in kinematics which summarizes all the ways performing loops parameterized by leg lengths can cause a mechanism to change poses.