# On relative grothendieck rings and algebraically constructible functions

## TL;DR

This paper explores Grothendieck rings in real geometry, focusing on arc-symmetric sets and algebraically constructible functions, analyzing duality, link operators, and motivic Milnor fibres with signs.

## Contribution

It introduces new insights into the structure of relative Grothendieck rings and their relation to algebraically constructible functions, extending previous work to the relative case.

## Key findings

- Analysis of duality and link operators in Grothendieck rings
- Behavior of these operators with motivic Milnor fibres
- Enhanced understanding of arc-symmetric sets in real geometry

## Abstract

We investigate Grothendieck rings appearing in real geometry, notably for arc-symmetric sets, and focus on the relative case in analogy with the properties of the ring of algebraically constructible functions defined by McCrory and Parusinski. We study in particular the duality and link operators, including its behaviour with respect to motivic Milnor fibres with signs.