# Higher degree S-lemma and the stability of quadratic modules

## TL;DR

This paper investigates a higher-degree generalization of the S-lemma related to Hilbert's theorem on ternary quartics, demonstrating its limitations through geometric and algebraic analysis within quadratic modules.

## Contribution

It introduces new tools to analyze the non-existence of a higher-degree S-lemma generalization, linking geometric and algebraic perspectives.

## Key findings

- Higher-degree S-lemma generalization is not possible without extra conditions.
- Established a connection between geometric and algebraic reasons in quadratic modules.
- Extended tools by Netzer to analyze positivity and stability in polynomial modules.

## Abstract

In this work we will investigate a certain generalization of the so called S-lemma in higher degrees. The importance of this generalization is, that it is closely related to Hilbert's 1888 theorem about tenary quartics. In fact, if such a generalization exits, then one can state a Hilbert-like theorem, where positivity is only demanded on some semi-algebraic set. We will show that such a generalization is not possible, at least not without additional conditions. To prove this, we will use and generalize certain tools developed by Netzer ([Ne]). These new tools will allow us to conclude that this generalization of the S-lemma is not possible because of geometric reasons. Furthermore, we are able to establish a link between geometric reasons and algebraic reasons. This will be accomplished within the framework of quadratic modules.