Philipp Jukic | Searcharxiv

arXiv2020

Maximal Spectra of rings consisting of regulated functions

In 1969 Höchster proved that for every quasi-compact T1-space $X$ we can find a commutative ring $R$ such that $X$ is homeomorphic to the maximal spectrum $\mathrm{Specm}(R)$ of $R$. This result implies the existence of a commutative ring $R$ that admits a non-Hausdorff and totally disconnected maximal spectrum $\mathrm{Specm}(R)$. However, there has not been an example of such a commutative ring yet. The aim of this paper is to investigate rings that almost admit a non-Hausdorff and totally disconnected maximal spectrum.

math.AC↗

arXiv2017

Higher degree S-lemma and the stability of quadratic modules

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.

math.AG↗