English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we analyze the theory of meromorphic $(1,0)$-forms$\omega\in\mathcal{M}\Omega^{(1,0)}(\mathbb{CP}^1).$ Hence, we show that on acompact Riemann surface of genus $g=0,$ isomorphic to $\mathbb{CP}^1,$ everynon-constant meromorphic function $f:X\to\mathbb{CP}^1$ has as many zeros aspoles, where each is counted according to multiplicities. Such an analysisgives rise to the following result. Invoking the Riemann-Roch theorem for acompact Riemann $X$ with canonical divisor $K,$ it follows that $deg(f)=0$ forany principal divisor $(f):=D$ on $X.$ More precisely,$\ell(D)-\ell(K-D)=deg(D)+1=1$ or $\ell(D)-\ell(K-D)-1=0.$ Furthermore, for adiffeomorphism $\eta:X\to\mathbb{CP}^1$ of a certain kind, a multistep programis implemented to show $X$ is a compact algebraic variety of dimension one,i.e. a non-singular projective variety. Hence, we adopt a group-theoreticapproach and provide a useful heuristic, that is, a set of technical conditionsto facilitate the algebro-geometric analysis of simply connected Riemannsurfaces $X.$

On the Algebro-Geometric Analysis of Meromorphic (1,0)-forms