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The Ramificant Determinant

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 05 Nov 2019Embargo end date: 01 Jan 2019 France Publisher:SIGMA (Symmetry, Integrability and Geometry: Methods and Application)Journal:Symmetry, Integrability and Geometry: Methods and Applications (eissn: 1815-0659,

Authors: Biswas, Kingshook; Pérez-Marco, Ricardo;

doi: 10.3842/sigma.2019.086 , 10.48550/arxiv.1903.06770

arXiv: http://arxiv.org/abs/1903.06770

The Ramificant Determinant

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Abstract

We give an introduction to the transalgebraic theory of simply connected log-Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves of genus $0$). We define the base vector space of transcendental functions and establish by elementary means some transcendental properties. We introduce the Ramificant Determinant constructed with transcendental periods and we give a closed-form formula that gives the main applications to transalgebraic curves. We prove an Abel-like Theorem and a Torelli-like Theorem. Transposing to the transalgebraic curve the base vector space of transcendental functions, they generate the structural ring from which the points of the transalgebraic curve can be recovered algebraically, including infinite ramification points.

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