Intro -- Preface -- Preface -- Part I: Riemann-Roch Spaces -- 1 Elements of Algebra -- 1.1 Domains -- 1.2 Rings and Fields -- 1.3 Norm and Trace -- 1.4 Group of Units -- 2 Function Fields and Curves -- 2.1 Algebraic Function Fields -- 2.2 Affine Curves -- 2.3 Projective Curves -- 2.4 Genus of a Curve -- 3 Riemann-Roch Spaces -- 3.1 Valuation Rings -- 3.2 Places and Divisors -- 3.3 Riemann-Roch Space -- 3.4 Holomorphy Rings -- 4 Integral Domains -- 4.1 Fractional Ideals -- 4.2 Localization -- 4.3 R-module V -- Part II: Computation -- 5 Computing Integral Bases -- 5.1 Discriminant -- 5.2 Ideal Basis -- 5.3 Idealizer -- 5.4 Radical -- 5.5 Computing the Discriminants Ideal Radical -- 5.6 Computing Idealizers and Inverse Ideals -- 5.7 Trager's Algorithm -- 5.8 Examples -- 6 Computing Riemann-Roch Spaces -- 6.1 Reduced Basis Algorithms -- 6.2 The Dimension of L(D) -- 6.3 Computing L(D) -- 7 Computing Resultant and Norm Form Equations -- 7.1 Resultant -- 7.2 Computing NL/K (x1a1+ @let@token + xnan)=k -- 7.3 Examples -- 8 Computing Integral Points on Rational Curves -- 8.1 Integral Points on Curves -- 8.2 Integral Points on Curves with | (C)| 3 -- 8.3 Integral Points on Curves with | (C)| =2 (i) -- 8.4 Integral Points on Curves with | (C)| =2 (ii) -- 8.5 Integral Points on Curves with | (C)| =1 -- Appendices -- A Unimodular Matrices -- A.1 Transformation Matrix -- A.2 Unimodular Transformations -- B Algorithm's Codes -- B.1 Integral Points 3 Algorithm's Code -- B.2 Integral Points 2i Algorithm's Code -- B.3 Integral Points 2ii Algorithm's Code -- B.4 Integral Points 1 Algorithm's Code -- Bibliography -- Index -- Index of Algorithms.

The book is focused on Riemann-Roch spaces and on the computation of algebraic structures connected to the Riemann-Roch theorem.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2022. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.