The Real Holomorphy Ring And Sums Of Powers In Rational And Elliptic Function Fields Over The Field Of Real Numbers

This thesis is devoted to a study of real holomorphy rings in R(x) and ellipticrnfunction _elds R(x)(prnf(x)); f is a monic polynomial of degree 3 with nornmultiple roots in C over R and sums of powers in R(x). The objects of studyrnare real valuation rings, real holomorphy rings and real places in the rationalrnfunction _elds R(x) and then investigate their extension to elliptic functionrn_elds which are quadratic extension of R(x). We write down the full list ofrnreal valuation rings to give the description of holomorphy ring in both _eldsrnand _nally we study applications to sums of powers.

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