Paraskevas Alvanos - Riemann-Roch Spaces and Computation

Besides the computation of integral points on curves, computation of Riemann-Roch spaces are used for the construction of Goppa codes (Goppa, 1981; Goppa, 1988), symbolic parametrizations of curves (Van Hoeij, 1995; Van Hoeij, 1997), integration of algebraic functions (Davenport, 1981) and a lot more (Hiren et.al, 2005). Riemann-Roch space arises from the classical Riemann-Roch theorem which computes the dimension of the field of functions with specific zeros and poles dim D = deg Dg + 1 + i(D). Here D is a divisor, g is the genus of the function field and i(D) an invariant of the function field. The inequality dim D deg Dg + 1 was initially proved by Riemann (1857) and was upgraded to equality by Roch (1865). The scope of the textbook is not to add original research to the literature but to give an educational perspective on Riemann-Roch spaces and the computation of algebraic structures connected to Riemann-Roch theorem. The proofs of theorems that use stiff techniques are avoided, and proofs with educational value (according to the authors opinion) are presented, allowing the reader to follow the book without many difficult computation.

In order to follow the textbook, the reader should be aware of the basic algebraic structures such as group, ring, prime ideal, maximal ideal, number field, modules, vector spaces and the basics of linear algebra, matrix theory and polynomial theory. The first part of the textbook consists of four chapters where algebraic structures and some of their properties are analysed. The second part of the textbook consists of four more chapters where algorithms and examples connected to Riemann-Roch spaces are presented. Throughout the textbook a variety of examples cover the majority of the cases. This textbook is a rsum of the authors work and his research the last 15 years. Therefore, the author would like to thank all of his colleagues and friends that helped him in every way during those years.

Especially, the author would like to express his deep gratitude to his beloved wife, K. Roditou, for her comments and her support during the writing of this textbook. The author would also like to thank very much the referees for their corrections and their very thoughtful and useful comments and K. Chatzinikolaou for his suggestions about the design of the cover. Last but not least, the author would like to express his ultimate respect to his former supervisor and current mathematical mentor prof. Poulakis.

Definitions are enriched with lots of examples and figures for better understanding. We will only deal with commutative rings and number fields and therefore every reference to a ring or a field implies a commutative ring and a number field respectively.

As we know the set of nn invertible matrices, where the identity element of the ring is the identity matrix and the zero element of the ring is the zero matrix, form a commutative ring. The product of two matrices A, B might be the zero matrix while neither of A and B are zero matrices. For instance Thus, the ring of nn diagonal matrices is not an integral domain, even though it is a commutative ring. All fields, the ring of rational integers and all the subrings of integral domains are integral domains. Proposition 1.1.1.If R is an integral domain then R[x] is also an integral domain. Proof.

In order to prove that R[x] is an integral domain it is equivalent to prove that for any two elements and if f(x)g(x) = 0 then f(x) = 0 or g(x) = 0. The product f(x)g(x) is Thus