The conic h(x, y) = [x.sup.2] + [cy.sup.2] -1 = 0, c [member of] R, is an invariant

algebraic curve of system (1).

The following examples explain how for the reader who is not familiar with the concepts of algebraic function fields (i.e.,

algebraic curves) as genus, rational points etc is referred to [11].

5.1 General position with respect to

algebraic curvesFor example, Bezout's theorem, that the number of points of intersection of two

algebraic curves is equal to the product of their degrees, would obviously suffer greatly if restricted to the real affine plane."

Although we have presented an algorithm to compute singular points of irreducible

algebraic curves in [17], the algorithm is almost experimental and related analysis of the algorithm is not provided, for instance, the feasibility and complexity.

[1] Bix, R., Conics and Cubics: A Concrete Introduction to

Algebraic Curves, Springer, 2006.

Stichtenoth, "Group codes on certain

algebraic curves with many rational points", Applicable Algebra in Engineering, Communication and Computing, vol.

Key words: Finite Fields,

Algebraic Curves, Algebraic Function Fields, Elementary Abelian p-Extensions, Rational Points.

Via the correspondence between

algebraic curves and floor diagrams [BM09, Theorem 2.5] these notions correspond literally to the respective analogues for

algebraic curves.

A selection of 10 papers from it consider such topics as self-dual codes and invariant theory; vector bundles in error-correcting for geometric Goppa codes; combinatorial designs and code synchronization; real and imaginary hyper-elliptic curve cryptography; divisibility, smoothness, and cryptographic applications; a variant of the Reidemeister-Schreier algorithm for the fundamental groups of Riemann surfaces; theta functions and

algebraic curves with automorphisms; enumerative geometry and string theory; and the cryptographical properties of extremal algebraic graphs.

Ramanathan: Moduli for principal bundles over

algebraic curves. I, Proc.

DEMMEL, Algorithms for intersecting parametric and

algebraic curves I.