Computational algebraic geometry

Algebraic geometry is a powerful combination of algebra and geometry, with a rich history and many applications, both theoretical and practical. It also has an intimidating reputation. Hence it is important to have a variety of introductory texts at different levels of sophistication. We will see that Schenck's book offers an interesting path into this wonderful subject. 1. What is algebraic geometry? Algebraic geometry goes back to the coordinate geometry of Descartes, which enables one to describe curves and surfaces by means of equations. When the equations involve only polynomials, then the algebra of the polynomials is deeply linked to the geometry of the corresponding curves and surfaces. But there is more to algebraic geometry than just equations and solutions, for the solutions often exhibit " extra structure ". Consider what happens when a line meets the circle x 2 + y 2 = 2: y = x y = x − 2 A B C The picture on the left has two points of intersection A and B, while on the right we have the single point of intersection C. However, since the line is tangent to the circle at this point, we say that C has multiplicity two. Multiplicity can be defined rigorously in various ways. At the point C, we can proceed as follows. Let R[x, y] C = f g f, g ∈ R[x, y], g(C) = 0 be the ring of rational functions in x, y that are defined at C. Inside this ring we have the ideal x 2 + y 2 − 2, y − x + 2 generated by x 2 + y 2 − 2 and y − x + 2. Then one can show that the quotient ring O C = R[x, y] C /x 2 + y 2 − 2, y − x + 2