More advanced questions in algebraic geometry concern relations between curves given by different equations and the topology of curves, and many other topics.Īlgebraic geometry grew significantly in the 20th century, branching into topics such as computational algebraic geometry, Diophantine geometry, and analytic geometry. Other common questions in algebraic geometry concern points of special interest such as singularities, inflection points and points at infinity - we shall see these throughout the catalogue. What does this imply?Īlgebraic geometry sets out to answer these questions by applying the techniques of abstract algebra to the set of polynomials that define the curves (which are then called "algebraic varieties"). The mathematics involved is inevitably quite hard, although it is covered in degree-level courses. What geometric properties can be inferred from the equation? What about more complicated polynomial equations such as the semicubical parabola $y^2 - x^3 = 0$, which has a cusp at its tip (such "singularities" are of great importance), or something less intelligible such as $x^2 -x^2y^2 + y^3 + xy -1 = 0$? And, intriguingly, what if we change the minus sign in the equation for a circle to a plus, so that it reads $x^2 + y^2 + 1 =0$? Now $x$ and $y$ have to be complex, or there are no solutions at all. A simple example is a circle of radius 1, which is the set of all points which are at a unit distance from its centre, but it is also the set of points $(x,y)$ satisfying $x^2 + y^2 -1 = 0$. As its name suggests, algebraic geometry deals with curves or surfaces (or more abstract generalisations of these) which can be viewed both as geometric objects and as solutions of algebraic (specifically, polynomial) equations.
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