slides - University of Birmingham
... x y + xy − kxy + k = 0 which is an elliptic curve of degree 3 amenable to the advanced methods of algebraic geometry. Every triangle with a given value of k (= s2/A) corresponds to a point on this elliptic curve in the region defined by x > 0, y > 0 and xy > 1 For the (3-4-5) triangle shown at the o ...
... x y + xy − kxy + k = 0 which is an elliptic curve of degree 3 amenable to the advanced methods of algebraic geometry. Every triangle with a given value of k (= s2/A) corresponds to a point on this elliptic curve in the region defined by x > 0, y > 0 and xy > 1 For the (3-4-5) triangle shown at the o ...
Filip Najman: Arithmetic geometry (60 HOURS) Arithmetic
... Arithmetic geomoetry is a branch of mathematics which uses the methods of algebraic geometry and applies it to problems coming from number theory. In arithmetic geometry, we study the properties of the set of solutions of a polynomial equation or a set of polynomial equations, but over ”arithmetical ...
... Arithmetic geomoetry is a branch of mathematics which uses the methods of algebraic geometry and applies it to problems coming from number theory. In arithmetic geometry, we study the properties of the set of solutions of a polynomial equation or a set of polynomial equations, but over ”arithmetical ...
Algebraic curve
In mathematics, an algebraic curve or plane algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.For example, the unit circle is an algebraic curve, being the set of zeros of the polynomial x2 + y2 − 1. Various technical considerations result in the complex zeros of a polynomial being considered as belonging to the curve. Also, the notion of algebraic curve has been generalized to allow the coefficients of the defining polynomial and the coordinates of the points of the curve to belong to any field, leading to the following definition.In algebraic geometry, a plane affine algebraic curve defined over a field k is the set of points of K2 whose coordinates are zeros of some bivariate polynomial with coefficients in k, where K is some algebraically closed extension of k. The points of the curve with coordinates in k are the k-points of the curve and, all together, are the k part of the curve.For example, (2,√−3) is a point of the curve defined by x2 + y2 − 1 = 0 and the usual unit circle is the real part of this curve. The term ""unit circle"" may refer to all the complex points as well to only the real points, the exact meaning usually clear from the context. The equation x2 + y2 + 1 = 0 defines an algebraic curve, whose real part is empty.More generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve that is not contained in some plane is called a skew curve. The simplest example of a skew algebraic curve is the twisted cubic. One may also consider algebraic curves contained in the projective space and even algebraic curves that are defined independently to any embedding in an affine or projective space. This leads to the most general definition of an algebraic curve:In algebraic geometry, an algebraic curve is an algebraic variety of dimension one.