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- Diophantine_geometry abstract "In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general commutative ring such as the integers. A single equation defines a hypersurface, and simultaneous Diophantine equations give rise to a general algebraic variety V over K; the typical question is about the nature of the set V(K) of points on V with co-ordinates in K, and by means of height functions quantitative questions about the "size" of these solutions may be posed, as well as the qualitative issues of whether any points exist, and if so whether there are an infinite number. Given the geometric approach, the consideration of homogeneous equations and homogeneous co-ordinates is fundamental, for the same reasons that projective geometry is the dominant approach in algebraic geometry. Rational number solutions therefore are the primary consideration; but integral solutions (i.e. lattice points) can be treated in the same way as an affine variety may be considered inside a projective variety that has extra points at infinity.The general approach of diophantine geometry is illustrated by Faltings' theorem (a conjecture of L. J. Mordell) starting that an algebraic curve C of genus g > 1 over the rational numbers has only finitely many rational points. The first result of this kind may have been the theorem of Hilbert and Hurwitz dealing with the case g = 0. The theory consists both of theorems and many conjectures and open questions.".
- Diophantine_geometry wikiPageExternalLink 1183526083.
- Diophantine_geometry wikiPageExternalLink 1183532391.
- Diophantine_geometry wikiPageID "502205".
- Diophantine_geometry wikiPageRevisionID "575789483".
- Diophantine_geometry hasPhotoCollection Diophantine_geometry.
- Diophantine_geometry id "d/d032630".
- Diophantine_geometry title "Diophantine geometry".
- Diophantine_geometry subject Category:Diophantine_geometry.
- Diophantine_geometry comment "In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general commutative ring such as the integers.".
- Diophantine_geometry label "Diophantine geometry".
- Diophantine_geometry label "Diophantische meetkunde".
- Diophantine_geometry label "Диофантова геометрия".
- Diophantine_geometry sameAs Diophantische_meetkunde.
- Diophantine_geometry sameAs m.0bwhyn7.
- Diophantine_geometry sameAs Q2179749.
- Diophantine_geometry sameAs Q2179749.
- Diophantine_geometry wasDerivedFrom Diophantine_geometry?oldid=575789483.
- Diophantine_geometry isPrimaryTopicOf Diophantine_geometry.