Rational or K-rational Points On Algebraic Varieties
Further information: Diophantine geometryLet V be an algebraic variety over a field K. When V is affine, given by a set of equations fj(x1, ..., xn)=0, j=1, ..., m, with coefficients in K, a K-rational point P of V is an ordered n-tuple (x1, ..., xn) of numbers from the field K that is a solution of all of the equations simultaneously. In the general case, a K-rational point of V is a K-rational point of some affine open subset of V.
When V is projective, defined in some projective space by homogeneous polynomials (with coefficients in K), a K rational point of V is a point in the projective space that is a common solution of all the equations .
Sometimes when no confusion is possible (or when K is the field of the rational numbers), we say rational points instead of K-rational points.
Rational (as well as K-rational) points that lie on an algebraic variety (such as an elliptic curve) constitute a major area of current research. For an abelian variety A, the K-rational points form a group. The Mordell-Weil theorem states that the group of rational points of an abelian variety over K is finitely generated if K is a number field.
The Weil conjectures concern the distribution of rational points on varieties over finite fields, where 'rational points' are taken to mean points from the smallest subfield of the finite field the variety has been defined over.
Read more about this topic: Rational Point
Famous quotes containing the words rational, points, algebraic and/or varieties:
“Plato—who may have understood better what forms the mind of man than do some of our contemporaries who want their children exposed only to “real” people and everyday events—knew what intellectual experience made for true humanity. He suggested that the future citizens of his ideal republic begin their literary education with the telling of myths, rather than with mere facts or so-called rational teachings.”
—Bruno Bettelheim (20th century)
“PLAIN SUPERFICIALITY is the character of a speech, in which any two points being taken, the speaker is found to lie wholly with regard to those two points.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“Now there are varieties of gifts, but the same Spirit; and there are varieties of services, but the same Lord; and there are varieties of activities, but it is the same God who activates all of them in everyone.”
—Bible: New Testament, 1 Corinthians 12:4-6.