Displaying similar documents to “Explicit descriptions of quadratic maps on ℙ¹ defined over a field K”

On ternary quadratic forms over the rational numbers

Amir Jafari, Farhood Rostamkhani (2022)

Czechoslovak Mathematical Journal

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For a ternary quadratic form over the rational numbers, we characterize the set of rational numbers represented by that form over the rational numbers. Consequently, we reprove the classical fact that any positive definite integral ternary quadratic form must fail to represent infinitely many positive integers over the rational numbers. Our proof uses only the quadratic reciprocity law and the Hasse-Minkowski theorem, and is elementary.

Birational geometry of quadrics

Burt Totaro (2009)

Bulletin de la Société Mathématique de France

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We construct new birational maps between quadrics over a field. The maps apply to several types of quadratic forms, including Pfister neighbors, neighbors of multiples of a Pfister form, and half-neighbors. One application is to determine which quadrics over a field are ruled (that is, birational to the projective line times some variety) in a larger range of dimensions. We describe ruledness completely for quadratic forms of odd dimension at most 17, even dimension at most 10, or dimension...

Dissident maps on the seven-dimensional Euclidean space

Ernst Dieterich, Lars Lindberg (2003)

Colloquium Mathematicae

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Our article contributes to the classification of dissident maps on ℝ ⁷, which in turn contributes to the classification of 8-dimensional real division algebras. We study two large classes of dissident maps on ℝ ⁷. The first class is formed by all composed dissident maps, obtained from a vector product on ℝ ⁷ by composition with a definite endomorphism. The second class is formed by all doubled dissident maps, obtained as the purely imaginary parts of the structures...