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This includes numerous results about quadratic number fields, which can often be translated into the language of binary quadratic forms, but also includes developments about forms themselves or that originated by thinking about forms, including Shanks's infrastructure, Zagier's reduction algorithm, Conway's topographs, and Bhargava's ...
Note that there is a close relation between reducing binary quadratic forms and continued fraction expansion; one step in the continued fraction expansion of a certain quadratic irrationality gives a unary operation on the set of reduced forms, which cycles through all reduced forms in one equivalence class.
A mapping q : M → R : v ↦ b(v, v) is the associated quadratic form of b, and B : M × M → R : (u, v) ↦ q(u + v) − q(u) − q(v) is the polar form of q. A quadratic form q : M → R may be characterized in the following equivalent ways: There exists an R-bilinear form b : M × M → R such that q(v) is the associated quadratic form.
In mathematics, in number theory, Gauss composition law is a rule, invented by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his Disquisitiones Arithmeticae, [1] a textbook on number theory published in 1801, in Articles 234 - 244.
Vieta jumping is a classical method in the theory of quadratic Diophantine equations and binary quadratic forms. For example, it was used in the analysis of the Markov equation back in 1879 and in the 1953 paper of Mills. [1]
Over F 2, the Arf invariant is 0 if the quadratic form is equivalent to a direct sum of copies of the binary form , and it is 1 if the form is a direct sum of + + with a number of copies of . William Browder has called the Arf invariant the democratic invariant [ 3 ] because it is the value which is assumed most often by the quadratic form. [ 4 ]
For binary quadratic forms there is a group structure on the set C of equivalence classes of forms with given discriminant. The genera are defined by the generic characters . The principal genus, the genus containing the principal form, is precisely the subgroup C 2 and the genera are the cosets of C 2 : so in this case all genera contain the ...
This L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form. It is a special case of a Hasse–Weil L-function. The natural definition of L(E, s) only converges for values of s in the complex plane with Re(s) > 3/2.