Indefinite quadratic polynomials

Quadratic Irrationals, Quadratic Ideals and Indefinite Quadratic Forms II

Let D = 1 be a positive non-square integer and let δ = √ D or 1+ √ D 2 be a real quadratic irrational with trace t = δ + δ and norm n = δδ. Let γ = P+δ Q be a quadratic irrational for positive integers P and Q. Given a quadratic irrational γ, there exist a quadratic ideal Iγ = [Q, δ + P ] and an indefinite quadratic form Fγ(x, y) = Q(x−γy)(x−γy) of discriminant Δ = t − 4n. In the first section,...

On Indefinite Binary Quadratic Forms and Quadratic Ideals

We consider some properties of indefinite binary quadratic forms F (x, y) = ax +bxy−y of discriminant ∆ = b +4a, and quadratic ideals I = [a, b−√∆ ]. AMS Mathematics Subject Classification (2000): 11E04, 11E12, 11E16

Neighbors of Indefinite Binary Quadratic Forms

In this paper, we derive some algebraic identities on right and left neighbors R(F ) and L(F ) of an indefinite binary quadratic form F = F (x, y) = ax + bxy + cy of discriminant Δ = b − 4ac. We prove that the proper cycle of F can be given by using its consecutive left neighbors. Also we construct a connection between right and left neighbors of F . Keywords—Quadratic form, indefinite form, cy...

On successive minima of indefinite quadratic forms

Dimensions of Anisotropic Indefinite Quadratic Forms, I

By a theorem of Elman and Lam, fields over which quadratic forms are classified by the classical invariants dimension, signed discriminant, Clifford invariant and signatures are exactly those fields F for which the third power IF of the fundamental ideal IF in the Witt ring WF is torsion free. We study the possible values of the uinvariant (resp. the Hasse number ũ) of such fields, i.e. the sup...