Inequalities for Indefinite Forms

Generalized Inequalities for Indefinite Forms

We establish abstract inequalities that give, as particular cases, many previously established Hölder-like inequalities. In addition to unifying the proofs of these inequalities, which, in most cases, tend to be technical and obscure, the proofs of our inequalities are quite simple and basic. Moreover, we show that sharper inequalities can be obtained by applying our results.

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...

Effective Estimates on Indefinite Ternary Forms

We give an effective proof of a theorem of Dani and Margulis regarding values of indefinite ternary quadratic forms at primitive integer vectors. The proof uses an effective density-type result for orbits of the groups SO(2, 1) on SL(3,R)/ SL(3,Z).