On the first Witt index of quadratic forms

We prove Hoffmanns conjecture determining the possible values of the first Witt index of anisotropic quadratic forms of any given dimension. The proof makes use of the Steenrod type operations on the modulo 2 Chow groups constructed by P. Brosnan. Mathematics Subject Classification (2000) 11E04, 14C25     

Witt index for Galois Ring valued quadratic forms

We calculate the Witt index for Galois Ring valued quadratic forms. We obtain such index depending on the invariant that classifies nonsingular Galois Ring valued quadratic forms together with the type of the corresponding bilinear form (alternating or not).     

A computation of the Witt index for rational quadratic forms

Summary This paper adds the finishing touches to an algorithmic treatment of quadratic forms over the rational numbers. The Witt index of a rational quadratic form is explicitly computed. When combined with a recent adjustment in the Haase invariants, this gives a complete set of invariants for rational quadratic forms, a set which can be computed and which respects all of the standard natural operations (including the tensor product) for quadratic forms. The overall approach does not use (at least explicitly) anyp-adic methods, but it does give the Witt ring of thep-adics as well as the Witt ring of the rationals.     

Degree four cohomological invariants for quadratic forms

Professor Jacob gratefuly acknowledges the Mathematical Sciences Research Institute, Berkeley California, for its support and excellent working conditions provided during 1986–1987. Additional support was provided by the National Science Foundation and the Research Office of Oregon State University     

On quadratic invariants and symplectic structure

We show that the theorems of Sanz-Serna and Eirola and Sanz-Serna concerning the symplecticity of Runge-Kutta and Linear Multistep methods, respectively, follow from the fact that these methods preserve quadratic integral invariants and are closed under differentiation and restriction to closed subsystems.     

On quadratic forms

The present paper is related to the Jordan—von Neumann characterization of inner product spaces, to the Halperin problem concerning quadratic forms, to some results of the present author on quadratic and sesquilinear forms and to recently obtained results of C. T. Ng and of J. Vukman.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

Witt kernels of quadratic forms for purely inseparable multiquadratic extensions in characteristic 2

The aim of this paper is to give a complete answer to the question of hyperbolicity of nonsingular quadratic forms over purely inseparable multiquadratic extensions in characteristic . This completes partial computations of Mammone and Moresi.

     

On the quotient ring by diagonal invariants

For a finite Coxeter group, W, and its reflection representation , we find the character and Hilbert series for a quotient ring of [^*] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman.     

On quasi-reduced quadratic forms

With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 ＋ bXY ＋ cY2 = （a, b, c） of discriminant △〉0（and of sign σ（f） equal to the sign of a）, the quadratic forms associated with f and defined by {〈a＋bu＋cu2,b＋2cu.c〉},with 1≤σ（f）u≤b/2｜c｜ （whenever they exist）, 〈c,-b-2cu,a＋bu＋cu2〉 with b/2｜c｜≤σ（f）u≤[w（f）]=[b＋√△/2｜c｜], are all different from one another and build a set I（f） whose cardinality is #I（f）={1＋[ω（f）],when（2c）｜b,[ω（f）],when （2c）｜b. If f and g are two different reduced quadratic forms, we show that I（f） ∩ I（g） = Ф. Our main result is that the set Q△ is given by the disjoint union of all I（f） with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #（Q△） involving sums of partial quotients of certain continued fractions.     

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅¹⁰⁹. We determine all these K's with dK?¹⁰⁶. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅¹⁰¹¹.     

On ternary quadratic forms

A result from 1988 on the square-free integers represented by a positive definite ternary quadratic form with integral coefficients is made uniform in the determinant of the form.     

Canonical forms and invariants of some classes of control systems

We study the canonical forms and invariants of linear and bilinear control systems and the properties of orbits of these classes of systems under similarity transformations, that is, the action of the group G = GL _n (a change of coordinates in the state space) on the spaces of systems.     

On ring derivations and quadratic functionals

Summary In this note the general form of ring derivations on a function algebra is obtained. In particular, we give a very simple proof of the following result of Nandakumar: a ring derivation on a function algebra is trivial provided that the Choquet boundary of the algebra contains a dense sequentially non-isolated set. Using this theorem we prove a result concerning representability of quadratic functionals by bilinear forms.This work was supported by the Research Council of Slovenia.     

Sets of arithmetical invariants in transfer Krull monoids

Transfer Krull monoids are a recently introduced class of monoids and include the multiplicative monoids of all commutative Krull domains as well as of wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number between 1 and the elasticity of the monoid can be realized as the elasticity of an element). In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals. Without the assumption on the distribution of prime divisors, arbitrary finite sets can be realized as sets of catenary degrees and as sets of tame degrees.