The fan-pall imbedding theorem over formally real fields

The converse of the Cauchy interlacing theorem, relating eigenvalues of a symmetric real matrix and eigenvalues of a principal submatrix, first proved by Fan and Pall, is extended to the case of symmetric matrices with entries in an arbitrary formally real field.     

On some Hasse Principles over formally real fields

Mathematische Zeitschrift -     

Quadratic forms over formally real fields with eight square classes

Complete classification of formally real fields with 8 square classes with respect to the behaviour of quadratic forms is given. Two fields F and K are equivalent with respect to quadratic forms if the quadratic form schemes of the two fields are isomorphic or in other words, if the Witt rings W(F) and W(K) are isomorphic. It is shown here that for formally real fields with 8 square classes there are exactly seven possible quadratic form schemes and for each of the seven schemes a formally real field with 8 square classes and the given scheme is constructed.     

Similarity to symmetric matrices over fields which are not formally real

We show that a matrix is similar to a symmetric matrix over a field of characteristic 2 if and only if the minimum polynomial of the matrix is not the product of distinct irreducible polynomials whose splitting fields are inseparable extensions. When the field is not of characteristic 2, a known theorem is generalized by considering k, the number of elementary divisors of odd degree of the n × n A: If ?1 is a sum of 2^v squares and n differs from a multiple of 2 ^{v + 1} by at most ±k, then A is similar to a symmetric matrix.     

Similarity to symmetric matrices over fields which are not formally real

We show that a matrix is similar to a symmetric matrix over a field of characteristic 2 if and only if the minimum polynomial of the matrix is not the product of distinct irreducible polynomials whose splitting fields are inseparable extensions. When the field is not of characteristic 2, a known theorem is generalized by considering k, the number of elementary divisors of odd degree of the n × n A: If -1 is a sum of 2^v squares and n differs from a multiple of 2^{v + 1} by at most ±k, then A is similar to a symmetric matrix.     

Extending orderings on formally real fields

Archiv der Mathematik -     

Invariants of formally real fields and quadratic forms

Let F be a formally real field. Connections between the invariants I(F), N(F), ud(F), ũd(F), β_F(i) (i ϵ [1, l(F) - 1]) are studied, using a new invariant C(F). The possibilities for β_F are described when l(F) ≤ 5 or ud(F) ≤ 4, without considering existence.     

On formally real modules

The aim of this paper is to develop the study of modules in real algebra. By introducing two kinds of reality for modules over commutative rings, we establish some results on semireal modules and real modules. Our results involve real radicals of submodules and orderings on modules.     

On the Orlicz-Sobolev imbedding theorem

The Orlicz space analog of the Sobolev imbedding theorem established for bounded domains by Donaldson and Trudinger is here extended to unbounded domains.     

The u-invariant of one-dimensional function fields over real power series fields

The u-invariant of a field K of characteristic not 2 is the largest dimension of an anisotropic torsion quadratic form over K. We extend the recent theorem by Harbater, Hartmann, and Krashen on the u-invariant of a one-dimensional function field over a complete discretely valued field to the case where the residue field can be ordered.     

A generalized Sard theorem on real closed fields

We work with semi‐algebraic functions on arbitrary real closed fields. We generalize the notion of critical values and prove a Sard type theorem in our framework.     

Companion forms over totally real fields

We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2 < k < p, which is ordinary at all primes dividing p and has tamely ramified Galois representation at all primes dividing p, then there is a “companion form” of parallel weight k′ := p + 1 − k. This work generalises results of Gross and Coleman–Voloch for modular forms over Q.     

Hermitian forms over division algebras over real function fields

We develop here the theory of (skew-) hermitian forms over division algebras over the real function field and its completions. In particular, a local and local-global classification for forms of all types are given and some Hasse principles are proved.     

2-universal -lattices over real quadratic fields

Let be a real quadratic field with m a square-free positive rational integer, and be the ring of integers in F. An -lattice L on a totally positive definite quadratic space V over F is called r-universal if L represents all totally positive definite -lattices l with rank r over . We prove that there exists no 2-universal -lattice over F with rank less than 6, and there exists a 2-universal -lattice over F with rank 6 if and only if m=2, 5. Moreover there exists only one 2-universal -lattice with rank 6, up to isometry, over .     

Jacobi forms over totally real number fields

We define Jacobi forms over a totally real algebraic number field K and construct examples by first embedding the group and the space into the symplectic group and the symplectic upper half space respectively. Then symplectic modular forms are created and Jacobi forms arise by taking the appropriate Fourier coefficients. Also some known relations of Jacobi forms to vector valued modular forms over rational numbers are extended to totally real fields.     

Witt equivalence of algebraic function fields over real closed fields

We examine the conditions for two algebraic function fields over real closed fields to be Witt equivalent. We show that there are only two Witt classes of algebraic function fields with a fixed real closed field of constants: real and non-real ones. The first of them splits further into subclasses corresponding to the tame equivalence. This condition has a natural interpretation in terms of both: orderings (the associated Harrison isomorphism maps 1-pt fans onto 1-pt fans), and geometry and topology of associated real curves (the bijection of points is a homeomorphism and these two curves have the same number of semi-algebraically connected components). Finally, we derive some immediate consequences of those theorems. In particular we describe all the Witt classes of algebraic function fields of genus 0 and 1 over the fixed real closed field. Received: 16 February 2000; in final form: 7 December 2000 / Published online: 18 January 2002