Isotropy of quadratic forms over the function field of a quadric

Hoffmann, Detlev W.

doi:10.1007/bf02572626

Cited by 78 publications

(60 citation statements)

References 7 publications

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“…One could take η to be a complementary form since φ α | k(φα) is hyperbolic. Here we see η| k(φα) is anisotropic by the following result of Hoffmann (Theorems 1 in [2]). If q, q are anisotropic forms over k such that dim(q) > 2 n−1 > dim(q ), then q | k(q) is anisotropic.…”

Section: Remarksmentioning

confidence: 99%

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Algebraic cobordisms of a Pfister quadric

Vishik

Yagita

2007

Journal of the London Mathematical Society

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Section: Remarksmentioning

confidence: 99%

“…For a quadratic form ξ let us denote as Q ξ the quadric defined by ξ, e.g., Q α = Q φα . Theorem 6.1 (Rost [11], Hoffmann [2]) Let ξ be a subform of the Pfister form φ α of dim(ξ) = 2 n−1 + s, s > 0 (i.e., ξ is a neighbour of φ α ). Let η be the complementary form (φ α = ξ ⊕ η).…”

Section: Glmentioning

confidence: 99%

Algebraic cobordisms of a Pfister quadric

Vishik

Yagita

2007

Journal of the London Mathematical Society

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“…In collaboration with Mammone [15] we extended to characteristic 2 a theorem of Hoffmann on the isotropy of quadratic forms [5]. Here is our result.…”

Section: Definition 55 Any Quadratic Form Of Dimension ≤ 1 Is Calledmentioning

confidence: 80%

On the generic splitting of quadratic forms in characteristic 2

Laghribi

2002

Mathematische Zeitschrift

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In [9] and [10] Knebusch established the basic facts of generic splitting theory of quadratic forms over a field of characteristic different from 2. This paper is related to [11] and [13] where Knebusch and Rehmann generalized partially this theory to a field of characteristic 2. More precisely, we begin with a complete characterization of quadratic forms of height 1 (we don't exclude anisotropic quadratic forms with quasi-linear part of dimension at least 1). This allows us to extend the notion of degree to characteristic 2. We prove some results on excellent forms and splitting tower of a quadratic form. Some results on quadratic forms of height 2 and degree 1 or 2 are given.

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“…We will use induction on m. If we are in the situation of (i), the anisotropy of ϕ F (ψ) follows from [H1,Theorem 1]. Note that if m = 0, then we are certainly in the situation of (i).…”

Section: Lemma 2 Let ϕ 0 Be a Pfister Neighbor Of Dimensionmentioning

confidence: 99%

“…If ψ ∈ C 2 (K, P ), then dim ϕ ≤ 2 k+1 < dim ψ and the anisotropy of ϕ K(ψ) follows from [H1,Theorem 1].…”

Section: Remarks and Examplesmentioning

confidence: 99%

Pythagoras numbers of fields

Hoffmann

1999

J. Amer. Math. Soc.

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A field F F of characteristic ≠ 2 \neq 2 is said to have finite Pythagoras number if there exists an integer m ≥ 1 m\geq 1 such that each nonzero sum of squares in F F can be written as a sum of ≤ m \leq m squares, in which case the Pythagoras number p ( F ) p(F) of F F is defined to be the least such integer. As a consequence of Pfister’s results on the level of fields, p ( F ) p(F) of a nonformally real field F F is always of the form 2 n 2^n or 2 n + 1 2^n+1 , and all integers of such type can be realized as Pythagoras numbers of nonformally real fields. Prestel showed that values of the form 2 n 2^n , 2 n + 1 2^n+1 , and ∞ \infty can always be realized as Pythagoras numbers of formally real fields. We will show that in fact to every integer n ≥ 1 n\geq 1 there exists a formally real field F F with p ( F ) = n p(F)=n . As a refinement, we will show that if n , m ≥ 2 n,m\geq 2 and k ≥ 1 k\geq 1 are integers such that 2 m ≥ 2 k ≥ n 2m\geq 2^{k}\geq n , then there exists a uniquely ordered field F F with p ( F ) = n p(F)=n and u ( F ) = u ~ ( F ) = 2 m u(F)=\tilde {u}(F)=2m (resp. u ( F ) = u ~ ( F ) = ∞ u(F)=\tilde {u}(F)=\infty ), where u u (resp. u ~ \tilde {u} ) denotes the supremum of the dimensions of anisotropic forms over F F which are torsion in the Witt ring of F F (resp. which are indefinite with respect to each ordering on F F ).

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Isotropy of quadratic forms over the function field of a quadric

Cited by 78 publications

References 7 publications

Algebraic cobordisms of a Pfister quadric

Algebraic cobordisms of a Pfister quadric

On the generic splitting of quadratic forms in characteristic 2

Pythagoras numbers of fields

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