Sums of values represented by a quadratic form

Grégory Berhuy (1) , Nicolas Grenier-Boley (2) , Mohammad Gholamzadeh Mahmoudi (3)

1 IF - Institut Fourier
2 LDAR (EA_4434) - Laboratoire de Didactique André Revuz
3 SUT - Sharif University of Technology [Tehran]

Len q be a quadratic form over a field K of characteristic different from 2. The properties of the smallest positive integer n such that −1 is a sums of n values represented by q are investigated. The relations of this invariant which is called the q-level of K, with other invariants of K such as the level, the u-invariant and the Pythagoras number of K are studied. The problem of determining the numbers which can be realized as a q-level for particular q or K is also studied. A special emphasis is given to the case where q is a Pfister form. We also observe that the q-level is naturally emerges when one tries to obtain a lower bound for the index of the subgroup of non-zero values represented by a Pfister form q.

Mots clés: Pfister form; Pythagoras; sums of squares; square classes; number; Level of a field; quadratic form; formally real fields; hermitian level; u-invariant

Domaines: Mathématiques [math]

Grégory Berhuy, Nicolas Grenier-Boley, Mohammad Gholamzadeh Mahmoudi. Sums of values represented by a quadratic form. Manuscripta mathematica, 2013, 140 (3-4), pp.531-556. ⟨10.1007/s00229-012-0551-4⟩. ⟨hal-01720654v2⟩