berhuy@fourier.ujf-grenoble.fr, nicolas.grenier-boley@univ-rouen.fr, mmahmoudi@sharif.ir

Submission: 2011, Oct 6

Let $q$ be a quadratic form over a field $K$ of characteristic different from $2$. We investigate the properties of the smallest positive integer $n$ %$n={\sn}_{q}(K)$ such that $-1$ is a sums of $n$ values represented by $q$ in several situations. We relate this invariant (which is called the $q$-level of $K$) to other invariants of $K$ such as the level, the $u$-invariant and the Pythagoras number of $K$. The problem of determining the numbers which can be realized as a $q$-level for particular $q$ or $K$ is studied. We also observe that the $q$-level 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 $q$. We highlight necessary and/or sufficient conditions for the $q$-level to be finite. Throughout the paper, special emphasis is given to the case where $q$ is a Pfister form.

2010 Mathematics Subject Classification: 11E04, 11E10, 11E25, 11E39, 11E81, 12D15

Keywords and Phrases: Level of a field, quadratic form, Pfister form, round form, Pythagoras number, sums of squares, formally real field, $u$-invariant, hermitian level, sublevel, weakly isotropic form, signature of a quadratic form, strong approximation property, ordering of a field.

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