G. Berhuy, N. Grenier-Boley, M. G. Mahmoudi: Sums of values represented by a quadratic form

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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