Sums of values represented by a quadratic form

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

1 IF - Institut Fourier
2 LDAR (EA_4434) - Laboratoire de Didactique André Revuz
3 Department of Mathematical Sciences, Sharif University of Technology

Len $q$ be a quadratic form over a field $K$ of characteristic different from $2$. The properties of the smallest positive integer $n$ %$n={\sn}_{q}(K)$ 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 $q$.

Domaines: Mathématiques générales [math.GM]

Grégory Berhuy, Nicolas Grenier-Boley, Mohammad Mahmoudi. Sums of values represented by a quadratic form. 2011. ⟨hal-01720654v1⟩