Sums of values represented by a quadratic form

International audienceLen 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