On generic quadratic forms in a power of the fundamental ideal

We work with non-degenerate quadratic forms over fields of characteristic \(\ne 2\) $\ne 2$ and write I for the fundamental ideal in the Witt ring of quadratic forms. For given \(n\geqslant 4\) $n⩾4$ and \(d\geqslant 2^n+2^{n-1}\) $d⩾2^n+2^{n-1}$ , a construction of a generic d-dimensional quadratic form \(q\in I^n\) $q\in I^n$ is not available. To remedy this issue, we introduce a sequence of d-dimensional quadratic forms in \(I^n\) $I^n$ , approximating q. It allows us to define and study certain invariants of q including the reduced Chow ring and the indexes of its grassmannians as well as the J-invariant of q. An extension of the results to characteristic 2 is also provided.