On the first Witt index of quadratic forms (Q1411968)

The first Witt index \(i_1(\phi)\) of an anisotropic quadratic form \(\phi\) over a field \(F\) of characteristic not 2 is the (usual) Witt index of \(\phi\) over the function field \(F(\phi)\) of the projective quadric given by \(\phi\). Trying to determine the possible values of the first Witt index for forms of a fixed dimension led D. Hoffmann to the following conjecture. If \(\dim (\phi) - 1\) is written as the sum of distinct and ascending two-powers, then the integer \(i_1(\phi) - 1\) is a proper partial sum. Only partial results confirming the conjecture had been known, and in the present paper the author proves the conjecture in its full generality. In the author's words, the proof makes use of the Steenrod type operations on the modulo 2 Chow groups constructed by \textit{P. Brosnan} [Trans. Am. Math. Soc. 355, 1869--1903 (2003; Zbl 1045.55005)].