On the holes in Iⁿ for symmetric bilinear forms in characteristic 2

Let F be a field. Following the resolution of Milnor's conjecture relating the graded Witt ring of F to its mod-2 Milnor K-theory, a major problem in the theory of symmetric bilinear forms is to understand, for any positive integer n, the low-dimensional part of Iⁿ (F), the nth power of the fundamental ideal in the Witt ring of F. In a 2004 paper, Karpenko used methods from the theory of algebraic cycles to show that if b is a non-zero anisotropic symmetric bilinear form of dimension < 2^n+1 representing an element of Iⁿ (F), then b has dimension 2^n+1 - 2ⁱ for some 1 i n. When i = n, a classical result of Arason and Pfister says that b is similar to an n-fold Pfister form. At the next level, it has been conjectured that if n 2 and i= n-1, then b is isometric to the tensor product of an (n-2) -fold Pfister form and a 6-dimensional form of trivial discriminant. This has only been shown to be true, however, when n = 2, or when n = 3 and char (F) 2 (another result of Pfister). In the present article, we prove the conjecture for all values of n in the case where char (F) =2. In addition, we give a short and elementary proof of Karpenko's theorem in the characteristic-2 case, rendering it free from the use of subtle algebraic-geometric tools. Finally, we consider the question of whether additional dimension gaps can appear among the anisotropic forms of dimension 2^n+1 representing an element of Iⁿ (F). When char (F) 2, a result of Vishik asserts that there are no such gaps, but the situation seems to be less clear when char (F) = 2.