Graded Witt kernels of the compositum of multiquadratic extensions with the function fields of Pfister forms

Let F be a field of characteristic 2 and Wq(F) be the Witt group of nonsingular quadratic forms over F. Let φ be a bilinear Pfister form over F and L be a multiquadratic extension of F of separability degree less than of equal to 2. In this paper we compute the kernel of the natural homomorphism H2m+1(F)⟶H2m+1(L(φ)), where H2m+1(F) is the cokernel of the Artin-Schreier operator ℘:ΩFm⟶ΩFm/dΩFm−1 given by xdx1x1∧⋯∧dxmxm↦(x2−x)dx1x1∧⋯∧dxmxm, where ΩFm is the space of m-differential forms over F, and F(φ) is the function field of the affine quadric given by the diagonal quadratic form associated to the bilinear form φ. As a consequence, we deduce the kernel of the natural homomorphisms Iqm+1‾(F)⟶Iqm+1‾(L(φ)) and Iqm+1(F)⟶Iqm+1(L(φ)), where Iqm+1‾(F) denotes the quotient Iqm+1(F)/Iqm+2(F) such that Iqm+1(F)=ImF⊗Wq(F) and ImF is the m-th power of the fundamental ideal IF of the Witt ring of F-bilinear forms. We also include some results concerning the case where φ is replaced by a bilinear Pfister neighbor or a quadratic Pfister form.