2024/09/03 by Stephen Scully, Scully, Stephen
Mathematics · Physics and Astronomy · #11E04 #14E05 #Algebraic Geometry (math.AG) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2409.02059
openalex publication_date 2024/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let p and q be anisotropic quasilinear quadratic forms over a field F of characteristic 2, and let i be the isotropy index of q after scalar extension to the function field of the affine quadric with equation p=0. In this article, we establish a strong constraint on i in terms of the dimension of q and two stable birational invariants of p, one of which is the well-known "Izhboldin dimension", and the other of which is a new invariant that we denote Δ(p). Examining the contribution from the Izhboldin dimension, we obtain a result that unifies and extends the quasilinear analogues of two fundamental results on the isotropy of non-singular quadratic forms over function fields of quadrics in arbitrary characteristic due to Karpenko and Karpenko-Merkurjev, respectively. This proves in a strong way the quasilinear case of a general conjecture previously formulated by the author, suggesting that a substantial refinement of this conjecture should hold.