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A bound for the index of a quadratic form after scalar extension to the function field of a quadric

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of a celebrated bound established in earlier work of Karpenko-Merkurjev and Totaro; on the other, it is a direct generalization of Karpenko's theorem on the possible values of the first higher isotropy index. We prove its validity in two important cases: (i) the case where $\mathrm{char}(F) \neq 2$, and (ii) the case where $\mathrm{char}(F) = 2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic-geometric, and the second being purely algebraic.