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 an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, 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 key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ 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.

中文翻译：

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二次型函数域标量扩展后二次型指数的界

让$q$是在一般域上定义的各向异性二次形式$F$. 在本文中，我们制定了各向同性指数的新上限$q$在对任意二次曲线的函数场进行标量扩展之后。一方面，这个界限提供了对 Karpenko-Merkurjev 和 Totaro 早期工作中建立的重要界限的改进；另一方面，它是卡尔彭科定理对第一个较高各向同性指数的可能值的直接推广。我们在两个关键案例中证明了它的有效性：（i）$\text{char}(F)\neq 2$, 和 (ii) 的情况$\文本{字符}(F)=2$和$q$是准线性（即，可对角化）。这两种情况分别使用完全不同的方法进行处理，第一种是代数几何，第二种是纯代数。