Anisotropy of quadratic forms over global fields of characteristic $\neq$ 2 is Diophantine

TL;DR

The paper proves that over a global field $K$ with $ ext{char}(K)\neq 2$, the set $\{(a_i)\in(K^{\times})^{m} : \sum_{i=1}^m a_i x_i^2 \text{ is anisotropic over } K\}$ is diophantine for every $m$, with $m\ge 5$ following from local-global principles and the genuinely new case being $m=4$. It blends Koenigsmann's framework with global class-field theory, diophantine definability of non-norms, and a quaternion-algebra encoding via $H_{a,b}$, together with a diagonalization procedure to reduce anisotropy to fixed diophantine systems. The authors also derive corollaries for diagonal quadratic forms and provide a uniform approach that covers number fields and global function fields of odd characteristic. Collectively, this advances the Diophantine definability landscape for natural arithmetic sets and sheds light on the isotropy behavior of quadratic forms across global fields.

Abstract

We prove that the set of anisotropic quadratic forms over global fields of characteristic different from 2 is a diophantine set. Our proof builds upon and extends the method of Koenigsmann, using tools from class field theory, local-global principles, and recent advances on the diophantine definability of non-norm sets over global fields.

Theorems & Definitions (45)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof