isAnisotropic(beta)
Recall a symmetric bilinear form $\beta$ is said to be isotropic if there exists a nonzero vector $v$ for which $\beta(v,v) = 0$. Witt's decomposition theorem implies that a nondegenerate symmetric bilinear form decomposes uniquely into an isotropic and an anisotropic part. Certifying (an)isotropy is then an important computational problem when working with the GrothendieckWitt ring.
Over $\mathbb{C}$, any form of rank two or higher contains a copy of the hyperbolic form, and hence is isotropic. Thus we can determine anisotropy simply by a consideration of rank.


Forms over $\mathbb{R}$ are anisotropic if and only if all its diagonal entries are positive or are negative.


Over finite fields, a form is anisotropic so long as it is nondegenerate, of rank $\le 2$ and not isomorphic to the hyperbolic form.


Over $\mathbb{Q}$ things become a bit more complicated. We can exploit the localtoglobal principle for isotropy (the HasseMinkowski principle), which states that a form is isotropic over $\mathbb{Q}$ if and only if it is isotropic over all its completions, meaning all the $p$adic numbers and $\mathbb{R}$ [L05, VI.3.1]. We note, however, the classical result that all forms of rank $\ge 5$ in $\mathbb{Q}_p$ are isotropic [S73, IV Theorem 6]. Thus isotropy in this range of ranks is equivalent to checking it over the real numbers.



For forms of rank $\le 4$ the problem reduces to computing the maximum anisotropic dimension of the form over local fields. Ternary forms are isotropic away from primes dividing the coefficients of the form in a diagonal basis by e.g. [L05, VI.2.5(2)], so there are only finitely many places to check. Over these relevant primes, isotropy of a form $\beta \in \text{GW}(\mathbb{Q})$ over $\mathbb{Q}_p$ is equivalent to the statement that $(1,\text{disc}(\beta))_p = H(\beta)$ where $H(\beta)$ denotes the HasseWitt invariant attached to $\beta$ and $(,)_p$ is the Hilbert Symbol.
A binary form $q$ is isotropic if and only if it is isomorphic to the hyperbolic form, which implies in particular that the rank, signature, and discriminant of $q$ agree with that of $\mathbb{H}=\langle 1,1\rangle$.
Citations:
The object isAnisotropic is a method function.