  In 2013, Marcus, Spielman, and Srivastava resolved the famous Kadison-Singer
conjecture. It states that for $n$ independent random vectors $v_1,\cdots, v_n$
that have expected squared norm bounded by $\epsilon$ and are in the isotropic
position in expectation, there is a positive probability that the determinant
polynomial $\det(xI - \sum_{i=1}^n v_iv_i^\top)$ has roots bounded by $(1 +
\sqrt{\epsilon})^2$. An interpretation of the Kadison-Singer theorem is that we
can always find a partition of the vectors $v_1,\cdots,v_n$ into two sets with
a low discrepancy in terms of the spectral norm (in other words, rely on the
determinant polynomial).
  In this paper, we provide two results for a broader class of polynomials, the
hyperbolic polynomials. Furthermore, our results are in two generalized
settings:
  $\bullet$ The first one shows that the Kadison-Singer result requires a
weaker assumption that the vectors have a bounded sum of hyperbolic norms.
  $\bullet$ The second one relaxes the Kadison-Singer result's distribution
assumption to the Strongly Rayleigh distribution.
  To the best of our knowledge, the previous results only support determinant
polynomials [Anari and Oveis Gharan'14, Kyng, Luh and Song'20]. It is unclear
whether they can be generalized to a broader class of polynomials. In addition,
we also provide a sub-exponential time algorithm for constructing our results.
