Roots of Gårding hyperbolic polynomials

We explore the regularity of the roots of Garding hyperbolic polynomials and real stable polynomials. As an application we obtain new regularity results of Sobolev type for the eigenvalues of Hermitian matrices and for the singular values of arbitrary matrices. These results are optimal among all Sobolev spaces.
13 pages; some details added and typos corrected, accepted for publication in Proc. Amer. Math. Soc