Determinantal representations of hyperbolic curves via polynomial homotopy continuation

Hermitian Determinantal Representations of Hyperbolic Curves

(1) f = det(xM1 + yM2 + zM3), where M1,M2,M3 are Hermitian d × d matrices. The representation is definite if there is a point e ∈ R for which the matrix e1M1+e2M2+e3M3 is positive definite. This imposes an immediate condition on the projective curve VC(f). Because the eigenvalues of a Hermitian matrix are real, every real line passing through e meets this hypersurface in only real points. A pol...

Determinantal representations of hyperbolic forms via weighted shift matrices

Determinantal representations of elliptic curves via Weierstrass elliptic functions

Helton and Vinnikov proved that every hyperbolic ternary form admits a symmetric derminantal representation via Riemann theta functions. In the case the algebraic curve of the hyperbolic ternary form is elliptic, the determinantal representation of the ternary form is formulated by using Weierstrass ℘-functions in place of Riemann theta functions. An example of this approach is given.

Roots of Bivariate Polynomial Systems via Determinantal Representations

Solving polynomial systems via homotopy continuation and monodromy

We develop an algorithm to find all solutions of a generic system in a family of polynomial systems with parametric coefficients using numerical homotopy continuation and the action of the monodromy group. We argue that the expected number of homotopy paths that this algorithm needs to follow is roughly linear in the number of solutions. We demonstrate that our software implementation is compet...