  Hidden-variable resultant methods are a class of algorithms for solving
multidimensional polynomial rootfinding problems. In two dimensions, when
significant care is taken, they are competitive practical rootfinders. However,
in higher dimensions they are known to miss zeros, calculate roots to low
precision, and introduce spurious solutions. We show that the hidden variable
resultant method based on the Cayley (Dixon or B\'ezout) matrix is inherently
and spectacularly numerically unstable by a factor that grows exponentially
with the dimension. We also show that the Sylvester matrix for solving
bivariate polynomial systems can square the condition number of the problem. In
other words, two popular hidden variable resultant methods are numerically
unstable, and this mathematically explains the difficulties that are frequently
reported by practitioners. Regardless of how the constructed polynomial
eigenvalue problem is solved, severe numerical difficulties will be present.
Along the way, we prove that the Cayley resultant is a generalization of
Cramer's rule for solving linear systems and generalize Clenshaw's algorithm to
an evaluation scheme for polynomials expressed in a degree-graded polynomial
basis.
