A solution to Meneguette's polynomial problem

In 1979, Jeltsch [Numer.

Math.

32 (1979), 167-181] conjectured that every zero-stable Brown method is stiffly stable.

For Brown $(K,L)$ methods, Meneguette subsequently reduced the relevant $A_0$-stability question to a zero-location property for a family of perturbed characteristic polynomials.

A closely related perturbation question already appeared in his 1987 Oxford doctoral thesis, and he later posed a broader purely polynomial version of the problem in 1994 [SIAM Review 36 (1994), 656-657].

We solve Meneguette's polynomial problem in its original form.

More precisely, we prove the stronger statement that, even when all the zeros of the original polynomial are allowed to lie in the closed unit disc, every positive perturbation of its leading coefficient yields a polynomial all of whose zeros lie in the open unit disc.

Our argument is entirely polynomial.

Combined with Meneguette's coefficient and strong-stability results, the theorem establishes Jeltsch's conjecture for the corresponding subclass of zero-stable Brown methods.