On stable cones of polynomials via reduced Routh parameters

On stable cones of polynomials via reduced Routh parameters

On finding complex roots of polynomials using the Routh array

We present a method, based on the Routh stability criterion, for transforming the task of finding the complex roots of a polynomial of even degree n with real coefficients to that of: (a) finding their real parts as the n/2 real roots of a polynomial of degree n(n−1)/2; (b) for each real part, calculating the corresponding imaginary part as a square root. We illustrate the method for polynomial...

h-Polynomials via Reduced Forms

The flow polytope F G̃ is the set of nonnegative unit flows on the graph G̃. The subdivision algebra of flow polytopes prescribes a way to dissect a flow polytope F G̃ into simplices. Such a dissection is encoded by the terms of the so called reduced form of the monomial ∏ (i,j)∈E(G) xij . We prove that we can use the subdivision algebra of flow polytopes to construct not only dissections, but als...

The Routh-Hurwitz array and realization of characteristic polynomials

Tutte polynomials of q-cones

We derive a formula for the Tutte polynomial t(G′; x, y) of a q-cone G′ of a GF (q)-representable geometry G in terms of t(G; x, y). We use this to construct collections of infinite sequences of GF (q)-representable geometries in which corresponding geometries are not isomorphic and yet have the same Tutte polynomial. We also use this to construct, for each positive integer k, sets of non-isomo...