666 research outputs found
Counterexamples to the Neggers-Stanley conjecture
The Neggers-Stanley conjecture (also known as the Poset conjecture) asserts
that the polynomial counting the linear extensions of a partially ordered set
on by their number of descents has real zeros only. We provide
counterexamples to this conjecture.Comment: 4 page
On operators on polynomials preserving real-rootedness and the Neggers-Stanley Conjecture
We refine a technique used in a paper by Schur on real-rooted polynomials.
This amounts to an extension of a theorem of Wagner on Hadamard products of
Toeplitz matrices. We also apply our results to polynomials for which the
Neggers-Stanley Conjecture is known to hold. More precisely, we settle
interlacing properties for -polynomials of series-parallel posets and
column-strict labelled Ferrers posets
Hyperbolicity cones of elementary symmetric polynomials are spectrahedral
We prove that the hyperbolicity cones of elementary symmetric polynomials are
spectrahedral, i.e., they are slices of the cone of positive semidefinite
matrices. The proof uses the matrix--tree theorem, an idea already present in
Choe et al.Comment: 9 pages. Some typos corrected. Details added. To appear in
Optimization Letter
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