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On the half-plane property and the Tutte group of a matroid
Combinatorics · Submitted 5 Jun 2009
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular.
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The abstract, sentence by sentence
A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular. We also prove that T_8 and R_9 fail to have the WHPP.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s1 | A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. | background 1.00 | 0.08 | 0.04 | asserted 0.98 |
| s2 | If the polynomial can be chosen with all of its nonzero coefficients equal to one then the matroid has the half-plane property (HPP). | background 1.00 | 0.08 | 0.05 | asserted 0.99 |
| s3 | We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. | method 0.66 | 0.95 | 0.18 | asserted 0.97 |
| s4 | This method makes use of the Tutte group of a matroid. | method 0.95 | 0.62 | 0.05 | asserted 0.99 |
| s5 | We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular. | result 1.00 | 0.94 | 0.42 | asserted 1.00 |
| s6 | We also prove that T_8 and R_9 fail to have the WHPP. | result 1.00 | 0.94 | 0.10 | asserted 1.00 |
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Original abstract
A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular. We also prove that T_8 and R_9 fail to have the WHPP.
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Let us recall that a polynomial $P$ with complex coefficients is said to have the half-plane property [ref] if either $P ≡0$ or else $P(x_1,…,x_n) ≠0$ whenever $x_1,…,x_n$ are complex numbers with strictly positive real part.[ A polynomial $P ≢0$ with the half-plane property is also termed Hurwitz stable. ] We shall show (Corollary [ref] below) that if $P$ is a polynomial with real coefficients that is strictly positive on $(0,∞)^n$ and such that $P^-β$ is completely monotone on $(0,∞)^n$ for at least one $β> 0$, then $P$ necessarily has the half-plane property (but not conversely).
Complete monotonicity for inverse powers of some combinatorially defined polynomials · Some further remarks · 0.82
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