scholarly journals Norm Bounds for Ehrhart Polynomial Roots

2007 ◽  
Author(s):  
Benjamin Braun
Keyword(s):  
Ehrhart Polynomial ◽  
Polynomial Roots

scholarly journals Ehrhart Polynomial Roots of Reflexive Polytopes

10.37236/7780 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Gábor Hegedüs ◽  
Akihiro Higashitani ◽  
Alexander Kasprzyk
Keyword(s):  
Recent Work ◽  
Ehrhart Polynomial ◽  
Polynomial Roots ◽  
Lattice Polytope ◽  
Reflexive Polytopes ◽  
Half Strip ◽  
Reflexive Polytope

Recent work has focused on the roots $z\in\mathbb{C}$ of the Ehrhart polynomial of a lattice polytope $P$. The case when $\Re{z}=-1/2$ is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when $\mathrm{dim}(P)\leq 7$. We also consider the "half-strip condition", where all roots $z$ satisfy $-\mathrm{dim}(P)/2\leq\Re{z}\leq \mathrm{dim}(P)/2-1$, and show that this holds for any reflexive polytope with $\mathrm{dim}(P)\leq 5$. We give an example of a $10$-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi–Shibata in dimension $34$.

scholarly journals On the Ehrhart Polynomial of Minimal Matroids

2021 ◽  
Author(s):  
Luis Ferroni
Keyword(s):  
Ehrhart Polynomial ◽  
Matroid Polytopes ◽  
Connected Matroid ◽  
Fixed Rank

AbstractWe provide a formula for the Ehrhart polynomial of the connected matroid of size n and rank k with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $$h^*$$ h ∗ -real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are $$h^*$$ h ∗ -real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.

On the structure of polynomial roots

2021 ◽  
Vol 105 (563) ◽  
pp. 253-262
Author(s):  
R. W. D. Nickalls
Keyword(s):  
Underlying Structure ◽  
Polynomial Roots ◽  
Polynomial Degree

This Article explores how root multiplicity and polynomial degree influence the structure of the roots of a univariant polynomial. After setting up the notation, we draw upon a result derived in [1], and show that all polynomial roots have a common underlying structure comprising just five parameters. Finally we present some examples involving the lower polynomials.

scholarly journals Bounds on polynomial roots using intercyclic companion matrices

2018 ◽  
Vol 539 ◽  
pp. 94-116
Author(s):  
Kevin N. Vander Meulen ◽  
Trevor Vanderwoerd
Keyword(s):  
Polynomial Roots ◽  
Companion Matrices

scholarly journals A Diagnostic for Seasonality Based Upon Polynomial Roots of ARMA Models

2021 ◽  
Vol 37 (2) ◽  
pp. 367-394
Author(s):  
Tucker McElroy
Keyword(s):  
Moving Average ◽  
Type I ◽  
Simulation Studies ◽  
Arma Models ◽  
Asymptotic Results ◽  
Global Test ◽  
Raw Data ◽  
Polynomial Roots ◽  
Oscillatory Character ◽  
Statistical Agencies

Abstract Methodology for seasonality diagnostics is extremely important for statistical agencies, because such tools are necessary for making decisions whether to seasonally adjust a given series, and whether such an adjustment is adequate. This methodology must be statistical, in order to furnish quantification of Type I and II errors, and also to provide understanding about the requisite assumptions. We connect the concept of seasonality to a mathematical definition regarding the oscillatory character of the moving average (MA) representation coefficients, and define a new seasonality diagnostic based on autoregressive (AR) roots. The diagnostic is able to assess different forms of seasonality: dynamic versus stable, of arbitrary seasonal periods, for both raw data and seasonally adjusted data. An extension of the AR diagnostic to an MA diagnostic allows for the detection of over-adjustment. Joint asymptotic results are provided for the diagnostics as they are applied to multiple seasonal frequencies, allowing for a global test of seasonality. We illustrate the method through simulation studies and several empirical examples.

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