On the number of roots of self-inversive polynomials on the complex unit circle

Vieira, R. S.

doi:10.1007/s11139-016-9804-2

Cited by 17 publications

(17 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For a proof see [25]. If λ = 0 this correspond to a result of Lakatos and Losonczi [16] which says that a self-inversive polynomial with non-zero discriminant has all roots on the unit circle if…”

Section: Self-inversive Polynomialsmentioning

confidence: 62%

See 1 more Smart Citation

Self-inversive polynomials, curves, and codes

Joyner¹,

Shaska²

2018

Higher Genus Curves in Mathematical Physics and Arithmetic Geometry

View full text Add to dashboard Cite

We study connections between self-inversive and self-reciprocal polynomials, reduction theory of binary forms, minimal models of curves, and formally self-dual codes. We prove that if X is a superelliptic curve defined over C and its reduced automorphism group is nontrivial or not isomorphic to a cyclic group, then we can write its equation as y n = f (x) or y n = xf (x), where f (x) is a self-inversive or self-reciprocal polynomial. Moreover, we state a conjecture on the coefficients of the zeta polynomial of extremal formally self-dual codes.arXiv:1606.03159v1 [math.CV]

show abstract

Section: Self-inversive Polynomialsmentioning

confidence: 62%

“…Primary 14Hxx; 11Gxx. c 0000 (copyright holder) [7,13,[16][17][18][19][20]22,25]. Further, we discuss the roots of the self-inversive polynomials.…”

Section: Introductionmentioning

confidence: 99%

Self-inversive polynomials, curves, and codes

Joyner¹,

Shaska²

2018

Higher Genus Curves in Mathematical Physics and Arithmetic Geometry

View full text Add to dashboard Cite

show abstract

“…We remark that if were possible to solve the Bethe Ansatz equations analytically, then we would obtain a complete analytical answer for the spectral problem considered by the Bethe Ansatz technique but, unfortunately, they are too complex for such an ambitious endeavor be accomplished by now -it is only up to the second excited state that a complete analytical solution of the Bethe Ansatz equations were obtained so far [13,14] -, reason by which the Bethe Ansatz equations are usually solved through numerical methods [15,16].…”

Section: The Solutions Of the Bethe Ansatz Equationsmentioning

confidence: 99%

The algebraic Bethe Ansatz and combinatorial trees

Vieira

Lima-Santos

2019

Journal of Integrable Systems

View full text Add to dashboard Cite

We present in this paper a comprehensive introduction to the algebraic Bethe Ansatz, taking as examples the six-vertex model with periodic and non-periodic boundary conditions. We propose a diagrammatic representation of the commutation relations used in the algebraic Bethe Ansatz, so that the action of the transfer matrix in the nth excited state gives place to labeled combinatorial trees. The analysis of these combinatorial trees provides in a straightforward way the eigenvalues and eigenstates of the transfer matrix, as well as the respective Bethe Ansatz equations. Several identities between the R-matrix elements can also be derived from the symmetry of these diagrams regarding the permutation of their labels. This combinatorial approach gives some insights about how the algebraic Bethe Ansatz works, which can be valuable for non-experts readers.

show abstract

“…In [15] Vieira, extending a result of Lakatos and Losonczi [7], presented a sufficient condition for a self-reciprocal polynomial to have a fixed number of roots on the complex unit circle U = {z ∈ C : |z| = 1}. Let p(z) = a d z d + a d−1 z d−1 + · · · + a 1 z + a 0 be a d-th degree self-reciprocal polynomial.…”

Section: Introductionmentioning

confidence: 99%