Self-inversive polynomials, curves, and
                    codes

Joyner, David; Shaska, Tony

doi:10.1090/conm/703/14138

Cited by 3 publications

(2 citation statements)

References 12 publications

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“…They are of some importance in other parts of mathematics. Here are some connections: they play a role in coding theory [11], symplectic geometry or knot theory. A symplectic polynomial is the characteristic polynomial of a symplectic matrix.…”

Section: Using the Sign Notation Of (1) These Two Statements Are Agai...mentioning

confidence: 99%

An Elementary Dyadic Riemann Hypothesis

Knill

2018

Preprint

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The connection zeta function of a finite abstract simplicial complex G is defined as ζ L (s) = x∈G λ −sx , where λ x are the eigenvalues of the connection Laplacian L defined by L(x, y) = 1 if x and y intersect and 0 else. (I) As a consequence of the spectral formula χ(G) =x (−1) dim(x) = p(G) − n(G), where p(G) is the number of positive eigenvalues and n(G) is the number of negative eigenvalues of L, both the Euler characteristic χ(G) = ζ(0) − 2iζ (0)/π as well as determinant det(L) = e ζ (0)/π can be written in terms of ζ. (II) As a consequence of the generalized Cauchy-Binet formula for the coefficients of the characteristic polynomials of a product of matrices we show that for every one-dimensional simplicial complex G, the functional equationis the Zeta function of the positive definite squared connection operator L 2 of G. Equivalently, the spectrum σ of the integer matrix L 2 for a 1-dimensional complex always satisfies the symmetry σ = 1/σ and the characteristic polynomial of L 2 is palindromic. The functional equation extends to products of one-dimensional complexes. (III) Explicit expressions for the spectrum of circular connection Laplacian lead to an explicit entire zeta function in the Barycentric limit. The situation is simpler than in the Hodge Laplacian H = D 2 case [21], where no functional equation was available. In the connection Laplacian case, the limiting zeta function is a generalized hypergeometric function which for an integer s is given by an elliptic integral over the real elliptic curve w 2 = (1 + z)(1 − z)(z 2 − 4z − 1), which has the analytic involutive symmetry (z, w) → (1/z, w/z 2 ).

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Section: Using the Sign Notation Of (1) These Two Statements Are Agai...mentioning

confidence: 99%

An Elementary Dyadic Riemann Hypothesis

Knill

2018

Preprint

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“…11.5], and a recent strengthening by Lalín and Smyth [27]) essentially moves that counting problem to that of counting the number of roots of a self-inversive polynomial in the unit circle, and to being able to verify whether such a polynomial is circle rooted. Vieira [40] surveys self-inversive and palindromic polynomials from a viewpoint that is entirely germane to this paper, while Joyner and Shaska [19] give some a priori reasons for considering these kinds of polynomials. Many of the methods and results to be presented here work for selfinversive polynomials and will be shown as such.…”

Section: Introductionmentioning

confidence: 99%

Dragging the roots of a polynomial to the unit circle

Mandel¹,

Robins²

2019

Preprint

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Several conditions are known for a self-inversive polynomial that ascertain the location of its roots, and we present a framework for comparison of those conditions. We associate a parametric family of polynomials p α to each such polynomial p, and define (p), (p) to be the sharp threshold values of α that guarantee that, for all larger values of the parameter, p α has, respectively, all roots in the unit circle and all roots interlacing the roots of unity of the same degree. Interlacing implies circle rootedness, hence (p) ≥ (p), and this inequality is often used for showing circle rootedness. Both (p) and (p) turn out to be semi-algebraic functions of the coefficients of p, and some useful bounds are also presented, entailing several known results about roots in the circle. The study of (p) leads to a rich classification of real selfinversive polynomials of each degree, organizing them into a complete polyhedral fan. We have a close look at the class of polynomials for which (p) = (p), whereas in general the quotient (p) (p) is shown to be unbounded as the degree grows. Several examples and open questions are presented.

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Roots of Polynomials and Umbilics of Surfaces

Guilfoyle,

Klingenberg

2023

Results Math

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Self-inversive polynomials, curves, and codes

Cited by 3 publications

References 12 publications

An Elementary Dyadic Riemann Hypothesis

An Elementary Dyadic Riemann Hypothesis

Dragging the roots of a polynomial to the unit circle

Roots of Polynomials and Umbilics of Surfaces

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