Zeros of self-inversive polynomials

Bonsall, F. F.; Marden, Morris

doi:10.1090/s0002-9939-1952-0047828-8

Cited by 31 publications

(28 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We remark that the proof given here of Cohn's theorem is similar to a proof in Bonsall and Marden [1] that was suggested by J. L. Walsh. Other information on positively turning curves may be found in Polya and Szego [4, part three, problems 103-111].…”

supporting

confidence: 77%

On Positively Turning Immersions

Quine¹

1976

Proceedings of the American Mathematical Society

View full text Add to dashboard Cite

Abstract.Let y: S1 -» C be a C2 immersion of the circle. Let k be the number of zeros of y and suppose da.tg,y(e )/d9 > 0 for y(e'e) # 0; then twny = k/2 + (2ir)~ Sa daxgy where twny is the tangent winding number, and A = S1 -y~'(0). This generalizes the theorem of Cohn that if p is a self-inversive polynomial, the number of zeros of p' in \z\ > 1 is the same as the number of zeros of p in \z\ > 1. For k = 0, this is a topological generalization of Lucas' theorem. We show how (2ir) fA rfargy represents a generalization of the notion of the winding number of y about 0.

show abstract

supporting

confidence: 77%

On Positively Turning Immersions

Quine¹

1976

Proceedings of the American Mathematical Society

View full text Add to dashboard Cite

show abstract

“…Since the class of self-inversive polynomials of degree n includes polynomials of degree n which have all their zeros on U , it is interesting to mention the condition for a self-reciprocal polynomial having all its zeros on U . There have been a number of literatures (see [1]- [2] and [4]- [12]) about the distribution of zeros of self-reciprocal polynomials.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On the Zeros of Self-Reciprocal Polynomials Satisfying Certain Coefficient Conditions

Kim¹,

Lee²

2010

Bulletin of the Korean Mathematical Society

View full text Add to dashboard Cite

show abstract

“…Suppose that P has a zero inside the unit circle. By another theorem of Cohn ([4], p. 113, Theorem IV, see also [2], Theorem 1) the polynomial z m−1 P (z −1 ) = By Lemma 2 for the right hand side R of (12) we have (13) |R| m(A m + B).…”

Section: Remark 3 From Cohn's Theorem It Follows Easily That Both Zementioning

confidence: 99%

“…. , m) are real then in (2) ε should also be real, thus such polynomials are self-inversive if and only if (j) either ε = 1 and A m−k = A k (k = 0, . .…”

Section: Introductionmentioning

confidence: 99%