Differentiators and the geometry of polynomials

Pereira, Rajesh

doi:10.1016/s0022-247x(03)00465-7

Cited by 40 publications

(54 citation statements)

References 12 publications

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“…When the paper was under review, R. Pereira [31] published a solution to the de Bruijn-Springer and Schoenberg conjectures, based on similar ideas. The author thanks the referee for informing him about it.…”

Section: Acknowledgementsmentioning

confidence: 99%

Inverse spectral problem for normal matrices and the Gauss-Lucas theorem

Malamud

2004

Trans. Amer. Math. Soc.

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Abstract. We establish an analog of the Cauchy-Poincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss-Lucas theorem and prove the old conjecture of de Bruijn-Springer on the location of the roots of a complex polynomial and its derivative and an analog of Rolle's theorem, conjectured by Schoenberg.

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Section: Acknowledgementsmentioning

confidence: 99%

Inverse spectral problem for normal matrices and the Gauss-Lucas theorem

Malamud

2004

Trans. Amer. Math. Soc.

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“…We use say that a function has a local extremum at a point if it has either a local maximum or a local minimum at that point. We note that the proof of this result is nearly identical to part of the proof of [12,Lemma 2.9]. We provide a proof here for the convenience of the reader.…”

Section: Introductionmentioning

confidence: 52%

The theory and applications of complex matrix scalings

Pereira

Boneng

2014

Special Matrices

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Abstract:We generalize the theory of positive diagonal scalings of real positive de nite matrices to complex diagonal scalings of complex positive de nite matrices. A matrix A is a diagonal scaling of a positive de nite matrix M if there exists an invertible complex diagonal matrix D such that A = D * MD and where every row and every column of A sums to one. We look at some of the key properties of complex diagonal scalings and we conjecture that every n by n positive de nite matrix has at most n− scalings and prove this conjecture for certain special classes of matrices. We also use the theory of complex diagonal matrix scalings to formulate a van der Waerden type question on the permanent function; we show that the solution of this question would have applications to nding certain maximally entangled quantum states.

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“…The proof of Theorem 2 given above is a slight modification of the corresponding arguments in [26] and [28].…”

Section: Lemma 3 With the Above Notations One Hasmentioning

confidence: 99%

Equilibrium points of logarithmic potentials induced by positive charge distributions. I. Generalized de Bruijn-Springer relations

Borcea

2007

Trans. Amer. Math. Soc.

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Abstract. A notion of weighted multivariate majorization is defined as a preorder on sequences of vectors in Euclidean space induced by the Choquet ordering for atomic probability measures. We characterize this preorder both in terms of stochastic matrices and convex functions and use it to describe the distribution of equilibrium points of logarithmic potentials generated by discrete planar charge configurations. In the case of n positive charges we prove that the equilibrium points satisfy n 2 weighted majorization relations and are uniquely determined by n − 1 such relations. It is further shown that the Hausdorff geometry of the equilibrium points and the charged particles is controlled by the weighted standard deviation of the latter. By using finiterank perturbations of compact normal Hilbert space operators we establish similar relations for infinite charge distributions. We also discuss a hierarchy of weighted de Bruijn-Springer relations and inertia laws, the existence of zeros of Borel series with positive l 1 -coefficients, and an operator version of the Clunie-Eremenko-Rossi conjecture.

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Differentiators and the geometry of polynomials

Cited by 40 publications

References 12 publications

Inverse spectral problem for normal matrices and the Gauss-Lucas theorem

Inverse spectral problem for normal matrices and the Gauss-Lucas theorem

The theory and applications of complex matrix scalings

Equilibrium points of logarithmic potentials induced by positive charge distributions. I. Generalized de Bruijn-Springer relations

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