scholarly journals Number of common roots and resultant of two tropical univariate polynomials

2018 ◽  
Vol 511 ◽  
pp. 420-439
Author(s):  
Hoon Hong ◽  
J. Rafael Sendra
Keyword(s):  
Univariate Polynomials

scholarly journals Rayleigh Quotient Methods for Estimating Common Roots of Noisy Univariate Polynomials

2019 ◽  
Vol 19 (1) ◽  
pp. 147-163 ◽  
Author(s):  
Alwin Stegeman ◽  
Lieven De Lathauwer
Keyword(s):  
Eigenvalue Problem ◽  
Simulation Study ◽  
Rayleigh Quotient ◽  
Tensor Algebra ◽  
Symbolic Algebra ◽  
Starting Values ◽  
New Methods ◽  
Sylvester Matrix ◽  
Univariate Polynomials ◽  
The Common

AbstractThe problem is considered of approximately solving a system of univariate polynomials with one or more common roots and its coefficients corrupted by noise. The goal is to estimate the underlying common roots from the noisy system. Symbolic algebra methods are not suitable for this. New Rayleigh quotient methods are proposed and evaluated for estimating the common roots. Using tensor algebra, reasonable starting values for the Rayleigh quotient methods can be computed. The new methods are compared to Gauss–Newton, solving an eigenvalue problem obtained from the generalized Sylvester matrix, and finding a cluster among the roots of all polynomials. In a simulation study it is shown that Gauss–Newton and a new Rayleigh quotient method perform best, where the latter is more accurate when other roots than the true common roots are close together.

scholarly journals Multiplication of Polynomials using Discrete Fourier Transformation

2006 ◽  
Vol 14 (4) ◽  
pp. 121-128
Author(s):  
Krzysztof Treyderowski ◽  
Christoph Schwarzweller
Keyword(s):  
Fourier Transformation ◽  
Discrete Fourier Transformation ◽  
Vandermonde Matrices ◽  
Univariate Polynomials

Multiplication of Polynomials using Discrete Fourier Transformation In this article we define the Discrete Fourier Transformation for univariate polynomials and show that multiplication of polynomials can be carried out by two Fourier Transformations with a vector multiplication in-between. Our proof follows the standard one found in the literature and uses Vandermonde matrices, see e.g. [27].

Sparse shifts for univariate polynomials

1996 ◽  
Vol 7 (5) ◽  
pp. 351-364 ◽  
Author(s):  
Y. N. Lakshman ◽  
B. David Saunders
Keyword(s):  
Univariate Polynomials

scholarly journals Spectra of overlapping Wishart matrices and the Gaussian free field

2018 ◽  
Vol 07 (02) ◽  
pp. 1850003 ◽  
Author(s):  
Ioana Dumitriu ◽  
Elliot Paquette
Keyword(s):  
Free Field ◽  
Covariance Structure ◽  
Height Function ◽  
Gaussian Free Field ◽  
Moment Conditions ◽  
Wishart Matrices ◽  
Wigner Random Matrices ◽  
Infinite Array ◽  
Univariate Polynomials ◽  
Upper Half Plane

Consider a doubly-infinite array of i.i.d. centered variables with moment conditions, from which one can extract a finite number of rectangular, overlapping submatrices, and form the corresponding Wishart matrices. We show that under basic smoothness assumptions, centered linear eigenstatistics of such matrices converge jointly to a Gaussian vector with an interesting covariance structure. This structure, which is similar to those appearing in [A. Borodin, Clt for spectra of submatrices of Wigner random matrices, Mosc. Math. J. 14(1) (2014) 29–38; A. Borodin and V. Gorin, General beta Jacobi corners process and the Gaussian free field, preprint (2013), arXiv:1305.3627; T. Johnson and S. Pal, Cycles and eigenvalues of sequentially growing random regular graphs, Ann. Probab. 42(4) (2014) 1396–1437], can be described in terms of the height function, and leads to a connection with the Gaussian Free Field on the upper half-plane. Finally, we generalize our results from univariate polynomials to a special class of planar functions.

scholarly journals A Generalization of a Theorem of Boyd and Lawton

2013 ◽  
Vol 56 (4) ◽  
pp. 759-768 ◽  
Author(s):  
Zahraa Issa ◽  
Matilde Lalín
Keyword(s):  
Analogous Result ◽  
Mahler Measure ◽  
Log P ◽  
Multivariate Polynomial ◽  
Univariate Polynomials

Abstract.The Mahler measure of a nonzero n-variable polynomial P is the integral of log |P| on the unit n-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of log |P| for possibly different P’s), multiple Mahler measure (involving products of log |P| for possibly different P’s), and higher Mahler measure (involving logk |P|).

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