The approximate irreducible factorization of a univariate polynomial

2009 ◽  
Author(s):  
Zhonggang Zeng
Keyword(s):  
Irreducible Factorization ◽  
Univariate Polynomial

scholarly journals Differential resolvents of minimal order and weight

2004 ◽  
Vol 2004 (54) ◽  
pp. 2867-2893
Author(s):  
John Michael Nahay
Keyword(s):  
Upper Bound ◽  
Minimal Order ◽  
Determinantal Formula ◽  
Differential Field ◽  
Nonzero Coefficient ◽  
Univariate Polynomial ◽  
Linear Differential ◽  
Indicial Equation

We will determine the number of powers ofαthat appear with nonzero coefficient in anα-power linear differential resolvent of smallest possible order of a univariate polynomialP(t)whose coefficients lie in an ordinary differential field and whose distinct roots are differentially independent over constants. We will then give an upper bound on the weight of anα-resolvent of smallest possible weight. We will then compute the indicial equation, apparent singularities, and Wronskian of the Cockleα-resolvent of a trinomial and finish with a related determinantal formula.

scholarly journals Hyperbolic Polynomials and Convex Analysis

2001 ◽  
Vol 53 (3) ◽  
pp. 470-488 ◽  
Author(s):  
Heinz H. Bauschke ◽  
Osman Güler ◽  
Adrian S. Lewis ◽  
Hristo S. Sendov
Keyword(s):  
Convex Function ◽  
Interior Point ◽  
Convex Analysis ◽  
Symmetric Functions ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
Real Polynomial ◽  
Hyperbolic Polynomials ◽  
Real Roots ◽  
Univariate Polynomial

AbstractA homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t ⟼ p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, Gårding proved in 1951 that the largest such root is a convex function of x, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize Gårding’s result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.

Extreme Points of the Vandermonde Determinant on Surfaces Implicitly Determined by a Univariate Polynomial

2020 ◽  
pp. 791-818
Author(s):  
Asaph Keikara Muhumuza ◽  
Karl Lundengård ◽  
Jonas Österberg ◽  
Sergei Silvestrov ◽  
John Magero Mango ◽  
...  
Keyword(s):  
Extreme Points ◽  
Vandermonde Determinant ◽  
Univariate Polynomial

Polynomial Solutions to the Displacement Analysis of 10-Link 1-DOF Mechanisms

1998 ◽  
Author(s):  
A. K. Dhingra ◽  
A. N. Almadi ◽  
D. Kohli
Keyword(s):  
Closed Form ◽  
Solution Process ◽  
Input Output ◽  
Analysis Problem ◽  
Displacement Analysis ◽  
Kinematic Chains ◽  
Polynomial Solutions ◽  
Elimination Procedure ◽  
Half Angle ◽  
Univariate Polynomial

Abstract This paper presents closed-form polynomial solutions to the displacement analysis problem of planar 10-link mechanisms with 1 degree-of-freedom (DOF). Using the successive elimination procedure presented herein, the input-output (I/O) polynomials as well as the number of assembly configurations for five mechanisms resulting from two 10-link kinematic chains are presented. It is shown that the displacement analysis problems for all five mechanisms can be reduced to a univariate polynomial devoid of any extraneous roots. This univariate polynomial corresponds to the I/O polynomial of the mechanism. In addition, one of the examples also illustrates how trigonometric manipulations in conjunction with tangent half-angle substitutions can lead to non-trivial extraneous roots in the solution process. Theoretical conditions for identifying and eliminating these extraneous roots are also presented.

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