A Multiplication Technique for the Factorization of Bivariate Quaternionic Polynomials

We consider bivariate polynomials over the skew field of quaternions, where the indeterminates commute with all coefficients and with each other. We analyze existence of univariate factorizations, that is, factorizations with univariate linear factors. A necessary condition for existence of univariate factorizations is factorization of the norm polynomial into a product of univariate polynomials. This condition is, however, not sufficient. Our central result states that univariate factorizations exist after multiplication with a suitable univariate real polynomial as long as the necessary factorization condition is fulfilled. We present an algorithm for computing this real polynomial and a corresponding univariate factorization. If a univariate factorization of the original polynomial exists, a suitable input of the algorithm produces a constant multiplication factor, thus giving an a posteriori condition for existence of univariate factorizations. Some factorizations obtained in this way are of interest in mechanism science. We present an example of a curious closed-loop mechanism with eight revolute joints.

The W. Thurston algorithm applied to real polynomial maps

This note will describe an effective procedure for constructing critically finite real polynomial maps with specified combinatorics.

Topological Properties of the Immediate Basins of Attraction for the Secant Method

We study the discrete dynamical system defined on a subset of $$R^2$$ R 2 given by the iterates of the secant method applied to a real polynomial p. Each simple real root $$\alpha $$ α of p has associated its basin of attraction $${\mathcal {A}}(\alpha )$$ A ( α ) formed by the set of points converging towards the fixed point $$(\alpha ,\alpha )$$ ( α , α ) of S. We denote by $${\mathcal {A}}^*(\alpha )$$ A ∗ ( α ) its immediate basin of attraction, that is, the connected component of $${\mathcal {A}}(\alpha )$$ A ( α ) which contains $$(\alpha ,\alpha )$$ ( α , α ) . We focus on some topological properties of $${\mathcal {A}}^*(\alpha )$$ A ∗ ( α ) , when $$\alpha $$ α is an internal real root of p. More precisely, we show the existence of a 4-cycle in $$\partial {\mathcal {A}}^*(\alpha )$$ ∂ A ∗ ( α ) and we give conditions on p to guarantee the simple connectivity of $${\mathcal {A}}^*(\alpha )$$ A ∗ ( α ) .

An Algorithm for the Factorization of Split Quaternion Polynomials

We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split quaternion polynomials can still be factorized and we describe how to find these real polynomials. Split quaternion polynomials describe rational motions in the hyperbolic plane. Factorization with linear factors corresponds to the decomposition of the rational motion into hyperbolic rotations. Since multiplication with a real polynomial does not change the motion, this decomposition is always possible. Some of our ideas can be transferred to the factorization theory of motion polynomials. These are polynomials over the dual quaternions with real norm polynomial and they describe rational motions in Euclidean kinematics. We transfer techniques developed for split quaternions to compute new factorizations of certain dual quaternion polynomials.

Factorization of Dual Quaternion Polynomials Without Study’s Condition

In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real (“motion polynomials”), we only require the absence of real polynomial factors in the primal part and factorizability of the norm polynomial over the dual numbers into monic quadratic factors. This obviously necessary condition is also sufficient for existence of factorizations. We present an algorithm to compute factorizations of these polynomials and use it for new constructions of mechanisms which cannot be obtained by existing factorization algorithms for motion polynomials. While they produce mechanisms with rotational or translational joints, our approach yields mechanisms consisting of “vertical Darboux joints”. They exhibit mechanical deficiencies so that we explore ways to replace them by cylindrical joints while keeping the overall mechanism sufficiently constrained.

STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING

Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss's first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

Level lines of a polynomial in the plane

We propose a method for computing the position of all level lines of a real polynomial in the real plane. To do this, it is necessary to compute its critical points and critical curves, and then to compute critical values of the polynomial (there are finite number of them). Now finite number of critical levels and one representative of noncritical level corresponding to a value between two neighboring critical ones enough to compute. We propose a scheme for computing level lines based on polynomial computer algebra algorithms: Gröbner bases, primary ideal decomposition. Software for these computations are pointed out. Nontrivial examples are considered.