Euclidean remainders for generalized polynomials
The paper deals with methods for computations with real polynomials in generalized form, i.e. with polynomials written as linear combinations of the elements of an orthogonal basis \(\{p_ i(\lambda)\}\) for \({\mathbb{R}}[\lambda]\) satisfying the relations \(p_ 0(\lambda)=1\), \(p_ 1(\lambda)=\alpha_ 1\lambda +\beta_ 1\), and \(p_ i(\lambda)=(\alpha_ i\lambda +\beta_ i)p_{i-1}(\lambda)- \gamma_{i-1}p_{i-2}(\lambda)\) for \(i\geq 2\), where \(\alpha_ i>0\) and \(\gamma_ i\geq 0\). Namely, if \(a(\lambda)=p_ n(\lambda)+a_ 1p_{n- 1}(\lambda)+...+a_ np_ 0(\lambda)\) and \(b(\lambda)=b_ 0p_ m(\lambda)+b_ 1p_{m-1}(\lambda)+...+b_ mp_ 0(\lambda)\) with \(b_ 0\neq 0\) and \(m<n\), the author shows how to apply the Euclidean algorithm for finding gcd of a(\(\lambda)\) and b(\(\lambda)\) without ever resorting to the ordinary representation of any of the polynomials involved in terms of the power basis \(\{\) \(\lambda\) \(i\}\). For this, the author introduces the comrade matrix A of the polynomial a(\(\lambda)\) and then reduces the block matrix [b(A),I], where I is the n-by-n identity matrix of the same size as A, to a quasitriangular form by elementary row operations. As a byproduct, solutions to a set of linear congruences are obtained.
scientific article; zbMATH DE number 4068174
scientific article