Invariance properties, and characterization of the greatest common divisor of a set of polynomials

Abstract

On m-sets of polynomials from R[s] of degree d, 𝓅m. d the notion of extended-R-equivalence (&E-R-E )ℯand the shifting operation are defined. The equivalence class of 𝓅md under ℯℯ( 𝓅m, d)is characterized by a complete invariant and a canonical form; it is shown that the greatest common divisor ℯ (g.c.d.) of 𝓅m d is an invariant of ℯ(𝓅m, d )For proper sets 𝓅m, d it is also shown that the g.c.d. is invariant under the combined action of E-R-E transformations and shifting operations. Finally, a systematic procedure for the computation of the g.c.d. of 𝓅mdis given. This procedure is based on the use of elementary row operations on real matrices and shifting operations.