A Bivariate Analogue to the Composed Product of Polynomials

摘要

The concept of a composed product for univariate polynomials has rnbeen explored extensively by Brawley, Brown, Carlitz, Gao,Mills, et al. Starting with these fundamental ideas andutilizing fractional power series representation(in particular, the Puiseux expansion) of bivariate polynomials, we generalize the univariate results. We define a bivariate composed sum,composed multiplication,and composed product (based on function composition). Further, we investigate the algebraic structure of certain classes of bivariate polynomials under these operations. We also generalize a result of Brawley and Carlitz concerningthe decomposition of polynomials into irreducibles.