Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: About division by 1. | About division by 1 Alain Lascoux IGM 77454 Marne La Vallee Cedex 2 FRANCE Submitted May 1 2001 Accepted September 19 2001. MR Subject Classifications 05E05 11A05 Abstract The Euclidean division of two formal series in one variable produces a sequence of series that we obtain explicitly remarking that the case where one of the two initial series is 1 is sufficiently generic. As an application we define a Wronskian of symmetric functions. The Euclidean division of two polynomials P z Q z in one variable z of consecutive degrees produces a sequence of linear factors the successive quotients and a sequence of successive remainders both families being symmetric functions in the roots of P and Q separately. Euclidean division can also be applied to formal series in z but it never stops in the generic case leaving time enough to observe the law of the coefficients appearing in the process. Moreover since the quotient of two formal series is also a formal series it does not make much difference if we suppose that one of the two initial series is 1. This renders the division of series simpler than that of polynomials in fact the latter could be obtained from the former. By formal series we mean a unitary series f z 1 C1Z C2Z2 . We shall moreover formally factorize it f z ơz Y Â 1 - za -1 X z S-U--- __ i 0 Written during the conference Applications of the Macdonald Polynomials at the Newton Institute in April 2001. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 N8 1 the alphabet A. being supposed to be an infinite set of indeterminates or of complex numbers the coefficients Si A being called the complete functions in A. Given two series dividing f-i z ơz A by f0 z ơz B means finding the unique coefficients a p such that p A - 1 az ơz B 1 z 2 1 p is a unitary series fi z ơz C . Dividing in turn f0 z by fi z one obtains f2 z and iterating one gets from the initial pair f-i f0 an infinite sequence of series
