About division by 1 Alain Lascoux

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 + c2z 2 + · · · . We shall moreover formally factorize it f(z) = σz(A ) := ∏ a∈A (1 − za)−1 = ∞ ∑ i=0 z Si(A ) , ∗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 ha l-0 06 22 72 8, v er si on 1 12 S ep 2 01 1 Author manuscript, published in "Electronic Journal of Combinatorics 8, 1 (2001) 10pp."