 
Summary: Vincent's Theorem from a modern point of view
Alberto Alesina  Massimo Galuzzi
June 9th
1998.
1 Introduction
In this paper, after having summarized the main results we obtained in [2], we
suggest some ideas which may lead to future developments.
The reader may wonder at the very particular nature of our subject, and
whether it is inconsistent with the generality of category theory; but as empha
sized in [6] the peculiar nature of mathematics resides exactly in the force it
gains by contrasting general ideas to `facts' in a never ending dialectics.
AndrŽe Weil loved to quote Euler's maxim: "nihil est in numerico quod non
est in algebraico".
In fact, even the most trivial numerical identity may be the starting point for
a deep understanding of the mathematical structure upon which it may depend
in a subtle and unforeseeable way. On the other hand, no abstract mathematical
structure is meaningful if it isn't able to generate concrete and particular results.
Vincent's theorem originally appeared as a sort of complement to Lagrange's
method to approximate the roots of algebraic equations via continued fractions.
We described in great detail this aspect of the theorem in [2]. In this paper we
