Of differential algebra, Gian-Carlo Rota wrote:
No elementary presentation of this beautiful subject has ever been attempted, to the best of my knowledge; Cohen’s book of the twenties is the closest, and it is still eagerly (and secretly) read today. Let me stick my neck out and propose that two results of differential algebra might be appreciated even by students in an elementary course. I will state one by way of end of this already long lesson, and reserve the second one for the next lesson. I have always felt excited when telling the students that even though there is no formula for the general solution of a second order linear differential equation, there is nevertheless an explicit formula for the Wronskian of two solutions. The Wronskian allows one to find a second solution if one solution is known (by the way, this is a point on which you will find several beautiful examples in Boole’s text). But there is a more fundamental fact, which I will state in a mathematical form that needs to be bowdlerized if we ever decide to try it out on an elementary class. It states that every differential polynomial in the two solutions of a second order linear differential equation which is independent of the choice of a basis of solutions equals a polynomial in the Wronskian and in the coefficients of the differential equation (this is the differential equations analogue of the fundamental theorem on symmetric functions, but keep it quiet).
"Keep it quiet"? This makes me think this was not a consensus among mathematicians, and possibly was not widely known.
How widely known is this proposition?
Among those who work in this area, is that proposition generally regarded as a "differential equations analogue of the fundamental theorem on symmetric functions"?
