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The symmetric polynomials

Albert Girard (1590-1633), a flemish mathematician, published in 1629, in Amsterdam, a book called Invention nouvelle en l'algèbre, in which clear relations between roots and coefficients of polynomials were stated for the first time. Let where . Girard had the brilliant idea of postulating that such a polynomial had to have n roots; whenever a polynomial had less than n real roots, one could extend the set of the roots to have order n by adjoining complex roots. Under these conditions, one could write where is the root. Hence the derivative is given by

hence

Notice that for each we have

Now, provided , we have

Now let . This means

which gives infinite relations between coefficients and roots. Take for example

Although it is possible to derive infinitely many such relations, to isolate one single root one needs at least a difference between some powers of the roots. In the case of the quadratic we have

hence we can isolate the roots and solve the equation.

Leo Liberti
Thu Feb 26 17:04:11 CET 1998