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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 tex2html_wrap_inline258 where tex2html_wrap_inline260. 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 rootsgif. Under these conditions, one could write tex2html_wrap_inline266 where tex2html_wrap_inline268 is the tex2html_wrap_inline270 root. Hence the derivative is given by tex2html_wrap_inline272
Notice that for each tex2html_wrap_inline278 we have
Now, provided tex2html_wrap_inline282, we have

Now let tex2html_wrap_inline288. This means
which gives infinite relations between coefficients and roots. Take for example tex2html_wrap_inline292

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 tex2html_wrap_inline298 we have
hence we can isolate the roots and solve the equation.

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