Symmetric polynomials for Putnam

Question

I was taking a look at Titu Andreescu's "Putnam and Beyond" book and came across a polynomial issue that involved the knowledge of unusual symmetrical polynomials. The places where I studied the subject address at a more basic level. Where can I find something about symmetric polynomials similar to the one shown?

Answer 1

See the section relation with roots of a monic univariate polynomial of "symmetric polynomials". Apply to $y^3+y^2-2y-1=0$, and its roots $X^3,Y^3$ and $Z^3$.

Viete's formulas give relations between the roots of a polynomial and its coefficients. In particular, the coefficients are the elementary symmetric polynomials in the roots.

That is about all that is used in the solution.

The two equations with symmetric polynomials are easy to verify. Those specific symmetric polynomials (cleverly) enable one to exploit the information gotten from Viete's formulas, and solve the problem. We know the left hand sides. And the right hand sides involve $X+Y+Z$, which is what we want.

The first is $(X+Y+Z)^3$, and the second $(XY+YZ+XZ)^3$, both using the pattern $(a+b+c)^3-(a^3+b^3+c^3)=3(a+b+c)(ab+bc+ac)-3abc$.

This pattern is also an instance of the fundamental theorem on symmetric polynomials, since $a^3+b^3+c^3$ is here represented by a polynomial in the elementary symmetric polynomials.