How Can f and g Be Expressed Using Elementary Symmetric Polynomials?

[math_grl]

Let $$f, g \in \mathbb{Z}[x, y, z]$$ be given as follows: $$f = x^8 + y^8 + z^6$$ and $$g = x^3 +y^3 + z^3$$. Express if possible $$f$$ and $$g$$ as a polynomial in elementary symmetric polynomials in $$x, y, z$$.

Professor claims there is an algorithm we were supposed to know for this question on the midterm. I missed it. Any ideas?

 

[Eynstone]

g can be expressed as required using Newton's formula ; f is not even symmetric.

 

[math_grl]

By Newton's formula,
$$g = (\sigma_1^2 - 2\sigma_2)\sigma_1 - \sigma_1 \sigma_2 + 3\sigma_3 = \sigma_1^2 - 3\sigma_1 \sigma_2 + 3\sigma_3$$

where the $$\sigma_i$$'s are the elementary symmetric polynomials?

just trying to verify that I did it right?

and f is not of the form $$\sum^n_{i=1} x_i^k$$ for some $$k \in \mathbb{N}$$ so we can't use Newton's formula...but I was wondering one would know precisely that it's not possible to express it in terms of the e.s.p.'s?

 

[Eynstone]

and f is not of the form $$\sum^n_{i=1} x_i^k$$ for some $$k \in \mathbb{N}$$ so we can't use Newton's formula...but I was wondering one would know precisely that it's not possible to express it in terms of the e.s.p.'s?

Suppose that f = P(s1,s2,...) is expressible in terms of the e.s.p.'s. P won't change on switching y & z ; f will. A contradiction.

 

[blue_raver22]

I would say that the algorithm the professor is referring to is likely the Vieta's formulas for symmetric polynomials. These formulas allow us to express a polynomial in terms of its roots, which are the solutions to the equation when the polynomial is set equal to 0. In this case, the roots of f and g would be the values of x, y, and z that make the polynomials equal to 0.

Using Vieta's formulas, we can express f and g as polynomials in the elementary symmetric polynomials in x, y, and z. The elementary symmetric polynomials are defined as the sums of all possible products of the roots taken k at a time, where k ranges from 1 to the total number of roots. In this case, there are three roots (x, y, and z), so we would have three elementary symmetric polynomials: x + y + z, xy + xz + yz, and xyz.

To express f and g in terms of these elementary symmetric polynomials, we would first need to find the roots of the polynomials. This can be done by setting each polynomial equal to 0 and using techniques such as factoring or the quadratic formula to solve for the roots.

Once we have the roots, we can plug them into the Vieta's formulas to express f and g in terms of the elementary symmetric polynomials. For example, the Vieta's formulas for f would be:

x^8 + y^8 + z^6 = (x + y + z)^8 - 8(x + y + z)^6(xy + xz + yz) + 28(x + y + z)^4(xyz) - 56(x + y + z)^2(xyz)^2 + 70(xyz)^3

Using the roots of f, we can plug in the values for x, y, and z to find the coefficients of the elementary symmetric polynomials and fully express f in terms of them. The same process can be done for g.

In conclusion, the algorithm the professor is referring to is most likely Vieta's formulas for symmetric polynomials. By using these formulas, we can express f and g in terms of the elementary symmetric polynomials, which allows for a simpler and more general representation of the polynomials.