Symmetric polynomial algorithm?

In summary, the problem involves expressing the given polynomials f and g as a polynomial in elementary symmetric polynomials in x, y, and z. It is possible to express g in this form using Newton's formula, while f is not even symmetric and cannot be expressed in this form. This can be proven by considering that if f were expressible in terms of the e.s.p.'s, it would not change when switching y and z, leading to a contradiction.
  • #1
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Let [tex]f, g \in \mathbb{Z}[x, y, z][/tex] be given as follows: [tex]f = x^8 + y^8 + z^6[/tex] and [tex]g = x^3 +y^3 + z^3[/tex]. Express if possible [tex]f[/tex] and [tex]g[/tex] as a polynomial in elementary symmetric polynomials in [tex]x, y, z[/tex].

Professor claims there is an algorithm we were supposed to know for this question on the midterm. I missed it. Any ideas?
 
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  • #2
g can be expressed as required using Newton's formula ; f is not even symmetric.
 
  • #3
By Newton's formula,
[tex]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[/tex]

where the [tex]\sigma_i[/tex]'s are the elementary symmetric polynomials?

just trying to verify that I did it right?

and f is not of the form [tex]\sum^n_{i=1} x_i^k[/tex] for some [tex]k \in \mathbb{N}[/tex] 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?
 
  • #4
math_grl said:
and f is not of the form [tex]\sum^n_{i=1} x_i^k[/tex] for some [tex]k \in \mathbb{N}[/tex] 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.
 
  • #5


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.
 

1. What is a symmetric polynomial algorithm?

A symmetric polynomial algorithm is a mathematical algorithm used to manipulate symmetric polynomials, which are polynomials that remain unchanged when variables are permuted. These algorithms are used in a variety of fields, including computer science, physics, and engineering.

2. What are the main applications of symmetric polynomial algorithms?

Symmetric polynomial algorithms have many applications, including but not limited to: finding roots of polynomials, solving systems of linear equations, optimization problems, and image processing.

3. How do symmetric polynomial algorithms differ from other polynomial algorithms?

Unlike other polynomial algorithms, symmetric polynomial algorithms take advantage of the symmetries of the polynomial to reduce the complexity of computations. This allows for more efficient and accurate solutions to problems involving symmetric polynomials.

4. What are some common techniques used in symmetric polynomial algorithms?

Some common techniques used in symmetric polynomial algorithms include the use of generating functions, Newton's identities, and the method of Lagrange interpolation. These techniques help to simplify and solve problems involving symmetric polynomials.

5. Can symmetric polynomial algorithms be applied to non-symmetric polynomials?

Yes, symmetric polynomial algorithms can be applied to non-symmetric polynomials by first transforming the polynomial into a symmetric form. This can be done using techniques such as substitution or polynomial interpolation.

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