Fundamental Theorem of Symmetric Polynomials, Newton’s Identities and Discriminants – Dokuz Eylül University Faculty of Science Department of Mathematics

Mustafa Eren Taşlı, Dokuz Eylül University. Date: 26th of December, 2024, Wednesday, Time: 15.00 – 16.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: We will define symmetric polynomials and the elementary symmetric polynomials in $n$ indeterminates over a field $F$ . The elementary symmetric polynomials in the indeterminates $x_1, x_2, \ldots, x_n$ are as follows:

$\begin{align*} \sigma_1=&x_1+x_2+\ldots+x_n \\ \sigma_2=&\sum_{1 \le i < j \le n} x_ix_j \\ \vdots\\ \sigma_n =&x_1 x_2 \ldots x_n \end{align*}$

The Fundamental Theorem of Symmetric Polynomials states that any symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials, that is:

Theorem. Let $f(x_1, x_2, \ldots, x_n)$ be a symmetric polynomial in the $n$ indeterminates $x_1, x_2, \ldots, x_n$ over a field $F$ . Then, there exists a polynomial $g(y_1, y_2, \ldots, y_n)$ in the $n$ indeterminates $y_1, y_2, \ldots, y_n$ such that

$f(x_1, x_2, \ldots, x_n) = g(\sigma_1, \sigma_2, \ldots, \sigma_n),$

where $\sigma_1, \sigma_2, \ldots, \sigma_n$ are the above elementary symmetric polynomials of the $n$ indeterminates $x_1, x_2, \ldots, x_n$ . Moreover, the polynomial $g(y_1, y_2, \ldots, y_n)$ is uniquely determined.

We will prove this theorem using the graded lexicographic order for multivariable polynomials.

Using the recurrence relation from the Newton Identities, we will learn how to express the sum of powers of the indeterminates, that is, the polyomials

$s_k = x_1^k + x_2^k + \ldots + x_n^k$

for a positive integer $k$ , as polynomials in terms of the elementary symmetric polynomials. We will reinforce this understanding with examples.

The discriminant in the indeterminates $x_1, x_2, \ldots, x_n$ over the field $F$ is given by:

$\Delta = \prod_{1 \leq i < j \leq n} (x_i - x_j)^2 \in F[x_1, \dots, x_n].$

The discriminant is a symmetric polynomial, and we will express it in terms of the elementary symmetric polynomials using determinants.

This seminar, as part of my graduation project titled Symmetric Polynomials, Newton’s Identities, Discriminants, and Resultants, serves as an introduction to a method for calculating the discriminant ( $\Delta$ ) of an $n$ -th degree polynomial without finding its roots.

Mustafa-Eren-Taşlı_Fundamental-Theorem-of-Symmetric-Polynomials_26-December-2024-Thursday_time-15-00_Classroom-B254