December 2024 – Dokuz Eylül University Faculty of Science Department of Mathematics

Fundamental Theorem of Symmetric Polynomials, Newton’s Identities and Discriminants

December 23 2024December 20 2024 by Engin Mermut

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.

Cardano’s Formula and Casus Irreducibilis

December 23 2024December 13 2024 by Engin Mermut

Çağdaş Çiğdemoğlu, Dokuz Eylül University. Date: 18th 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 start with a general monic cubic equation in the form

$x^3 + bx^2 + cx + d = 0,$

and transform it into the following form

$y^3 + py + q = 0,$

using a substitution. Then, we will construct the Cardano Formulas to find the roots of the equation. The roots are expressed as follows:

$y_1 = \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}},$

$y_2 = \omega \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \omega^2 \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}},$

$y_3 = \omega^2 \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \omega \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}},$

where $\omega = e^{i \frac{2\pi}{3}} = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$ is a primitive cube root of unity and where the above cube roots are one of the three complex cube roots whose product is $-\frac{p}{3}$ and these are fixed in the above formulas.

We will learn about the discriminant $\Delta$ of the above monic cubic polynomial, understand its significance, and examine how the roots change depending on the value of the discriminant.

For the monic cubic polynomial in the form $y^3 + py + q$ , the discriminant is expressed by:

$\Delta = -27q^2 - 4p^3 = (y_1-y_2)^2(y_1-y_3)^2(y_2-y_3)^2.$

For the general monic cubic polynomial $x^3 + bx^2 + cx + d$ , whose roots are $x_1, x_2, x_3$ , the discriminant is expressed by:

$\Delta = b^2c^2 + 18bcd - 4c^3 - 4b^3d - 27d^2 = (x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2.$

We will also discuss Casus Irreducibilis, which occurs when the coefficients $b, c, d$ (or $p, q$ ) are in a subfield of $F$ of $\mathbb{R}$ , the cubic polynomial is irreducible over $F$ (equivalently, the cubic equation has no roots in $F$ ) and when the discriminant is positive. In this case, the cubic equation has three distinct real roots that cannot be expressed using real radicals.

This seminar serves as an introduction to my project, which focuses on understanding the proof of Casus Irreducibilis using Galois Theory.

Cryptography and Mathematics

December 11 2024 by Celal Cem Sarıoğlu

This month’s career event theme is “Cryptography and Mathematics”. We are pleased to host a talk on this subject with Beste AKDOĞAN KÖSEMEN, a 2016 graduate of our department and currently pursuing her Ph.D. in Cryptography at METU Institute of Applied Mathematics. The event is open to all mathematics department students and anyone interested. (Non-mathematics department participants are kindly requested to contact the moderator for attendance.)

Speaker: Res. Asst. Beste AKDOĞAN KÖSEMEN (Ankara Yıldırım Beyazıt University, Mathematics Department & METU Institute of Applied Mathematics)
Topic: Mathematics and Cryptography
Date & Time: December 23, 2024, 12:00 PM
Venue: online.deu.edu.tr
Channel: DEUMatematikKARİYER
Moderator: Asst.Prof.Dr. Celal Cem SARIOĞLU