# Two-part lecture series: Development of the Theory of Equations | Polynomials, Matrices, and Linear Recurrences over Galois Fields

By Sagnik Saha in Talk Series

March 29, 2022

After a long hiatus, it gives us immense pleasure to announce our next speaker in the ‘Two-part Lecture Series’, jointly hosted by CMIT and the School of Mathematics, IISER Thiruvananthapuram.

Kindly note that the two-part lectures will be delivered in a manner to cover the necessary basics in the first part of the talk, while the second part builds upon the same and continues.

The following two-part lecture will be delivered by Prof Sudhir Ghorpade, Indian Institute of Technology, Bombay, on March 30, 2022, at 5:00 PM and on April 1, 2022, at 11:30 AM. Please find the talk details below.

**Speaker**: Prof. Sudhir Ghorpade, Indian Institute of Technology (IIT) Bombay.

**Part I** (Organised and hosted by CMIT)

Title: Development of the Theory of Equations from Shreedharacharya to Galois.

Venue and time: Google Meet (online) | 05:00 PM (GMT +5:30), Wednesday, March 30, 2022.

Abstract: Beginning with the classical formula (that can be ascribed to Shreedharacharya) for the roots of a quadratic equation and the related notion of the discriminant of a quadratic polynomial, I will trace the development of some of the major ideas in the theory of equations culminating in the phenomenal work of Galois. This will be laced with some amusing stories, mainly from the medieval and renaissance period, of some remarkable persons and events that played a major role in the development of the subject. Along the way, we will try to answer the kind of basic questions which smart high‐school or undergraduate students of mathematics are likely to ask themselves or their teachers, but often in vain; for example, is there a formula for the roots of a cubic equation, a quartic equation, and in general, an equation of any degree? What is (or what should be) the discriminant of a general polynomial of any degree? We will also see how these questions led to some of the modern aspects of algebra.

Chair for the Session: Chaitanya Joglekar, BSMS 4th Year, School of Mathematics, IISER Thiruvananthapuram.

**Part II** (Organised and hosted by SoM, IISER Thiruvananthapuram)

Title: Polynomials, Matrices, and Linear Recurrences over Galois Fields.

Venue and time: Google Meet (online) | 11:30 PM (GMT +5:30), Friday, April 01, 2022.

Abstract: Let $F$ be a Galois field, that is, a field whose cardinality is finite. We will discuss a variety of questions concerning the enumeration of certain classes of polynomials with coefficients in $F$, matrices with entries $F$, and homogeneous linear recurrence sequences of elements of $F$. Here is a sample of the questions, in their simplest form, that we will consider: (1) What is the number of nonsingular matrices of a given size whose order is maximum in the corresponding general linear group? (2) What is the probability that two polynomials of positive degrees are relatively prime? (3) What is the number of Hankel matrices of a given rank? (4) What is the number of primitive linear recurrence sequences of a given order?

We will attempt to outline some results, both old and new, and some conjectures concerning these questions and their extensions and generalisations. Connection of these with combinatorics, cryptography, and algebra may also be indicated. Parts of this talk are a joint work with Sartaj Ul Hasan and Meena Kumari, with Samrith Ram, and with Mario Garcia Armas and Samrith Ram.

Chair for the Session: Dr. Geetha T, School of Mathematics, IISER Thiruvananthapuram.

**Registration**: Students of IISER Thiruvananthapuram do not require registration for the talks. Participants from outside IISER Thiruvananthapuram can register here: link

All are cordially invited to attend the talks over Google Meet. Please join us in large numbers, we hope to see you all there! For any further queries, mail us at mathsclub@iisertvm.ac.in.