Theorems on Roots of Polynomials in Geometry, Topology, Complex Analysis and Functional Analysis

 Roots of Polynomials
Prof. V. Kannan
Dept. of Mathematics
SRM University-AP
23 February 2022

Abstract: "We discuss four classical theorems on roots of polynomials. They relate to four
different branches of Mathematics, namely, Geometry, Topology, Complex Analysis and
Functional Analysis. The theorems are stated below:
Theorem 1 (Lucas). Every root of the derivative of a polynomial lies within the convex
polygon formed by the roots of that polynomial.
Theorem 2: The roots of a polynomial vary continuously with its coefficients.
Theorem 3 (Cauchy): If all the coefficients of a polynomial lie in the unit disc D, then all
the roots lie in the disc 2D.
Theorem 4 (Bernstein): The norm of the linear operator D (namely the differentiation) on
(The space of polynomials of degree at most n on D) is exactly n."

Evaluation of Answer scripts - A nightmare

First time when I did it after my MSc at Andhra Loyola college for nearly 300 copies I catched fever.

Now, I had to do the same but luckily on electronic version but more or less similar size of 270 copies.

As I couldn't do much during Boston MIT trip, I had to start it at home after reaching.

Today , at last I could finish it off somehow.

 I have done good correction for only the first section.

And the rest of sections received inflated marks for simple contrinbutions which definitely a bad practice in the making of engineers for Future India.

Some of the copies are so sweet and most of them are blunders.

I don't understand what these just saved pupil from getting failed will do the technological fields.

Will they write proper codes for bank or MRI machines, heart pacemakers, will they build strong flyovers and skyscrappers ?Future of India is very gleam and despairing.

The day when AI is adopted to evaluate scripts the pass percentages of these future engineers will go down to fractions that we never have seen before.
 How badly these kids are motivated to do engineering in spite of the pressures from their parents and relatives.
Special mention: out of these 270 50% are done by our Teaching Assistant Alekhya single handedly.
And the rest were done by me and my sister Nagamani . She helped me with the task of entering marks while I was doing the marks allocation on a plane sheet of paper.

At last a great sigh of relief.

My bad experience with BIET.

I was unemployed for nearly one and half year after my 1.5 year ppost doctoral work.
Then after my marriage my sisters was happened.
Now, I went to join Bharath Institute of Technology at Ibrahimpatnam Hyderabad for an offer of 65k per month.

After submitting my certificates to BIET I joined .
First day was good. Everyone surprised at how come I with high profile of research ended up at BIET trap.
Slowly I started to understand how bad the place uis though the buildings are modern and polished but the minds of BIET managerial body is mere rowdism.

I used to travel to BIET everyday from Leela's apartment at jodimetla cross in the way to ghatkesar , to Ibrahimpatnam by first going up to uppal on Auto and then the Institute bus from there.

During the week , I got an interview letter from NIT Andhra for asst, prof ship in Mathematics.
In the same week I was informed that I am selected for SRMAP and I told them I will join by Sept 15th-2018. A fortnight ago, for which I already gave my interview at SRMAP on july 27th paneled by Prof. Vijaysekhar, Prof. Kannan, Prof. Balasubramanian Ramanchandran.

Despite my repeated mails about the status of interview at SRM I didn't get any reply . So I headed to BIET, Hyd for employment which I felt a good one to start with. But very soon I realized it is a graveyard when I tried to get back my certificates in the name of NIT Andhra interview.
They gave me the certificates after a big drama and aslo took 10000INR from me with nearly 2lakh worth cheques of SBi account, which I later executed with stop cheque order with  minimum amount of fee at my Gunadala Eluru road branch.

I made through the interview  and with that reason I resigned at BIET and left to home. BUt also mailed NIT Andhra that I can't join due to personal requirements at home. They replaced my candidacy with the next prospective.

On Aug 1st I joined SRMAP and it is a nice place to start with along with other reputed international faculty.
My Mathematics dept was chaired by Prof. Kannan with whom we have very closer conversations without any professional  barriers.

Later during the MITx trip I come to know about Prof. Vijaysekhar how friendly person he is at the same time how deeply he is motivated by the Math applications and Theoretical flavors.


During the interview I wa asked about my PhD research and as well as PDF research.
When I was asked the complexity of matrix determinant I made few hurries attempts, Alas , I endedup with the right answer. At the end I was asked another one complexity on product of two matrices which I guessed it wrongly but corrected by Prof. Kannan.

In the last semester
I taught Single variable calculus for CSE-B-F and ECE-A-C-2018-19.
Out of which the most attentive batch were ECE-A and next CSE-B. But CSE-B more calm and better performers.