Amongst the cool breeze, under shadows dancing around neon lights, city folk hurry home - to their cozy bedrooms while the gentle rain hums on the periphery of all concrete. The purple sky ushers the chronicles of lore unheard of; as if even the supreme had an epiphany about the vastness of this world, about the idiosyncratic sequences that cluster at all works of valour that are unrecognized. Senn takes a quick nap while people are getting ready around him; the session shall commence in half an hour. He dreams of deserts and black holes and weddings, which eventually converge into a wet, fertile rainforest owing to the nice petrichor of the outside world. Slowly the buzz of insects come to a standstill, the forest stops prancing in merry and the canopy opens a narrow path for sunlight to rush down on a small struggling sprout, which spreads its arms joy never seen before.

Thirty minutes pass. Senn is back at his desk, getting things ready. He has some idea how to work things out, but the most of it will have to be picked up on the run. Evenings that scream freedom, yet not from the confines of your house - could mean movies, coition or math.

“Greetings everyone! I hope I don’t seem alien or queer in faith and family to any of you out there. As you might have been informed of, Pura is on an extended leave for her own sake - which is why I am continuing her work. She is not gone forever, and should ideally be back in a few months. Another resolution could be to conduct the session from somewhere more cozy, for which we shall debate. But let’s begin with our session; it might be a fun trip through something we all did as high school students - monomials, binomials, trinomials … Polynomials!

To begin we shall understand what a polynomial is in a pretty lax manner. Suppose you have the uni-variable set $X={1,x,x^2,x^3,\ldots}$. Take any finite subset of this set, say $X\supset S={x^{r_1},x^{r_2},\ldots,x^{r_l}}$, where without loss of generality, you can assume the order ${r_l}>r_{l-1}>\ldots>r_2>r_1$ since the original set has an implicit order on the indices.

Then a linear combination of the elements of this set, say

with $k_i\in\mathbb{R}$ and $k_l\neq 0$ is a polynomial of degree $r_l$ with real coefficients. In actuality, you can draw the coefficients from any field $F$. Also, keep in mind that $x$ itself is a variable, i.e. it can be defined over any field (even rings, if you know them). For now, we will only look at real valued polynomials with real coefficients.

In general, you can define a polynomial of degree $n\in\mathbb{N}_0$ by the sum

where the coefficient of the highest order term is non-zero, i.e. $a_n\neq 0$.

I know how many of you might have much to say. Polynomials can be understood from a very elementary level, and so there are hundreds of easily comprehendible results out there. I shall avoid dipping in populated waters, since they might be polluted waters! Let’s try prying into some less lay, more play territory.

Consider a binomial $(1+x)$ exponentiated by a natural number $n$. Then we define

Clearly $p(x)$ is a polynomial of degree $n$, and by the binomial expansion formula, you can check it term by term as

where the $r^\text{th}$ term is $T_r = \binom{n}{r}x^{n-r}$.

Given any polynomial one can vary, often a question comes in mind - what is the largest term of the polynomial? By this, I don’t mean the degree; just the largest term if you were to put some arbitrary value of $x$. Obviously, your largest term will depend on $x$ in that case.

As $x$ goes off towards infinity and the coefficients $a_i$ are constants, we can safely assume that the leading order term ($x^n$ where $n$ is the degree) dominates the behaviour of the polynomial, i.e. we can write the approximation

However for $x$ comparable to (or much less than) the coefficients $a_i$, this might not be the case. The polynomial $p(x)=10^{370}x+2x^2$ is basically dominated by the $10^{370}x$ term for a long time (e.g. $p(10)=10^{371}+200\approx 10^{371}$).

(i) Given the polynomial $p_{\beta_n}(x)$ above, find the largest term $T_r$ for arbitrary $x$ and $n$.

(ii) Show that if you fix $n$ and increase $x$ indefinitely, then the largest term will eventually be the leading order term $T_0=\binom{n}{0}x^n=x^n$.

(iii) What is the minimum value of $x$ for which $x^n$ is the SOLE largest term?

“Were you able to get the results with proof?” Senn asked, as he turned a leaf of Pura’s scribble book. He was never fond of preparing for a session - that killed his natural dynamic and creative nature. Nevertheless, math wasn’t one where you could dump straight facts right out of the blue; and even if he could, they would be too obscure for anyone to remotely understand. Teaching math is a hard profession to walk into, but greatly fulfilling when it works.

“If you have studied binomial coefficients before, you might know of a nice result. It says that the middle order binomial coefficients $\binom{n}{r}$ (for $r=(n-1)/2$ and $(n+1)/2$) are equal to each other when $n$ is odd. This derives from the symmetry of the coefficients,

Clearly $\binom{n}{r}x^{n-r}$ is not symmetric as you vary $r$ - but do they show any adjacent term equality like in the normal coefficient case?

In $p_{\beta_n}(x)$ keep $x$ fixed as some natural number and vary $n$ from $1$ to $t$ for some large number $t$. Count the number of cases where there were TWO LARGEST TERMS of the series, i.e. $\operatorname{max}{T_i}=T_r=T_{r+1}$ for some $r$. Call such values of $n$ as bimodal.

(iv) Find the density of bimodal states as $n$ varies in ${1,2,\ldots,t}$, $t\gg 1$. Density is equal to the ratio of bimodal states to all states.

(v) Let $\rho_{n,x}(t)$ be the ratio of bimodal states. What value(s) of $t$ can you take so that $\rho_{n,x}(t)$ is independent of $t$?

Once you have chosen some value of $t$ such that $\rho_{n,x}(t)$ is independent of $t$, call this the standard bimodal density $\rho_{n,x}$.

(vi) Show that $\rho_{n,x}$ is a decreasing function of $x\in\mathbb{N}$.

Now that we have worked on a binomial generated polynomial, let’s go to some other niceties.

Let us talk of generating functions. They are ubiquitously used in various fields of math and science, and can vastly simplify non-trivialities. In an elementary sense, the generating function of a sequence $(a_0,a_1,a_2,\ldots)$ of reals is given by the infinite degree polynomial

A generating function can be written in a nice closed form if the infinite series converges. If the sequence is finite, then we can be more assured of its convergence.

Consider the function $f(x,t)=e^{xt-\frac{1}{2}t^2}$. This has a valid power series expansion for all real $x$ and $t$ (even complex, in that case) given by

which is a generating function of the sequence ${w_n(x)}$. It should not be too much work to see that we can get each term of the sequence through the relation

(vii) Show that $w_n(x)$ is a polynomial of degree $n$ on $x$.

(viii) Show that $w_n(x)$ is monic, i.e. the leading coefficient $a_n=1$.

Often terms in a generated sequence can be found out using a series of operations on the generating function. Note that different operations need not commute, so they are generally treated on a right to left manner, same when we compose functions or functions and operations. For example,

In case of $w_n(x)$ though, we can apply one operator many times to find out the term. Suppose we define an operation on the number $1$ $n$ times given as

where $A$ and $B$ are positive reals. It is easy to find out the first few iterations.

(ix) If we claim that $\mathcal{O}_n(x)=w_n(x)$, find the values of $A$ and $B$.

Given these values of $A$ and $B$, and the relation $\frac{d}{dx}w_n(x)=nw_{n-1}(x)$, we can try to formulate recurrence relations describing $w_n(x)$. Consider a Fibonacci-esque recurrence relation given by

(x) Find $\alpha(x)$ and $\beta(x)$. Reason as to why the recurrence relation is not linear.

“I believe these were not too trivial for anyone!” Senn exclaimed, closing the scribble book. “Also, it’d be great to know that many were introduced to new stuff today, things that aren’t already known. Indeed, polynomials do extend beyond their elementary treatment and can prop up in the most unexpected situations, such as on a chess board or MS Paint.”

The more you know, the more Oh,no!mials.

Submitting

Send in your solutions at maths.club@iiserkol.ac.in.