Definition of principal term of polynomials

Question

In "Introduction to functional differential equations" by Hale and Lunel (1993), I found the following definition of principal term.

Let $P(z,w)$ be a polynomial of the form

\begin{equation} P(z,w)=\sum_{m=0}^{r}\sum_{n=0}^{s} a_{mn} z^m w^n \; . \end{equation}

We call $a_{rs}z^rw^s$ the principal term of the polynomial if $a_{rs}\not=0$ and, if, for each other term $a_{mn}z^mw^n$ with $a_{mn}\not=0$, we have either $ r > m$, $s > n$ or $r = m$, $s>n$, or $r > m$, $s = n$.

I honestly do not understand this definition; if we write the polynomial as above with the assumption that $a_{rs}\not=0$, doesn't this means the principal term is exactly $a_{rs}$?

Answer 1

The polynomial $P(z,w)=z+w$ does not have a principal term; to write it in the form \begin{equation} P(z,w)=\sum_{m=0}^{r}\sum_{n=0}^{s} a_{mn} z^m w^n \;, \end{equation} we must have $r,s\geq1$. The only nonzero coefficients are $a_{10}=a_{01}=1$, so in particular $a_{rs}=0$. This means the polynomial has no principal term.

And indeed the definition is needlessly roundabout. Given a polynomial \begin{equation}\tag{1} P(z,w)=\sum_{m=0}^{r}\sum_{n=0}^{s} a_{mn} z^m w^n \; , \end{equation} it is immediate that for every coefficient $a_{mn}$ of $P(z,w)$ we have $0\leq m\leq r$ and $0\leq n\leq s$. So for each coefficient other than $a_{rs}$ indeed the condition

...either $r>m$, $s>n$ or $r=m$, $s>n$, or $r>m$, $s=n$.

is satisfied. Hence the latter part of the definition is redundant.

What the definition seems to aim at, is the fact that the representation of the form $(1)$ is not unique. Indeed one might as well write $$P(z,w)=\sum_{m=0}^{r+1}\sum_{n=0}^{s+1} a_{mn} z^m w^n \; ,$$ and define $a_{mn}=0$ if $m=r+1$ or $n=s+1$ or both. The definition fails, however, by incorrect double usage of the variables $r$ and $s$. This can be fixed as follows:

Le t $P(z,w)$ be a polynomial of the form \begin{equation} P(z,w)=\sum_{m=0}^{r}\sum_{n=0}^{s} a_{mn} z^m w^n \; . \end{equation} We call $a_{uv}z^uw^v$ the principal term of the polynomial if $a_{uv}\neq0$, and for each other coefficient $a_{mn}$ with $a_{mn}\neq0$ we have $u\geq m$ and $v\geq n$.

Alternatively, and slightly less concretely, one might define it as follows:

A polynomial $P(z,w)$ is said to have principal term $a_{rs}z^rw^s$ if it has a representation of the form \begin{equation} P(z,w)=\sum_{m=0}^{r}\sum_{n=0}^{s} a_{mn} z^m w^n \;, \end{equation} with $a_{rs}\neq0$.

Answer 2

I think I have got the point now.

The fact is that the representation of a polynomial in the form

\begin{equation} P(z,w)=\sum_{m=0}^{r}\sum_{n=0}^{s} a_{mn} z^m w^n \; . \end{equation}

is not unique. For instance, we could write also

\begin{equation} P(z,w)=\sum_{m=0}^{r+1}\sum_{n=0}^{s+1} a_{mn} z^m w^n \; . \end{equation}

by setting to zero the new coefficients.

So, the correct definition is that the principal term of a polynomial is the term $a_{ij}z^iw^j$ if $a_{ij}\not=0$ and, for any other term $a_{mn}\not=0$, either $i>m$, $j>=n$ or $i=m$, $j>n$.

However, if we agreed on "the minimal representation" of a polynomial, I think we could just define the principal term as $a_{rs}z^rw^s$ if $a_{rs}\not=0$.