Maximum number of inflections of a polynomial division

Question

I've come across a quote from an economics paper and cannot give a proof of it. I would like to know if anyone can provide it or some reference.

The quote goes as follow: "The wage-curve is a ratio between a polynomial of the mth degree and one of the (m - 1)th degree in r. By a known property such rational functions admit up to (3m-6) points of inflexion. (Cf., e.g. Enriques[6])."

The reference provided by the author is a very old italian textbook. My italian is far from good enough to read an entire math book..

I don't know if it is needed, but I shall provide more information. Let w be the wage rate; r the profit rate; l a m-dimensional row vector; d a m-dimensional column vector; A a mxm matrix. All the elements in l, d and A are known non-negative real numbers. Then w and r will be related in the specific way:

$$w={det(\mathbf{I}-\mathbf{A}(1+r)) \over \mathbf{l}.adj(\mathbf{I}-\mathbf{A}(1+r)).\mathbf{d}}$$

It is clear enough that the numerator is a polynomial of mth degree in r and the denominator a (m-1)th degree in r. I would like to know the maximum number of inflections of this expression. I've tried taking the second derivative of it but still got nowhere. I still get a ratio between two (complicated) polynomials and cannot decide its degree and so the number of roots it has. I don't need to determine if it is an inflection or not. I guess it amounts to determining the degree of the resulting polynomial.

Thank you, Pedro.

Answer 1

Let $f = \frac{P}{Q}$ with $P$ of degree $m$ and $Q$ of degree $m-1$. By polynomial long division:

$$f(x) = a\,x + b + \frac{R(x)}{Q(x)}$$

where $R$ has degree $\le m-2$. Then:

$$ f'' = \left(\,\frac{R}{Q}\,\right)'' = \left(\,\frac{R'\,Q-R\,Q'}{Q^2}\,\right)' = \frac{(R''Q+RQ'-R'Q'-RQ'')Q - 2 (R'\,Q-R\,Q') Q'}{Q^3} $$

By inspection, the degree of the numerator is $\le (m-4) + (m-1) + (m-1) = 3m-6$.

Answer 2

Rather than $3m-6$ it appears that there could be $3m-4$ inflection points. This is rather simple so there is little room for error. Can someone point out an error?

Let $w(r)=\dfrac{P(r)}{Q(r)}$ where $P$ and $Q$ are polynomials in $r$ of degrees $m$ and $m-1$, respectively.

Then

\begin{equation} \dfrac{dw}{dr}=\dfrac{P^\prime(r)Q(r)-P(r)Q^\prime(r)}{Q^2(r)}=\dfrac{U(r)}{Q^2(r)} \end{equation}

where $U$ has degree $2m-2$.

\begin{eqnarray} \dfrac{d^2w}{dr^2}&=&\dfrac{U^\prime(r)Q^2(r)-2U(r)Q(r)Q^\prime(r)}{Q^4(r)}\\ &=&\dfrac{U^\prime(r)Q(r)-2U(r)Q^\prime(r)}{Q^3(r)} \end{eqnarray}

with numerator of degree $3m-4$