Help with operator $f(x^q)=\frac{1}{q+1}x^q$.

This question is somewhat related to this.

I am looking for an operator $f:\mathbb{R}[x]\to\mathbb{R}[x]$, that is, $f$ is an operator that maps polynomials in one variable to polynomials in one variable.

The requirement of $f$ is that it must map the monomial $x^q$ to the monomial $\frac{1}{q+1}x^q$, that is $f(x^q)=\frac{1}{q+1}x^q$. This operator should be independent of $q$.

I first thought on using integration, however I do not want this (see below).

My proposal is the following:

Let $P\in\mathbb R[x]$. We define $\phi:\mathbb R[x]\to\mathbb R[x]$ by $\phi(P)=\frac{d}{dx}(xP)$ (I think I can freely use $x$).

Next we define $f:\mathbb R[x]\to\mathbb R[x]$ by $f(P)=\frac{P^2}{\phi(P)}$. Note that in the particular case $P=x^q$ we do satisfy our requirement $$f(x^q)=\frac{x^{2q}}{(q+1)x^{q}}=\frac{1}{q+1}x^q.$$

--

Now, let us extend the previous idea to operators $f:\mathbb R[x_1,\ldots,x_n]\to\mathbb R[x_1,\ldots,x_n]$ .

Let $P\in \mathbb R[x_1,\ldots,x_n]$. Following the above stuff we define a family of maps $\phi_k:\mathbb R[x_1,\ldots,x_n]\to\mathbb R[x_1,\ldots,x_n]$ by $\phi_k(P)=\frac{d}{dx_k}(x_kP)$ (again I think I can freely use $x_k$, also $\phi_k(\cdot)=\phi(\cdot,k)$ with $k\in\mathbb N\backslash \left\{ 0 \right\}$).

Next, we define similarly $f_k(P)=\frac{P^2}{\phi_k(P)}$.

In the particular case that $P=x^Q=x_1^{q_1}\cdots x_n^{q_n}$, we have $$f_k(x^Q)=\frac{(x_1^{2q_1}\cdots x_k^{2q_k}\cdots x_n^{2q_n}) }{ (q_k+1)(x_1^{q_1}\cdots x_k^{q_k}\cdots x_n^{q_n}) }=\frac{1}{q_k+1}x^Q.$$

What do you think?, any criticism is appreciated. Specially: is it true that if I define an operator on $P$, I can use $x_k$?

If $x\in\mathbb{R}$ it is OK to define $f(x^q)=x\int x^q dx=\frac{1}{q+1}x^q+c$.

However if $x\in\mathbb{R}^n$ then we would have $f_k(x^q)=x_k\int x^q dx_k=\frac{1}{q+1}x^q+\phi(x_1,\cdots,x_{k-1},x_{k+1},\cdots,x_n)$, where $\phi$ is a function of all terms $x_j$ except $x_k$.

It seems to me that the operator you seek would be: $$f(x^p)=\frac{1}{x}\int_0^x t^p dt$$ this has the desired properties.

Note that this operator is actually independant of $p$, and can be written more generally, for some $g\in\Bbb R[x]$ $$f(g)=\frac{1}{x}\int_0^x g(t)dt$$

• This is indeed good for $g\in\mathbb R[x]$ as you say. What about $g\in\mathbb R[x_1,\ldots,x_n]$? ... is it possible to define $f_k(g)=\frac{1}{x_k}\int_0^{x_k}g(x_1,\ldots,x_{k-1},t,x_{k+1}\ldots,x_n)dt$ ? – PepeToro Jul 27 '14 at 17:35
• If we consider that a monomial would now be $g(x_1,...,x_n)=x_1^{p_1}...x_n^{p_n}$ the let $$f(g)=\frac{1}{x^n}\int_0^{x_1}...\int_0^{x_n}g(t_1,...,t_n)dt_1...dt_n$$ but this depends on what you consider a monomial to now be, in honesty ive never considered what a multi variable monomial would be – Ellya Jul 27 '14 at 17:48
• Just did a quick google, and what i have put for g is a multivariable monomial. – Ellya Jul 27 '14 at 17:51
• Thanks, I think this helps a lot. I'll accept the answer once I've checked everything carries on well in my project. – PepeToro Jul 27 '14 at 18:05

EDIT: I'm quite sure that I misunderstood the OP's intention. However, the comments to this answer may still be of interest and so have made this a wiki answer.

No such function exists. We want $f(x^q)=x^q/(q+1)$ for all $q$; consider in particular $q=6$. Then we require \begin{align} f(x^{6}) &=f((x^2)^3))=f((x^3)^2))\\ &=\frac{x^{6}}{7}=\frac{(x^{2})^3}{3}=\frac{(x^{3})^2}{4}\end{align} which cannot be true since $7\neq 4\neq 3$.

• Thanks. Do you think there is some way to require that the function "acts on $x^q$ and not in factorisations"? Or something like that? Does it makes a difference that I am not defining $f(P^q)=P/(q+1)$ but requiring that in the particular case that the argument is something "exactly like" $x^q$ I then obtain $x^q/(q+1)$ ? I set $f(P)=P^2/\phi(P)$, this is well defined right? – PepeToro Jul 27 '14 at 16:40
• @user58533: I suspect the problem is that you're wanting to work with $f(x)$ as part of a polynomial ring, rather than as a function unto itself. That is: you want to find an operator $\hat{f}$ which acts on the monomials $x^q$, not a function which receives numerical inputs. Somehow that needs to be clear at the level of your definition, and I'm not sure what the right way is. – Semiclassical Jul 27 '14 at 16:43
• that is correct, I am assuming/thinking of $f$ as an operator that receives polynomials, not numbers. – PepeToro Jul 27 '14 at 16:45
• @user58533: Probably it's better to think of $\mathbb{R}[x]$ explicitly as a vector space. In those terms, you want an operator that maps the basis vector $e_q$ to $(q+1)^{-1} e_q.$ But what we're wrestling with is a notational issue. – Semiclassical Jul 27 '14 at 16:46
• I need to think on this as I do not understand very well your argument. I thought I was sure I was working on "maps from the space of polynomials onto itself". But it seems I am doing something wrong. Thanks. – PepeToro Jul 27 '14 at 16:51