0
$\begingroup$

Let $P(x_1,\ldots,x_n)\in \mathbb{C}[x_1,\ldots,x_n]$ a polynomial in complex $n$-variables. There is a method to find out if this polynomial can be written as $$P(x_1,\ldots,x_n)=m(x_1)\ldots m(x_n)$$ with $m(x)$ a polynomial in one variable?

$\endgroup$
1
  • $\begingroup$ Observing $P(x,y)=x^2+y^2=(x+iy)(x-iy)$, we found out that the claim is not valid always. $\endgroup$
    – Qurultay
    Jan 20 at 13:58
0
$\begingroup$

The polynomial has the seprated form $$P(x_1,\dots,x_n) = p_1(x_1)\cdots p_n(x_n), $$ if and only if the PDE $$P^{n-1}\frac{\partial^n}{\partial x_1 \cdots \partial x_n} P = \frac{\partial P}{\partial x_1} \cdots \frac{\partial P}{\partial x_n} $$ holds. Conditional on that fact, you may also check whether the polynomial is invariant under all permutations in $S_n$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.