# Name of operation on polynomials

1. Given 2 polynomials $p(x)=\prod_{i=1}^{r}(x-r_i), q(x)=\prod_{j=1}^{s}(x-r'_j)$, what is the name of the operation that constructs $f\#g(x) = \prod_{i=1}^{r}_{j=1}^{s} (x-r_i r'_j)$ from $p$ and $q$?

2. Let's say $p, q \in \mathbb{Z}[x]$. Is there an algorithm for computing the monic polynomial in $\mathbb{Z}[x]$ of least degree such that it vanishes on the set $\{ r_i r'_j \}_{i,j}$? I know that $p\# q$ has integral coefficients (by symmetric polynomials arguments), but I have examples where some elements occur more than once in the multiset $\{r_i r'_j \}$, and I guess it may effect the degree of the polynomial I seek.

The case $p=q$ is of big interest to me.

Motivation: given 2 linear recurrence (with integer coefficients) sequences $\{ a_n\}, \{b_n \}$, their Hadamarad product $\{a_n b_n \}$ also satisfies a linear recurrence with integer coefficients. Specifically, if $a_n$ satisfies a linear recurrence with characteristic polynomial $p(x)$ and $b_n$ with $q(x)$, then $\{a_n b_n \}$ satisfies a linear recurrence with characteristic polynomial $p \# q$ - but it may not be the minimal one, and that's why I am investigating.

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Re: 2, why not just divide $p\#q$ by the greatest common divisor of $p\#q$ and $(p\#q)'$? Multiple root elimination –  user31373 Jun 19 '12 at 16:42

1. I don't know if there is a widely accepted name, but I would call it the Kronecker product. The reason is that if $M, N$ are two matrices with eigenvalues $m_i, n_i$ then their Kronecker product $M \otimes N$ has eigenvalues $m_i n_j$.
2. Let $M, N$ be the companion matrices of your polynomials. Compute the characteristic polynomial of $M \otimes N$, then divide it by the gcd with its formal derivative as Leonid says in the comments. This is divisible by the polynomial you want and from here I think you just have to factor it.