Revision history [back]

The problem is quite vague as there exist lots of polynomials satisfying the conditions. Here is just one way to get some of them.

First, decide what factors of $AB(A+B)$ divide what polynomials. For example, if we fix that $A\mid P1$, $B\mid P2$, and $(A+B)\mid P3$, then we can generate them as follows:

1. Take any polynomial $Q$ and set $P3 = (A+B)Q$. Initially set both $P1$ and $P2$ equal to $0$.
2. For each term $c A^i B^j$ in $P3$:
3. (i) if $i=0$, add $m$ to $R2$
4. (ii) if $j=0$, add $m$ to $R1$
5. (iii) otherwise partition $c = c_1 + c_2$ (which can be done in many ways), and add $c_1A^i B^j$ to $P1$ and add $c_2A^i B^j$ to $P2$
6. If $\gcd(P1,P2)=1$, report $P1$, $P2$, and $P3$.

The problem is quite vague as there exist lots of polynomials satisfying the conditions. Here is just one way to get some of them.

First, decide what factors of $AB(A+B)$ divide what polynomials. For example, if we fix that $A\mid P1$, $B\mid P2$, and $(A+B)\mid P3$, then we can generate them as follows:

1. Take any polynomial $Q$ and set $P3 = (A+B)Q$. Initially set both $P1$ and $P2$ equal to $0$.
2. For each term $c A^i B^j$ in $P3$:
3. (i) if $i=0$, add $m$ to $R2$$P2$
4. (ii) if $j=0$, add $m$ to $R1$$P1$
5. (iii) otherwise partition $c = c_1 + c_2$ (which can be done in many ways), and add $c_1A^i B^j$ to $P1$ and add $c_2A^i B^j$ to $P2$
6. If $\gcd(P1,P2)=1$, report $P1$, $P2$, and $P3$.

The problem is quite vague as there exist lots of polynomials satisfying the conditions. Here is just one way to get some of them.

First, decide what factors of $AB(A+B)$ divide what polynomials. For example, if we fix that $A\mid P1$, $B\mid P2$, and $(A+B)\mid P3$, then we can generate them as follows:

1. Take any polynomial $Q$ in $A$ and $B$, and set $P3 = (A+B)Q$. Initially set both $P1$ and $P2$ equal to $0$.
2. For each term $c A^i B^j$ in $P3$:
3. (i) if $i=0$, add $m$ to $P2$
4. (ii) if $j=0$, add $m$ to $P1$
5. (iii) otherwise partition $c = c_1 + c_2$ (which can be done in many ways), and add $c_1A^i B^j$ to $P1$ and add $c_2A^i B^j$ to $P2$
6. If $\gcd(P1,P2)=1$, report $P1$, $P2$, and $P3$.