Ask Your Question

Revision history [back]

click to hide/show revision 1
initial version

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 $P3$.