Generating Reducible polynomials of a particular type

Question

I am working on a paper and I need to know if there exists any REDUCIBLE bivariate polynomials of the following form:

$Ax^4+2Ax^3+2Bx^2y-(2C+1)y+D=0$

Where A,B,C,D are all positive integers

I would be obliged, if anyone could provide a solution.

Answer 1

Just by computer search, one can see that for example $A=16,B=6,C=1,D=3$ gives $$ P_{A,B,C,D}(x,y)=16x^4+32x^3+12x^2y-3y+3=\\(2x+1)(8x^3+12x^2+6xy-6x-3y+3). $$ There are more, for example keeping same $A,D$ and having $B=10,C=2$, or $B=14,C=3$, and so on. So it seems that $A=16,B=4k+2,C=k,D=3$ for positive integer $k$ is one such family, each of which is divisible by $2x+1$. We might go further and find all polynomials which have this factor, these will satisfy $P_{A,B,C,D}(-1/2,y)=0$ for all $y$, so

$$ A(-1/2)^4+2A(-1/2)^3+2B(-1/2)^2y-(2C+1)y+D=0 $$ and by comparing coefficients this means $A/16-A/4+D=0$, $B/2-2C-1=0$. First equation simplifies to $16D=3A$, so $A=16l, D=3l$ for some integer $l$. Second equation simplifies to $B=4C+2$. So, all polynomials $P_{16l,4k+2,k,3l}(x,y)$ are divisible by $(2x+1)$.

You can go even further, for example using the same method as above, one can show that $(4x+1)$ divides $P_{256k,16l+8,l,7k}(x,y)$, $(2x+3)$ divides $P_{16k,4l+2,9l+4,27k}(x,y)$, and so on...

As requested in comments, here is a simple brute force code in Maple that does the search.

for A to 20 do:
for B to 20 do: 
for C to 20 do:
for DD to 20 do:
  f := A*x^4+2*A*x^3+2*B*x^2*y-(2*C+1)*y+DD:
  if not irreduc(f) then:
    print(f, A, B, C, DD):
  fi:
od:
od:
od:
od: