Factoring a multivariate polynomial

Question

For the polynomial $p(x,y) = ax^2 + bxy + cy^2$, where $a, b, c \in \mathbb{R}$, what are the conditions on the coefficients such that $p$ can be factored into the product of polynomials in $x$ and $y-x$, i.e. $p(x,y) = p_1(x)p_2(y-x)$? Are $x^2$, $x(y-x)$ and $(y-x)^2$, together with their possible scalings, the only examples? What about factorizations into functions that are not polynomials, i.e. $p(x,y) = f(x)g(y-x)$, where $f$ and $g$ are not necessarily polynomials and we assume the equality holds on some subset of $\mathbb{R}\times\mathbb{R}$? Are these possible, and, if so, is there some literature devoted to this topic? If the answer is "yes" references are also welcome.

Answer 1

$p(x,y) = p_1(x)p_2(y-x) \implies p(x,x)=p_1(x)p_2(0)=(a+b+c)x^2$

• if $a+b+c \ne 0$ it follows that $p_2(0)\ne0$ so $p_1(x)=\frac{a+b+c}{p_2(0)}x^2$ and, by consideration of degree, $p_2$ must be a constant non-zero polynomial, so in the end $p(x,y) = \lambda x^2$ for some $\lambda \ne 0\,$

• if $a+b+c = 0$ then $b=-a-c$ so $p(x)=ax^2-(a+c)xy+cy^2=(ax-cy)(x-y)$

  • if $a=c$ then $p(x)=a(x-y)^2$

  • if $a \ne c$ then $p(x) = \lambda(x - \mu y)(x-y)$ for some constants $\lambda \ne 0,\mu \ne 1$

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[ EDIT ]   For the second part of the question, and assuming sufficiently smooth $f,g\,$, taking the derivatives in $y$ twice would imply $p_{yy}^{\,''}(x,y)=2c=f(x) g''(y-x)\,$, which in turn would imply that both $f$ and $g''$ are constant functions, so the problem then reduces to the first case.