I have two (or more) polynomials $p(x,y)$, $q(x,y)$ going through $[0,0]$. Can I easily produce another irreducible polynomial having at the origin singularity having same qualities as $p(x,y)\cdot q(x,y)$? Nice candidate is $p(x,y)\cdot q(x,y) + x^{(deg_x p + deg_x q)}y^{(deg_y p + deg_y q)}$, but I cannot prove it.