The greatest integer common divisor of the three polynomials is the same as the $gcd$ of their combined coefficients

Question

I have three polynomials $f(x,y,z), g(x,y,z),h(x,y,z)$ in three integer variables $x,y,z$ and with integer coefficients.

I am searching for conditions in which the three polynomials have a common prime divisor. I came across with this problem when solving a system of algebraic equations involving the equilibrium points of a dynamical system. By "prime divisor" I mean an integer prime divisor not an irreducible integer polynomial divisor. I read that the greatest integer common divisor of the three polynomials is the same as the $gcd$ of their combined coefficients. But I am not understanding the meaning of this phrase, i.e., combined coefficients.

Answer 1

If $\,f,g,h\in\Bbb Z[x]\,$ with coef's $\,f_i,g_i,h_i,\,$ and $\,d\in \Bbb Z,\,$ then we have $$d\mid f\ \,{\rm in}\,\ \Bbb Z[x]\iff f/d\in \Bbb Z[x]\iff \{f_i/d\}\subset \Bbb Z\iff \forall\,i\!:\, d\mid f_i \iff d\mid \gcd\{f_i\}$$ So $\,d\mid f,g,h\!\iff\! d\mid \gcd\{f_i\},\gcd\{g_i\},\gcd\{h_i\}\! \!\iff\! d\mid \gcd\{f_i, g_i, h_i\}\,$ by gcd is associative.

Answer 2

Not if one polynomial is the negative of the sum of the other two.