Conditions in which the three polynomials have a common prime divisor 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.
 A: One way to interpret this is, how can we check whether three polynomials $f,g,h \in \mathbb{Z}[x,y,z]$ have a common prime divisor.  This is a mathematically sensible problem, and in the case we are looking for a common factor that is a nonconstant polynomial, not trivial.
There are a couple of things about the Question's phrasing which create doubt whether the OP will find answers to that interpretation satisfactory.  First was the description of $x,y,z$ as integer variables.  While this might seem natural enough, it suggested a concern about values of the variables and evaluation of the polynomials, rather than a concern about the divisors of the polynomials themselves.  Second was the motivating application to "solving a system of algebraic equations" because this also emphasizes the evaluation of the polynomials as functions.
In any case our staring point is to note that integer polynomials $\mathbb{Z}[x,y,z]$ in those three variables constitute a unique factorization domain.  This means that any irreducible factor is indeed a prime factor.  It follows that if $f,g,h$ have any nontrivial common factor (i.e. a factor other than $\pm 1$), then they would necessarily have a common prime divisor.
In a Comment on the Question I offered the OP a choice of interpretations for "common prime divisor".  One is to ask for an integer prime that divides all three polynomials.  Checking for this is easily done.  The the greatest integer common divisor of the three polynomials is the same as the gcd of their combined coefficients.  So consider the integer ideal $I\subseteq \mathbb{Z}$ generated by all those coefficients.  Since $\mathbb{Z}$ is a principal ideal domain, $I$ must be generated by an integer $k$, and if $I$ is a proper ideal, then $k$ will have prime divisors, any one of which is a common prime divisor of the three polynomials.

It is less trivial to determine whether two or three polynomials have a common polynomial divisor of degree greater than zero (as constant divisors would be considered polynomials of degree zero or less).  There are good algorithms for finding the greatest common divisor of two or three polynomials, but the difficulty over doing this for integers is that the gcd usually is not expressed as a "linear combination" of the terms.  For example, while $\gcd(2,5) = 1 = (-2)2+ 5$, we have $\gcd(x^2+1,2x) = 1$ but no way to generate $1$ as a combination of $x^2+1$ and $2x$ using integer polynomial coefficients.
A good starting point to read about algorithms for finding the greatest common divisor of polynomials is the Wikipedia article, Polynomial greatest common divisor.  The special facts in this Question is that we have multivariate polynomials with integer coefficients*.
If one were to consider a narrow version of this problem, it would simply be to ask whether the three polynomials $f,g,h$ have a common factor (with degree greater than zero) without demanding that such a factor be found.  Taking this approach calls for a bit of wariness, however, because working with pairs of the three polynomials could lead into this "trap":  each pair may have a nontrivial common factor without all three having a common factor.  (We can illustrate this already with three integers, such as $6,10,15$, which do not share a common factor greater than one even though each pair out of the three do.)
Computing the actual GCD (greatest common divisor) of a pair of integer polynomials will avoid that difficulty, because in a unique factorization domain (UFD):
$$ \gcd(f,g,h) = \gcd(\gcd(f,g),h) $$
Thus by twice computing the GCD of a pair of polynomials, one is guaranteed to get the GCD of a triple of polynomials, and so on if necessary.
Methods for finding a greatest common divisor of two polynomials profitably draw upon the rich ideas involved in polynomial factorization, a very active area of research through the end of the 20th century.
One of these ideas is to reduce the number of variables involved by evaluating the polynomials partially with an integer value for one (or more) of the variables.  After all, if $d(x,y,z)$ divides $f(x,y,z),g(x,y,z),h(x,y,z)$ as integer polynomials, then setting (say) $z=k$ as an integer value will produce bivariate polynomial divisor $d(x,y,k)$ of $f(x,y,k),g(x,y,k),h(x,y,k)$ respectively.  If there are enough integer arguments $k$ where the values $f(x,y,k),g(x,y,k),h(x,y,k)$ are collectively coprime, then these $d(x,y,k)$ will have to be $\pm 1$.  
Knowing that the degree of such a polynomial $d(x,y,k)$ must be bounded by the least degree of $f(x,y,k),g(x,y,k),h(x,y,k)$ in $x,y$ will often allow us to construct the possible candidates $p(x,y,z)$ by interpolation.
Another set of ideas arises from choosing a prime modulus $p\in \mathbb{Z}$ and computing with the coefficients of $f,g,h$ modulo $p$.  That makes the ring of coefficients a finite field rather than simply a PID or UFD, and with the coefficients taken in that way, univariate polynomials form a Euclidean domain in which GCD's may be computed by the more-or-less familiar Euclidean algorithm.
Connecting these two set of ideas is the premise that we can achieve a computation of GCD in $\mathbb{Z}[x,y,z]$ by working recursively through rings of bivariate and univariate polynomials.  A classroom note by R. Freund delves more deeply into the details here, with references to the literature for Readers who want to know more.
