Number of integer solutions of a multivariate polynomial Given a single-variable polynomial, we all know that the number of its roots is bounded in terms of its degree. A polynomial here is a polynomial with integer coeffecients, and a root of a polynomial is a real root.
Given a multivariate polynomial, this phenomenon, in its general form, stops working. However, by a well-known theorem by Bezout, we still can say that either it has an infinite number of roots, or it has only a finite number which is bounded above in terms of its degree.
My questions is about the number of integer roots: Does there exist a functions $B=B_n:\mathbb N\rightarrow \mathbb N$ such that given a polynomial $P(x_1,...,x_n)$ of degree $d$, we can say that the set of INTEGER roots of $P$ contains at most $B(d)$ elements, or otherwise it is infinite?
Thanks in advance.

There is no such bound.
Given an equation $X$ over $\mathbf{Q}$ with infinitely many rational points, you can always scale the coefficients so that any finite subset of the rational points all become integral points. (Put them all under a common denominator $N$, then replace each variable $x$ by $x/N$.)  Take an elliptic curve $E$ with positive rank, that is, $E(\mathbf{Q})$ is infinite. (They exist: for example, $E: y^2 = x^3 - 2$.) After scaling, you can find an elliptic curve with as many integral points as you like. (You can take $y^2 = x^3 - 2 N^6$ for the greatest common denominator $N$ of the finite set of rational points.) If such a bound in your question existed, then, since these curves all have the same degree, at some point these scaled elliptic curve would have to have infinitely many integral points. But $E(\mathbf{Z})$ is always finite for any elliptic curve by a theorem of Siegel.
