Bounds on integer solutions to multivariate polynomials The following problem came up while I was doing a related homework problem.
Imagine we have $f_1,\ldots,f_m$ be multivariate polynomials in variables $x_1,\ldots, x_n$ with rational coefficients. Can we bound the possible size of the smallest integer solution (that is to say find some $M$ such that if an integer solution exists, then there is an integer solution $a_1,\ldots,a_n$ such that $|a_i|<M$). If such a bound exists I imagine it depends on the number of $f_i$, their degree, and the size of their coefficients.
I am asking more about the existence of such a bound than what it is. Also it seems reasonable to me that such a bound would exist, but in my attempts to show this I have been getting hung up on something in each of the directions I have tried to go to prove it.
Thank you for any help or direction
 A: Building on earlier work of Julia Robinson, Martin Davis, and Hilary Putnam, Yuri Matiyasevich's theorem proved that Hilbert's tenth problem has a negative answer: there is no general procedure for deciding if a diophantine equation has a solution in (rational) integers.
If there were a procedure such as specified in this Question, applying it to a diophantine equation would produce a bound on the smallest integer solution (involving multiple variables), and a procedure of checking all possibilites less than that bound would then decide whether any integer solution exists.
Many specific diophantine problems, however, may be attacked in this way.  Computing effective bounds on integer solutions of elliptic curves is one such research of area.
Added:  Multiplicity of equations is not essential, since a sum of squares of rational polynomials being zero implies each of them is zero (at any real (specifically rational integer) point of evaluation).  However the multivariate aspect of the problem is essential for undecidability (lack of constructive bounds).  A univariate rational polynomial equations can be converted to one with integer coefficients by clearing denominators, and in this form the Rational Root Theorem implies any integer solution would divide the constant term.  In particular the (absolute value of the) constant term gives an upper bound on the smallest integer solution as asked for.
