I would like to ask your advice for the following couple of questions.
It is known that, if a system Ax = b of linear equations -- where the coefficients of A and b are rational numbers -- has an integral solution x, then it has an integral solution y of size polynomially bounded by the sizes of A and b. [Schriver, Theory of Linear and Integer Programming, Corollary 5.2a.]
Questions:
1) Does the result above still hold if we suppose that the components of x are non negative integers? That is, in this case, does there exist an integral solution y, whose component are all in N, and such that its size is polynomially bounded by the sizes of A and b?
2) What happens if we consider the same question (1) for a Diophantine system formed by two linear equations, together with a quadratic one? Can we get a similar result?
Thank you in advance for your expertise.