How one can transform a statement to a system of Diophantine conditions
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It is well known that for the following statement: 8y < b9x1; : : : ; xm[F(a; y; x1; : : : ; xm) = 0]:::::::::::::::::::
1) where a is the parameter(s) of the polynomial F, the Chinese remainder theorem method results in the following system of Diophantine conditions solvable in the unknowns q; w; z0; : : : ; zm provided that (1) holds: F(a; z0; z1; : : : ; zm) q b z0 = q b! (b + w + B(a; b; w))! divide q + 1 q b divide z1 w .... q b divide zm w where the polynomial B(a; b; w) is obtained from F(a; y; x1; : : : ; xm) by changing the signs of all its negative coe¢ cients and systematically replacing y by b and x1; : : : ; xm by w. The statement (1) is hold for all y < b; i.e., 0 y < b: Now, my question is: How one can transform the statement: 8c < y < b9x1; : : : ; xm[F(a; y; x1; : : : ; xm) = 0]:::::::::::::::::::2) to a system of Diophantine conditions where c > 0.