Matiyasevich theorem/Range of unknowns

Besides exhibiting a Diophantine equation itself, one has to specify what are the admissible values of the unknowns. The same equation may define different sets depending on whether the unknowns range over all integers or only over non-negative integers. However, such restriction doesn't effect the scope of the whole class of Diophantine sets. Indeed, the set defined by a Diophantine equation

<math dioeq>P(a,x_1,\dots,x_m)=0</math>

with integer-valued unknowns is also defined by the Diophantine equation \[P(a,y_1-z_1,\dots,y_m-z_m)=0\] with unknowns restricted to non-negative integers. Similar, the set defined by the same equation (<ref>dioeq</ref>) but with unknowns restricted to non-negative integers is also defined by the equation \[P(a,p_1^2+q_1^2+r_1^2+s_1^2,\dots,p_m^2+q_m^2+r_m^2+s_m^2)=0\] with arbitrary integer values for the unknowns (thanks to Lagrange's four-square theorem stating that every non-negative integer is the sum of four squares).