For instance, if is a subring of, then agrees with the restriction of to. However, they are often closely related to each other. If are two different rings, then the polynomial functions and arising from interpreting a polynomial form in these two rings are, strictly speaking, different functions. But this turns out not to be the case if one considers interpretations in all rings simultaneously, as we shall now discuss. The above examples show that if one only interprets polynomial forms in a specific ring, then some information about the polynomial could be lost (and some features of the polynomial, such as roots, may be “invisible” to that interpretation). Similarly, the linear form has no roots when interpreted in the integers, but has roots when interpreted in the rationals.
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