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... If multiplication results in poly. of degree > n-1 Reduce it by modulo some irreducible poly. m(x)  f(x) = q(x) m(x) + r(x) => r(x) = f(x) mod m(x) ...
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A system of polynomial equations is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in several variables, say x1, ..., xn, over some field k.Usually, the field k is either the field of rational numbers or a finite field, although most of the theory applies to any field.A solution is a set of the values for the xi which make all of the equations true and which belong to some algebraically closed field extension K of k. When k is the field of rational numbers, K is the field of complex numbers.
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