Systems of polynomial equations
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.
A trigonometric equation is an equation g = 0 where g is a trigonometric polynomial. Such an equation may be converted into a polynomial system by expanding the sines and cosines in it, replacing sin(x) and cos(x) by two new variables s and c and adding the new equation s2 + c2 − 1 = 0.
For example, the equation
is equivalent to the polynomial system
When solving a system over a finite field k with q elements, one is primarily interested in the solutions in k. As the elements of k are exactly the solutions of the equation xq − x = 0, it suffices, for restricting the solutions to k, to add the equation xiq − xi = 0 for each variable xi.
The elements of a number field are usually represented as polynomials in a generator of the field which satisfies some univariate polynomial equation. To work with a polynomial system whose coefficients belong to a number field, it suffices to consider this generator as a new variable and to add the equation of the generator to the equations of the system. Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers.
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