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Actually elementary questions about systems of polynomial equations
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2) look up the term "groebner basis". it's essentially gaussian elimination but for any polynomials instead of just linear equations. example:

x^(2)+3y+x-5 = 0 and x^(2)+y^(2)-xy-7x-3 = 0. mathematica gives me the groebner basis y^(4)-2y^(3)+61y^(2)+78y-300=0 and y^(3)-10y^(2)+51y+90x-60=0. so you just have to solve that degree 4 equation in one variable, and substitute the roots into the second equation and solve that for x to get all the solutions.

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