Algebraic geometry has its origin in the study of systems of polynomial equations f (x, . . ., x )=0, 1 1 n . . . f (x, . . ., x )=0. r 1 n Here the f ? k X, . . ., X ] are polynomials in n variables with coe?cients in a ?eld k. i 1 n n ThesetofsolutionsisasubsetV(f, . . ., f)ofk . Polynomialequationsareomnipresent 1 r inandoutsidemathematics, andhavebeenstudiedsinceantiquity. Thefocusofalgebraic geometry is studying the geometric structure of their...