Algebraic geometry

Main theorem of elimination theory

In algebraic geometry, the main theorem of elimination theory states that every projective scheme is proper. A version of this theorem predates the existence of scheme theory. It can be stated, proved, and applied in the following more classical setting. Let k be a field, denote by the n-dimensional projective space over k. The main theorem of elimination theory is the statement that for any n and any algebraic variety V defined over k, the projection map sends Zariski-closed subsets to Zariski-closed subsets. The main theorem of elimination theory is a corollary and a generalization of Macaulay's theory of multivariate resultant. The resultant of n homogeneous polynomials in n variables is the value of a polynomial function of the coefficients, which takes the value zero if and only if the polynomials have a common non-trivial zero over some field containing the coefficients. This belongs to elimination theory, as computing the resultant amounts to eliminate variables between polynomial equations. In fact, given a system of polynomial equations, which is homogeneous in some variables, the resultant eliminates these homogeneous variables by providing an equation in the other variables, which has, as solutions, the values of these other variables in the solutions of the original system. (Wikipedia).

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From playlist Solve a System of Equations Using Elimination | Hard

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From playlist Solve a System of Equations Using Elimination | Medium

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From playlist Solve a System of Equations Using Elimination | Medium

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From playlist Solve a System of Equations Using Elimination | Medium

Related pages

Prime ideal | Zariski topology | Monomial basis | Monomial | Elimination theory | Algebraic variety | Homogeneous polynomial | Affine plane | Commutative algebra | Projective space | Hilbert's Nullstellensatz | Francis Sowerby Macaulay | Polynomial ring | Direct product | System of polynomial equations | Field (mathematics) | Ring homomorphism | Algebraic geometry | Noetherian ring | Generic property | Hyperbola | Scheme (mathematics) | Gröbner basis | Principal ideal