Applications of Elimination theory

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Elimination Theory deals with elimination of a subset of variables from a given set of polynomial equations and finding the reduced set of polynomial equations (not involving the eliminated variables).What is the importance and applications of Elimination theory?

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Elimination Theory for polynomial equations is useful in Computer Aided Geometric Design. See for instance

"Algebraic Methods for Computer Aided Geometric Design" by Thomas W. Sederberg and Jianmin Zheng, a chapter in Handbook of Computer Aided Geometric Design.