Hilbert's Nullstellensatz

Relation between algebraic varieties and polynomial ideals

In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros" or, more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. It was proven by David Hilbert in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem) and became a foundational result of algebraic geometry. There are several formulations of the Nullstellensatz, the most elementary of which deal with conditions for the existence of solutions to systems of multivariate polynomial equations over an algebraically closed field (such as the complex numbers C {\displaystyle \mathbb {C} }).

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