Hilbert's Nullstellensatz

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 ${\displaystyle \mathbb {C} }$). The weak Nullstellensatz is a corollary (or a lemma, depending which is proved first) of the Nullstellensatz which can be stated as follows. Consider a system of polynomial equations

$${\displaystyle {\begin{cases}f_{1}(x_{1},\dots ,x_{n})=0\\\vdots \\f_{m}(x_{1},\ldots ,x_{n})=0\end{cases}}}$$

in the variables ${\displaystyle x_{1},\dots ,x_{n}}$, where ${\displaystyle f_{1},\dots ,f_{m}\in K[x_{1},\dots ,x_{n}]}$ are multivariate polynomials over an algebraically closed field ${\displaystyle K}$. If the system does not have a solution ${\displaystyle (x_{1},\dots ,x_{n})\in K^{n}}$, then there is "algebraic reason" for this situation: namely, this occurs precisely when there exist polynomials ${\displaystyle g_{1},\dots ,g_{m}\in K[x_{1},\dots ,x_{n}]}$ such that

$${\displaystyle g_{1}f_{1}+\cdots +g_{m}f_{m}=1.}$$

Since the expression on the left-hand side must evaluate to 0 at any ${\displaystyle (x_{1},\dots ,x_{n})}$ that solves the system of equations, it is obvious from the inconsistency that no solution can exist if this condition holds. Informally, the (weak) Nullstellensatz asserts that in the absence of such an inconsistency, a solution to the system of equations must exist.

The full Nullstellensatz is the following refinement: If ${\displaystyle f\in K[x_{1},\dots ,x_{n}]}$ is a polynomial such that every solution of the system of equations is also a solution of

$${\displaystyle f(x_{1},\dots ,x_{n})=0,}$$

then there is a similar type of "algebraic reason" for this occurrence: this occurs precisely when there exist a natural number ${\displaystyle r}$ and polynomials ${\displaystyle g_{1},\dots ,g_{m}}$ such that

$${\displaystyle f^{r}=g_{1}f_{1}+\cdots +g_{m}f_{m}.}$$