2.3 Hilbert's Nullstellensatz

Hilbert's Nullstellensatz is a game-changer in algebraic geometry. It links algebraic varieties and ideals in polynomial rings over algebraically closed fields. This connection lets us study geometric shapes using algebra and vice versa.

The theorem comes in two flavors: weak and strong. Both versions help us understand when polynomial equations have solutions and how varieties relate to ideals. This knowledge is key for solving problems in algebra, geometry, and even robotics.

Hilbert's Nullstellensatz

Statement and Interpretation

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• Hilbert's Nullstellensatz establishes a correspondence between algebraic varieties and ideals in polynomial rings over algebraically closed fields
• The weak Nullstellensatz states that if $k$ is an algebraically closed field and $I$ is an ideal in the polynomial ring $k[x₁, ..., xₙ]$, then the variety $V(I)$ is empty if and only if $I = (1)$, meaning $I$ contains a nonzero constant polynomial
• The strong Nullstellensatz states that if $k$ is an algebraically closed field and $I$ is an ideal in $k[x₁, ..., xₙ]$, then the ideal of all polynomials vanishing on the variety $V(I)$ is equal to the radical of $I$, i.e., $I(V(I)) = √I$
  • The radical of an ideal $I$, denoted by $√I$, is the set of all polynomials $f$ such that some power of $f$ lies in $I$
• The Nullstellensatz establishes a bijective correspondence between the set of all radical ideals in $k[x₁, ..., xₙ]$ and the set of all algebraic varieties in the affine space $kⁿ$, where $k$ is an algebraically closed field (complex numbers)

Implications and Consequences

• The Nullstellensatz implies that every maximal ideal in $k[x₁, ..., xₙ]$ is of the form $(x₁ - a₁, ..., xₙ - aₙ)$ for some point $(a₁, ..., aₙ)$ in $kⁿ$, and every point in $kⁿ$ corresponds to a maximal ideal
• The Nullstellensatz provides a method for determining the irreducibility of an algebraic variety by examining the primality of its corresponding ideal
• The Nullstellensatz can be used to prove that a system of polynomial equations has a solution in an algebraically closed field if and only if the ideal generated by the polynomials is not equal to $(1)$
  • Example: The system $x^2 + y^2 = 1$ and $x + y = 0$ has a solution in $ℂ^2$ because the ideal $⟨x^2 + y^2 - 1, x + y⟩$ does not equal $(1)$
• The bijective correspondence established by the Nullstellensatz allows for the study of geometric properties of varieties using algebraic tools, and vice versa

Ideals and Varieties

Correspondence between Ideals and Varieties

• Given an ideal $I$ in $k[x₁, ..., xₙ]$, the Nullstellensatz allows us to determine the corresponding algebraic variety $V(I)$ by finding the common zeros of all polynomials in $I$
  • Example: For the ideal $I = ⟨x^2 + y^2 - 1⟩$, the corresponding variety $V(I)$ is the unit circle in $ℂ^2$
• Conversely, given an algebraic variety $V$ in $kⁿ$, the Nullstellensatz allows us to determine the corresponding ideal $I(V)$ by finding all polynomials that vanish on every point of $V$
  • Example: For the variety $V = \{(x, y) ∈ ℂ^2 | x^2 + y^2 = 1\}$, the corresponding ideal $I(V) = ⟨x^2 + y^2 - 1⟩$

Operations on Ideals and Varieties

• The Nullstellensatz can be applied to study the intersection and union of algebraic varieties by considering the sum and product of their corresponding ideals
  • The intersection of varieties corresponds to the sum of ideals: $V(I) ∩ V(J) = V(I + J)$
  • The union of varieties corresponds to the product of ideals: $V(I) ∪ V(J) = V(IJ)$
• The Nullstellensatz can be used to determine the dimension of an algebraic variety by examining the height of its corresponding prime ideal
  • The dimension of a variety $V$ is equal to the Krull dimension of its coordinate ring $k[V]$, which is the quotient ring $k[x₁, ..., xₙ]/I(V)$

Proof of the Nullstellensatz

Weak Nullstellensatz

• The proof of the weak Nullstellensatz relies on the fact that in an algebraically closed field, every non-constant polynomial has a root
  • If $I ≠ (1)$, then $I$ does not contain a nonzero constant polynomial, and one can construct a maximal ideal containing $I$, which corresponds to a point in the variety $V(I)$
• The construction of the maximal ideal uses Zorn's Lemma and the fact that every ideal in a polynomial ring over a field is contained in a maximal ideal

Strong Nullstellensatz

• The proof of the strong Nullstellensatz involves showing that $I(V(I)) ⊇ √I$ and $I(V(I)) ⊆ √I$
  • To show $I(V(I)) ⊇ √I$, use the fact that if $f ∈ √I$, then $f^m ∈ I$ for some $m ≥ 1$, and thus $f$ vanishes on $V(I)$
  • To show $I(V(I)) ⊆ √I$, use the Rabinowitsch trick, which introduces a new variable to reduce the problem to the weak Nullstellensatz
• The Rabinowitsch trick involves considering the ideal $J = I + ⟨1 - yf⟩$ in $k[x₁, ..., xₙ, y]$, where $f ∈ I(V(I))$, and showing that $J ≠ (1)$ using the weak Nullstellensatz

Bijective Correspondence

• The proof of the bijective correspondence between radical ideals and algebraic varieties relies on the properties of the operations $V$ and $I$, such as $V(I(V)) = V$ and $I(V(I)) = √I$
• To show that the correspondence is bijective, one must prove that the maps $I ↦ V(I)$ and $V ↦ I(V)$ are inverses of each other when restricted to radical ideals and algebraic varieties
  • This involves showing that for any radical ideal $I$, $I(V(I)) = I$, and for any algebraic variety $V$, $V(I(V)) = V$

Applications of the Nullstellensatz

Solving Systems of Polynomial Equations

• The Nullstellensatz can be used to prove that a system of polynomial equations has a solution in an algebraically closed field if and only if the ideal generated by the polynomials is not equal to $(1)$
• This has applications in computer algebra, robotics, and computer vision, where finding solutions to polynomial equations is a fundamental problem
  • Example: In robotics, the forward and inverse kinematics problems involve solving systems of polynomial equations to determine the position and orientation of a robot's end effector

Studying Local Properties of Varieties

• The Nullstellensatz can be utilized to study the local properties of algebraic varieties, such as tangent spaces and singularities, by examining the localization of their corresponding ideals
  • The tangent space to a variety $V$ at a point $p$ corresponds to the kernel of the Jacobian matrix of the generators of $I(V)$ evaluated at $p$
  • Singularities of a variety can be characterized by the properties of the localized ring $k[V]_p$, such as its dimension and regularity

Decomposition of Varieties

• The Nullstellensatz can be used to prove the correspondence between the irreducible components of an algebraic variety and the minimal prime ideals containing its corresponding ideal
  • An algebraic variety is irreducible if and only if its corresponding ideal is prime
  • The minimal prime ideals containing $I(V)$ correspond to the irreducible components of $V$
• This correspondence allows for the study of the decomposition of varieties into irreducible components using algebraic methods, such as primary decomposition of ideals

Dimension and Degree of Varieties

• The Nullstellensatz can be used to determine the dimension of an algebraic variety by examining the height of its corresponding prime ideal
  • The dimension of a variety $V$ is equal to the Krull dimension of its coordinate ring $k[V]$, which is the supremum of the lengths of chains of prime ideals in $k[V]$
• The degree of a variety can be defined as the number of intersection points with a generic linear subspace of complementary dimension
  • The degree of a variety can be computed algebraically using the Hilbert polynomial of its corresponding ideal