Real NullstellensatzEdit
Real Nullstellensatz is a cornerstone of real algebraic geometry, describing how the algebra of polynomials over the real numbers interacts with their sets of real solutions. While the classical Nullstellensatz (over algebraically closed fields like the complex numbers) ties radical ideals to vanishing sets, the real version must acknowledge that not all polynomials that vanish on every real solution fit neatly into an algebraic radical. Instead, the right notion is the real radical, which captures vanishing on real solutions in a way that also respects positivity constraints. This leads to a precise correspondence between the ideal of polynomials that vanish on all real common zeros and a suitably defined real radical of the original ideal.
The key object is the real radical of an ideal, denoted √R(I). This is defined inside the polynomial ring over the real numbers, and it encodes certificates that a polynomial will vanish on every real point where the generators of I vanish, taking into account sums of squares. Concretely, a polynomial f belongs to the real radical √R(I) if there exist a natural number m and polynomials g1, ..., gs such that f^{2m} + g1^2 + ... + gs^2 ∈ I. Equivalently, f vanishes on V_R(I) in a way compatible with positivity constraints that arise from squaring, so the real radical filters out polynomials that would otherwise vanish only on part of the complexification or after sign considerations.
The Real Nullstellensatz states that, for an ideal I in the real polynomial ring R[x1, ..., xn], the ideal of all polynomials vanishing on the real zero set V_R(I) is exactly the real radical of I: I(V_R(I)) = √R(I). This equality mirrors the spirit of Hilbert’s Nullstellensatz, but it is adapted to the realities of real solutions and the role of sums of squares. It has both a “strong” form, giving an exact description of I(V_R(I)) in terms of √R(I), and a “weak” form, which yields qualitative information about when the real zero set is empty (the situation that forces √R(I) to be the whole ring).
The real root-structure and certificates of feasibility or infeasibility emerge naturally from this framework. If V_R(I) is empty, then √R(I) equals the whole ring, which provides a real-algebraic certificate of infeasibility. In practical terms, one can certify that a system of real polynomial equations has no real solution by producing a representation of the form f^{2m} + ∑ h_j^2 ∈ I for suitable choice of f and h_j, demonstrating that the supposed real zeros cannot exist. This is where the interplay with positivity comes into play: sums of squares serve as algebraic witnesses to nonnegativity and to the obstruction of real roots.
Historically, the Real Nullstellensatz emerged from the work of investigators exploring how positivity, sums of squares, and semialgebraic geometry interact. Important contributions came from the development of the theory of sums of squares and related certificates, which connect to modern optimization techniques and semidefinite programming. The broader family of results surrounding these ideas is often presented under the umbrella of real algebraic geometry, with key milestones tied to the development of the real radical concept and its certificates.
In addition to the core equality I(V_R(I)) = √R(I), the landscape includes abundant connections to positivity certificates and semialgebraic representations. The real Nullstellensatz serves as a bridge to the Positivstellensatz, which describes when a polynomial is positive on a given semialgebraic set via representations as sums of squares with polynomials enforcing the defining inequalities of the set. Notable results in this vein include the Krivine–Stengle Positivstellensatz and various Putinar-type theorems, which provide robust and computationally friendly certificates for nonnegativity on semialgebraic sets Krivine–Stengle Positivstellensatz and Putinar's Positivstellensatz and relate to the broader framework of sums of squares and semialgebraic set theory. The real Nullstellensatz thus sits at a crossroads of algebra, geometry, and optimization, with practical impact in areas such as real numbers-based modeling, control theory, and polynomial optimization.
A few representative ideas and examples help ground the theory. If I = ⟨x^2 + 1⟩ in R[x], then V_R(I) is empty, and consequently I(V_R(I)) = R[x] while √R(I) is the entire ring, reflecting the infeasibility of solving x^2 + 1 = 0 over the real numbers. By contrast, if I defines a real curve with real points, the real Nullstellensatz tells us which polynomials vanish on all those points and which can be expressed via the real radical relation above. The framework also explains why a polynomial nonnegative on a real algebraic set does not necessarily belong to the radical; positivity certificates require the broader apparatus of sums of squares and Positivstellensatz representations.
See also