Positivstellensatz
E761266
The Positivstellensatz is a fundamental result in real algebraic geometry that characterizes when a polynomial that is positive on a semialgebraic set can be represented using sums of squares and polynomial inequalities.
All labels observed (7)
How this entity was disambiguated
This entity first appeared as the object of triple T8850262 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Positivstellensatz Context triple: [Hilbert’s seventeenth problem, relatedTo, Positivstellensatz]
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A.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
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B.
Hilbert’s seventeenth problem
Hilbert’s seventeenth problem is a famous question in real algebraic geometry asking whether every nonnegative polynomial can be represented as a sum of squares of rational functions.
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C.
Chevalley–Warning theorem
The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
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D.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
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E.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Positivstellensatz Target entity description: The Positivstellensatz is a fundamental result in real algebraic geometry that characterizes when a polynomial that is positive on a semialgebraic set can be represented using sums of squares and polynomial inequalities.
-
A.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
B.
Hilbert’s seventeenth problem
Hilbert’s seventeenth problem is a famous question in real algebraic geometry asking whether every nonnegative polynomial can be represented as a sum of squares of rational functions.
-
C.
Chevalley–Warning theorem
The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
-
D.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
E.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
result in real algebraic geometry
ⓘ
theorem ⓘ |
| appliesTo |
polynomials with real coefficients
ⓘ
semialgebraic sets ⓘ |
| assumes | positivity or nonnegativity of a polynomial on a semialgebraic set ⓘ |
| characterizes | positivity of polynomials on semialgebraic sets ⓘ |
| concerns |
nonnegativity on basic closed semialgebraic sets
ⓘ
representation of positive polynomials ⓘ |
| concludes | existence of algebraic representation using sums of squares and defining inequalities ⓘ |
| connectedTo |
preorderings in polynomial rings
ⓘ
quadratic modules ⓘ real Nullstellensatz ⓘ |
| field | real algebraic geometry ⓘ |
| framework |
ordered rings
ⓘ
real spectra of rings ⓘ |
| generalizes | classical results on positive polynomials ⓘ |
| hasVariant |
Archimedean Positivstellensatz
ⓘ
linked to:
Positivstellensatz
Krivine–Stengle Positivstellensatz ⓘ
linked to:
Positivstellensatz
Putinar’s Positivstellensatz ⓘ
linked to:
Positivstellensatz
Schmüdgen’s Positivstellensatz ⓘ
linked to:
Positivstellensatz
|
| historicalContext | 20th century development in real algebraic geometry ⓘ |
| implies | existence of sum of squares decompositions under suitable conditions ⓘ |
| influenced |
Lasserre hierarchy in optimization
ⓘ
linked to:
Positivstellensatz
modern polynomial optimization methods ⓘ |
| involves |
polynomial inequalities
ⓘ
sums of squares of polynomials ⓘ |
| language | German ⓘ |
| provides |
algebraic certificates for positivity
ⓘ
conditions for representing positive polynomials as sums of squares and constraints ⓘ |
| relatedTo |
Hilbert’s 17th problem
ⓘ
linked to:
Hilbert’s seventeenth problem
moment problems ⓘ optimization theory ⓘ real closed fields ⓘ semialgebraic geometry ⓘ sum of squares representations ⓘ |
| translation | positivity theorem ⓘ |
| typicalDomain | polynomial rings over the reals ⓘ |
| usedFor |
constructing infeasibility certificates for systems of polynomial inequalities
ⓘ
deriving hierarchies of semidefinite relaxations ⓘ |
| usedIn |
algebraic certificates of infeasibility
ⓘ
certification of nonnegativity of polynomials ⓘ polynomial optimization ⓘ semidefinite programming ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Positivstellensatz Description of subject: The Positivstellensatz is a fundamental result in real algebraic geometry that characterizes when a polynomial that is positive on a semialgebraic set can be represented using sums of squares and polynomial inequalities.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
Tarski’s theorem on the completeness of elementary algebra and geometry
→
field
→
real algebraic geometry
ⓘ
linked to: Positivstellensatz
linked to: Positivstellensatz
linked to: Positivstellensatz
linked to: Positivstellensatz
linked to: Positivstellensatz
linked to: Positivstellensatz