Hilbert’s seventeenth problem
E210619
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hilbert’s seventeenth problem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1859191 — 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: Hilbert’s seventeenth problem Context triple: [Hilbert problems, hasPart, Hilbert’s seventeenth problem]
-
A.
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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B.
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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C.
Hilbert’s tenth problem
Hilbert’s tenth problem is a famous unsolved question in mathematics that asked for a general algorithm to determine whether any given Diophantine equation has an integer solution, and whose negative answer helped establish fundamental limits of computability.
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D.
Hilbert problems
The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
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E.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
- 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: Hilbert’s seventeenth problem Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Hilbert’s tenth problem
Hilbert’s tenth problem is a famous unsolved question in mathematics that asked for a general algorithm to determine whether any given Diophantine equation has an integer solution, and whose negative answer helped establish fundamental limits of computability.
-
D.
Hilbert problems
The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
-
E.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
Hilbert problem
ⓘ
mathematical problem ⓘ problem in real algebraic geometry ⓘ |
| asksWhether |
every nonnegative polynomial is a sum of squares of rational functions
ⓘ
every polynomial that takes only nonnegative values over the reals can be represented as a sum of squares of rational functions ⓘ |
| concerns |
positive semidefinite polynomials
ⓘ
rational functions ⓘ representation of nonnegative polynomials ⓘ sums of squares ⓘ |
| field |
algebraic geometry
ⓘ
real algebra ⓘ real algebraic geometry ⓘ |
| hasCanonicalFormulation | Given a polynomial with real coefficients that takes only nonnegative values for all real inputs, is it a sum of squares of rational functions with real coefficients? ⓘ |
| hasConsequence |
existence of nonnegative polynomials that are not sums of squares of polynomials
ⓘ
representation of nonnegative polynomials as sums of squares of rational functions ⓘ |
| historicalImportance |
influential in development of real algebra
ⓘ
major milestone in real algebraic geometry ⓘ |
| implies | every nonnegative polynomial over a real closed field is a sum of squares of rational functions ⓘ |
| involvesConcept |
nonnegative polynomial
ⓘ
ordered field ⓘ positive semidefinite form ⓘ rational function ⓘ real closed field ⓘ sum of squares ⓘ |
| numberInHilbertList | 17 ⓘ |
| originalLanguage | German ⓘ |
| partOf |
Hilbert problems
ⓘ
surface form:
Hilbert’s problems
|
| posedAtEvent |
International Congress of Mathematicians
ⓘ
surface form:
International Congress of Mathematicians 1900
|
| posedAtLocation | Paris ⓘ |
| posedBy | David Hilbert ⓘ |
| posedInYear | 1900 ⓘ |
| relatedTo |
Hilbert’s nineteenth problem
ⓘ
Hilbert’s sixteenth problem ⓘ Positivstellensatz ⓘ moment problem ⓘ quadratic forms ⓘ real closed field ⓘ sum of squares decomposition ⓘ |
| solutionMethod |
Artin–Schreier theory
ⓘ
real algebraic methods ⓘ theory of formally real fields ⓘ |
| solutionPublishedInYear | 1927 ⓘ |
| solutionYear | 1927 ⓘ |
| solvedBy | Emil Artin ⓘ |
| status | solved ⓘ |
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: Hilbert’s seventeenth problem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.