Hilbert’s seventeenth problem

E210619

Hilbert problem mathematical problem problem in real algebraic geometry

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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All labels observed (1)

Label	Occurrences
Hilbert’s seventeenth problem canonical	1

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 ⓘ

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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.

Hilbert problems → hasPart → Hilbert’s seventeenth problem ⓘ