Positivstellensatz

E761266

result in real algebraic geometry theorem

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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (2)

Label	Occurrences
Positivstellensatz canonical	1
real algebraic geometry	1

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 NERFINISHED ⓘ Krivine–Stengle Positivstellensatz NERFINISHED ⓘ Putinar’s Positivstellensatz NERFINISHED ⓘ Schmüdgen’s Positivstellensatz NERFINISHED ⓘ
historicalContext	20th century development in real algebraic geometry ⓘ
implies	existence of sum of squares decompositions under suitable conditions ⓘ
influenced	Lasserre hierarchy in optimization NERFINISHED ⓘ 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 NERFINISHED ⓘ 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 (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hilbert’s seventeenth problem → relatedTo → Positivstellensatz ⓘ

Tarski’s theorem on the completeness of elementary algebra and geometry → field → Positivstellensatz ⓘ

this entity surface form: real algebraic geometry