Hermite–Biehler theorem
E502194
The Hermite–Biehler theorem is a result in complex analysis and control theory that characterizes when a complex polynomial has all its zeros in the open upper half-plane in terms of the interlacing of zeros of two associated real polynomials.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hermite–Biehler class of entire functions | 1 |
| Hermite–Biehler theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5191868 — 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: Hermite–Biehler theorem Context triple: [Charles Hermite, hasConceptNamedAfter, Hermite–Biehler theorem]
-
A.
Routh–Hurwitz stability criterion
The Routh–Hurwitz stability criterion is a mathematical test in control theory that determines whether all roots of a system’s characteristic polynomial lie in the left half of the complex plane, ensuring system stability without explicitly computing the roots.
-
B.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
-
C.
Runge approximation theorem
The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
-
D.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
-
E.
Hadamard three-circle theorem
The Hadamard three-circle theorem is a result in complex analysis that describes how the maximum modulus of a holomorphic function behaves logarithmically between three concentric circles in the complex plane.
- 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: Hermite–Biehler theorem Target entity description: The Hermite–Biehler theorem is a result in complex analysis and control theory that characterizes when a complex polynomial has all its zeros in the open upper half-plane in terms of the interlacing of zeros of two associated real polynomials.
-
A.
Routh–Hurwitz stability criterion
The Routh–Hurwitz stability criterion is a mathematical test in control theory that determines whether all roots of a system’s characteristic polynomial lie in the left half of the complex plane, ensuring system stability without explicitly computing the roots.
-
B.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
-
C.
Runge approximation theorem
The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
-
D.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
-
E.
Hadamard three-circle theorem
The Hadamard three-circle theorem is a result in complex analysis that describes how the maximum modulus of a holomorphic function behaves logarithmically between three concentric circles in the complex plane.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in control theory ⓘ |
| appliesTo |
Hurwitz stable polynomials
ⓘ
univariate complex polynomials ⓘ |
| assumes | polynomial has no zeros on the real axis in standard form ⓘ |
| characterizes | when a complex polynomial has all zeros in the open upper half-plane ⓘ |
| concerns |
distribution of zeros of entire functions in some extensions
ⓘ
relationship between real and imaginary parts of a polynomial on the real axis ⓘ |
| conditionOnCoefficients | complex polynomial has real coefficients in many standard formulations ⓘ |
| conditionOnZeros |
associated real polynomials must have only real zeros
ⓘ
zeros of the two associated real polynomials must interlace on the real axis ⓘ |
| equivalentTo | a real-axis interlacing condition on two real polynomials ⓘ |
| field |
complex analysis
ⓘ
control theory ⓘ polynomial theory ⓘ |
| givesCriterionFor |
all zeros of a polynomial lying in the open left half-plane via change of variables
ⓘ
all zeros of a polynomial lying in the open upper half-plane ⓘ |
| hasGeneralization |
Hermite–Biehler class of entire functions
ⓘ
linked to:
Hermite–Biehler theorem
|
| historicalPeriod | late 19th to early 20th century mathematics ⓘ |
| implies | sign alternation properties of associated real polynomials on intervals between zeros ⓘ |
| involves |
decomposition of a complex polynomial into real and imaginary parts
ⓘ
two associated real polynomials ⓘ |
| mathematicalDomain |
complex function theory
ⓘ
real algebraic geometry ⓘ |
| namedAfter |
Charles Hermite
ⓘ
Ludwig Bieberbach Biehler ⓘ |
| relatedConcept |
entire functions of bounded type in a half-plane
ⓘ
interlacing of real roots ⓘ real-rooted polynomials ⓘ self-inversive polynomials ⓘ |
| relatedTo |
Hurwitz stability criterion
ⓘ
linked to:
Routh–Hurwitz stability criterion
Routh–Hurwitz theorem ⓘ
linked to:
Routh–Hurwitz stability criterion
stability of linear time-invariant systems ⓘ |
| subject |
complex polynomials
ⓘ
interlacing of zeros ⓘ location of zeros ⓘ real polynomials ⓘ upper half-plane ⓘ |
| typeOfCriterion | frequency-domain stability criterion ⓘ |
| usedFor |
analyzing root loci of polynomials
ⓘ
characterizing stable polynomials ⓘ designing stable feedback systems ⓘ |
| usedIn |
control theory
ⓘ
filter design ⓘ signal processing ⓘ stability analysis of dynamical systems ⓘ |
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: Hermite–Biehler theorem Description of subject: The Hermite–Biehler theorem is a result in complex analysis and control theory that characterizes when a complex polynomial has all its zeros in the open upper half-plane in terms of the interlacing of zeros of two associated real polynomials.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hermite–Biehler theorem