{ "model": { "rank": 3105, "code": "thing-q43260", "model_id": "vr.tr.polynomial", "name": "polynomial", "purpose": "Let an agent explain polynomials by degree, operations, roots, factorisation and applications, for learners and practitioners.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q43260/", "tier": 2, "score": 91, "payload": { "layer": "wikidata", "aliases": [ "invariant polynomial", "Hilbert polynomial", "trinomial", "monomial", "binomial", "knot polynomial", "matching polynomial", "characteristic polynomial", "constant polynomial", "Conway polynomial", "Conway–Alexander polynomial", "Vandermonde polynomial", "symmetric polynomial", "rook polynomial", "power sum symmetric polynomial", "elementary symmetric polynomial", "Schur polynomial", "complete homogeneous symmetric polynomial", "Hall–Littlewood polynomials", "Jack function", "LLT polynomial", "Stanley symmetric function", "zonal polynomial", "orthogonal polynomials", "cyclotomic polynomial", "sparse polynomial", "associated Laguerre polynomial", "falling factorial", "real polynomial", "complex polynomial", "subresultant", "Narayana polynomials", "Mononom", "primitive part", "Pochhammer symbol", "stable polynomial", "generator polynomial", "Hurwitz polynomial", "reciprocal polynomial", "homogeneous polynomial" ], "aliasCount": 113, "merged": 113, "knownIn": 91, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q43260", "url": "https://www.wikidata.org/wiki/Q43260", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T12:34:42Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q43260", "registryId": "vr.tr.polynomial", "name": "polynomial", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q43260/", "model": { "registry_id": "vr.tr.polynomial", "name": "polynomial", "purpose": "Let an agent explain polynomials by degree, operations, roots, factorisation and applications, for learners and practitioners.", "definition": "A mathematical expression built from variables and coefficients using addition, subtraction, multiplication and non-negative integer powers, such as monomials, binomials and trinomials; its degree is the highest power, and by the fundamental theorem of algebra a nonconstant polynomial with complex coefficients has as many complex roots as its degree, counted with multiplicity.", "what_it_is_for": "Algebra, modelling, approximation and computation.", "affordances": [ "add, multiply and divide polynomials", "find roots and factorise", "use polynomials for interpolation and approximation", "explain key theorems" ], "distinguishing_features": [ "Non-negative integer exponents", "Has a degree", "Roots determined by degree", "Closed under addition and multiplication" ], "appearance": "Not physical; expressions such as 3x^2 + 2x - 5 and their graphs.", "visual_identification": [ "Sums of terms with whole-number powers", "Graphs such as parabolas and cubics", "Expressions with negative or fractional powers are not polynomials" ], "physical_properties": [], "families_and_kinds": [ "monomials, binomials and trinomials", "linear, quadratic and cubic polynomials", "multivariate polynomials", "orthogonal polynomials", "invariant and knot polynomials" ], "related_models": [ { "relation": "is a kind of", "target": "algebraic expression", "why": "category" }, { "relation": "has", "target": "root", "why": "property" }, { "relation": "is studied in", "target": "algebra", "why": "field" }, { "relation": "is related to", "target": "invariant", "why": "related concept" } ], "identifiers": [], "standards_and_regulation": [ "No specific regulation" ], "failure_modes_and_hazards": [ "Numerical instability with high-degree fitting", "Sign and exponent errors" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "basics", "name": "Basics", "description": "Definitions.", "rationale": "Terms and degree.", "layers": [ { "id": "identify", "name": "Identify", "description": "Is it a polynomial.", "findings": [ { "id": "identify-finding", "name": "Identify", "description": "Identification.", "questions": [ { "text": "Is this expression a polynomial, and what is its degree?", "kind": "boundary" }, { "text": "What are its coefficients?", "kind": "definition" } ] } ] }, { "id": "operations", "name": "Operations", "description": "Arithmetic.", "findings": [ { "id": "operations-finding", "name": "Operations", "description": "Operations.", "questions": [ { "text": "What is the result of adding, multiplying or dividing these polynomials?", "kind": "measurement" }, { "text": "Is there a remainder?", "kind": "measurement" } ] } ] } ] }, { "id": "roots", "name": "Roots", "description": "Solving.", "rationale": "Roots follow theorems.", "layers": [ { "id": "solve", "name": "Solve", "description": "Finding roots.", "findings": [ { "id": "solve-finding", "name": "Solve", "description": "Finding roots.", "questions": [ { "text": "What are the roots of this polynomial?", "kind": "measurement" }, { "text": "Which method is appropriate, such as the quadratic formula?", "kind": "definition" } ] } ] }, { "id": "theorems", "name": "Theorems", "description": "Key results.", "findings": [ { "id": "theorems-finding", "name": "Theorems", "description": "Key theorems.", "questions": [ { "text": "What do the fundamental theorem of algebra and Abel-Ruffini theorem say?", "kind": "definition" }, { "text": "Why is there no general formula beyond degree four?", "kind": "definition" } ] } ] } ] }, { "id": "applications", "name": "Applications", "description": "Uses.", "rationale": "Polynomials model data.", "layers": [ { "id": "fitting", "name": "Fitting", "description": "Interpolation.", "findings": [ { "id": "fitting-finding", "name": "Fitting", "description": "Interpolation and fitting.", "questions": [ { "text": "How can a polynomial be fitted to these data points?", "kind": "action" }, { "text": "Is there a risk of overfitting?", "kind": "boundary" } ] } ] }, { "id": "computing", "name": "Computing", "description": "Algorithms.", "findings": [ { "id": "computing-finding", "name": "Computing", "description": "Algorithms.", "questions": [ { "text": "How are polynomials evaluated efficiently, such as by Horner method?", "kind": "definition" }, { "text": "Which libraries help?", "kind": "provenance" } ] } ] } ] }, { "id": "learning", "name": "Learning", "description": "Teaching.", "rationale": "Learners need practice.", "layers": [ { "id": "explain", "name": "Explain", "description": "Explaining.", "findings": [ { "id": "explain-finding", "name": "Explain", "description": "Explanation.", "questions": [ { "text": "How can factorising be explained step by step?", "kind": "action" }, { "text": "Which misconceptions are common?", "kind": "provenance" } ] } ] }, { "id": "graphs", "name": "Graphs", "description": "Graphing.", "findings": [ { "id": "graphs-finding", "name": "Graphs", "description": "Graphs.", "questions": [ { "text": "How does the degree affect the shape of the graph?", "kind": "definition" }, { "text": "Where are the turning points?", "kind": "measurement" } ] } ] } ] } ] }, "openQuestions": [ "Should polynomial families be separate entries?", "How should computer algebra tools be linked?", "How should teaching materials be linked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a polynomial, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain polynomials by degree, operations, roots, factorisation and applications, for learners and practitioners.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: invariant polynomial, Hilbert polynomial, trinomial, monomial, binomial, knot polynomial, matching polynomial, characteristic polynomial, constant polynomial, Conway polynomial, Conway–Alexander polynomial, Vandermonde polynomial.", "113 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 91 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a polynomial, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "invariant polynomial", "Hilbert polynomial", "trinomial", "monomial", "binomial", "knot polynomial", "matching polynomial", "characteristic polynomial", "constant polynomial", "Conway polynomial", "Conway–Alexander polynomial", "Vandermonde polynomial" ] } ], "standing": "Described in 91 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 113 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one polynomial from another, and a polynomial from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a polynomial, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a polynomial, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a polynomial is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a polynomial, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a polynomial, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a polynomial is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a polynomial sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a polynomial, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a polynomial, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a polynomial is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a polynomial from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a polynomial, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a polynomial, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a polynomial is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a polynomial passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a polynomial between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a polynomial, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a polynomial, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a polynomial is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a polynomial, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a polynomial is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a polynomial, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a polynomial, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a polynomial is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What polynomial is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of polynomial are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is polynomial operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of polynomial and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of polynomial?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as polynomial?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a polynomial came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a polynomial and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a polynomial, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a polynomial, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a polynomial is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a polynomial is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a polynomial, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a polynomial, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a polynomial is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }