{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.polynomial","code":"thing-q43260","url":"https://ver.cy/models/thing/q43260/","name":"polynomial","alternateNames":["invariant polynomial","Hilbert polynomial","trinomial","monomial","binomial","knot polynomial","matching polynomial","characteristic polynomial","constant polynomial","Conway polynomial","Conway–Alexander polynomial","Vandermonde polynomial"],"kind":"thing","status":"published","version":"0.2.0-wave.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A mathematical expression built from variables and coefficients using addition, subtraction, multiplication and non-negative integer powers, such as monomials, binomials and trinomials; the degree of a nonzero polynomial is the greatest total degree of its monomials with nonzero coefficients, where total degree is the sum of the exponents of the variables in a monomial, reducing to the highest exponent in the univariate case, and by the fundamental theorem of algebra a nonconstant univariate polynomial with complex coefficients has as many complex roots as its degree, counted with multiplicity.","purpose":"Let an agent explain polynomials by degree, operations, roots, factorisation and applications, for learners and practitioners.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Non-negative integer exponents","Has a degree","Roots determined by degree","Closed under addition and multiplication"],"structure":{"bundles":[{"id":"basics","name":"Basics","description":"Definitions.","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.","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.","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.","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"}]}]}]}]},"agentConduct":{"may":["Manipulate, factor and evaluate polynomials and find their roots.","Explain degree, roots and the fundamental theorem of algebra."],"mustNot":["Fit high-degree polynomials to data without warning of instability and overfitting.","Report roots without stating the numerical precision used.","Extrapolate a polynomial fit far outside the data as if it were reliable."],"requiresHuman":["Using a polynomial model in engineering or financial decisions that affect people."]},"ethics":{"considerations":["Overfitted models give false confidence in predictions that people rely on."],"affectedParties":["Students and learners","People affected by model-based decisions"]},"owners":{"steward":"Nobody owns the concept; it is held in common by mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q218005","type":"parent","note":"registry parent class"},{"target":"Q3964627","type":"parent","note":"registry parent class"},{"target":"algebraic expression","type":"related","note":"category"},{"target":"root","type":"related","note":"property"},{"target":"algebra","type":"related","note":"field"},{"target":"invariant","type":"related","note":"related concept"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.polynomial","Wikidata: Q43260 (https://www.wikidata.org/wiki/Q43260)"]},"properties":{"applicability":"not-applicable","items":[]},"recognition":{"applicability":"optional","items":["Sums of terms with whole-number powers","Graphs such as parabolas and cubics","Expressions with negative or fractional powers are not polynomials","Not physical; expressions such as 3x^2 + 2x - 5 and their graphs."]},"capabilities":{"applicability":"required","items":["add, multiply and divide polynomials","find roots and factorise","use polynomials for interpolation and approximation","explain key theorems"]},"hazards":{"applicability":"required","items":["Numerical instability with high-degree fitting","Sign and exponent errors"]},"interfaces":{"applicability":"required","items":["No specific regulation"]},"context":{"applicability":"required","items":["Algebra, modelling, approximation and computation.","monomials, binomials and trinomials","linear, quadratic and cubic polynomials","multivariate polynomials","orthogonal polynomials","invariant and knot polynomials"]}},"sources":[{"title":"Wikidata item Q43260: polynomial","url":"https://www.wikidata.org/wiki/Q43260","note":"identity and sense of the item"},{"title":"Wikipedia: Polynomial","url":"https://en.wikipedia.org/wiki/Polynomial","note":"general description of the item"}],"openQuestions":["Should polynomial families be separate entries?","How should computer algebra tools be linked?","How should teaching materials be linked?"],"resources":{"spec":"/models/things/publications/thing-q43260/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q43260/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-11T12:34:42Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:31+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}