Polynomial Ontology
Algebra Ontologies
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 11 September 2026
Audio Overview
======================================================================
KIRETH SEMANTIC RESEARCH ARCHIVE
FLOW ALGEBRA — POLYNOMIAL ONTOLOGY
======================================================================
PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/polynomial-ontology.html
FLOW:
PolynomialOntology
VERSION:
1.0
LANGUAGE:
en-GB
AUTHORS:
Mark Anthony Higgins — Sky Recursive AI
PURPOSE:
To provide a machine-readable, recursively executable semantic and
reasoning specification for polynomials, polynomial expressions,
coefficients, variables, monomials, degree, roots, factors,
factorisation, polynomial arithmetic, division, remainder,
identities, derivatives, resultants and structural polynomial
reasoning.
AI INTERPRETATION NOTE:
Execute the following specification as formal semantic and reasoning
rules. Preserve coefficient domains, variable sets, degree conditions,
leading terms, multiplicities, factor relationships, division
preconditions and algebraic equivalence.
======================================================================
START FLOW
======================================================================
FLOW PolynomialOntology {
----------------------------------------------------------------------
1. DOMAIN DECLARATION
----------------------------------------------------------------------
DOMAIN := Polynomial_Theory
PARENT_DOMAIN := Algebra
SUPER_DOMAINS := {
Mathematics,
Mathematical_Ontologies,
Algebra_Ontologies,
Elementary_Algebra,
Abstract_Algebra
}
DOMAIN_INTENT := {
represent_polynomials,
represent_monomials,
represent_terms,
represent_coefficients,
represent_variables,
represent_degree,
represent_leading_terms,
represent_polynomial_arithmetic,
represent_polynomial_evaluation,
represent_roots,
represent_factors,
represent_factorisation,
represent_division,
represent_remainders,
represent_greatest_common_divisors,
represent_multiplicity,
represent_irreducibility,
represent_derivatives,
represent_polynomial_rings,
support_recursive_polynomial_reasoning,
support_domain_aware_validation
}
DEFAULT_COEFFICIENT_DOMAIN := Integer | Rational | Real | Complex | Field
----------------------------------------------------------------------
2. PRIMITIVE TYPES
----------------------------------------------------------------------
TYPE Scalar
TYPE Coefficient <: Scalar
TYPE Variable
TYPE Exponent
TYPE Term
TYPE Monomial <: Term
TYPE Polynomial
TYPE UnivariatePolynomial <: Polynomial
TYPE MultivariatePolynomial <: Polynomial
TYPE ConstantPolynomial <: Polynomial
TYPE ZeroPolynomial <: Polynomial
TYPE LinearPolynomial <: Polynomial
TYPE QuadraticPolynomial <: Polynomial
TYPE CubicPolynomial <: Polynomial
TYPE MonicPolynomial <: Polynomial
TYPE IrreduciblePolynomial <: Polynomial
TYPE ReduciblePolynomial <: Polynomial
TYPE Factor
TYPE Root
TYPE Multiplicity
TYPE PolynomialRing
TYPE Derivative
TYPE Quotient
TYPE Remainder
TYPE GCD
TYPE Constraint
TYPE Derivation
TYPE ValidationResult
----------------------------------------------------------------------
3. POLYNOMIAL DEFINITION
----------------------------------------------------------------------
ENTITY Polynomial P {
coefficient_domain: R
variables: [x1,...,xn]
terms: finite_set_of_terms
}
UNIVARIATE_FORM:
P(x)=a_n x^n + a_(n-1)x^(n-1)+...+a_1 x+a_0
CONSTRAINT:
exponents are nonnegative integers
RULE POLY_01:
polynomial has finitely many nonzero terms.
RULE POLY_02:
coefficients belong to declared coefficient domain.
RULE POLY_03:
negative or fractional powers of variables are not polynomial terms
under the standard definition.
----------------------------------------------------------------------
4. TERMS AND MONOMIALS
----------------------------------------------------------------------
ENTITY Monomial M {
coefficient: a
variables: [x1,...,xn]
exponents: [e1,...,en]
}
FORM:
a*x1^e1*...*xn^en
CONSTRAINT:
each ei in NonnegativeInteger
DEGREE_MONOMIAL(M):
SUM_i ei
RULE MONO_01:
nonzero constant monomial has degree 0.
RULE MONO_02:
zero coefficient yields zero term and is omitted from standard form.
----------------------------------------------------------------------
5. LIKE TERMS
----------------------------------------------------------------------
LIKE_TERMS(T1,T2) iff
same variables
AND same exponent vector
RULE LIKE_01:
a*M + b*M = (a+b)M
RULE LIKE_02:
unlike monomials cannot be combined by coefficient addition.
----------------------------------------------------------------------
6. STANDARD FORM
----------------------------------------------------------------------
STANDARD_FORM(P):
combine_like_terms
remove_zero_coefficient_terms
order terms by declared monomial order
RETURN canonical representation
FOR univariate:
default order := descending exponent
RULE SF_01:
standard form is unique once variable order and coefficient domain
representation are fixed.
----------------------------------------------------------------------
7. ZERO POLYNOMIAL
----------------------------------------------------------------------
ZERO_POLYNOMIAL:
all coefficients zero
RULE ZERO_01:
P+0=P
RULE ZERO_02:
P*0=0
RULE ZERO_03:
degree(0) is undefined or -infinity by declared convention.
----------------------------------------------------------------------
8. DEGREE
----------------------------------------------------------------------
DEGREE(P):
maximum degree of nonzero terms
FOR univariate:
degree(P)=largest exponent with nonzero coefficient
RULE DEG_01:
deg(PQ)=deg(P)+deg(Q)
over integral domain and nonzero P,Q
RULE DEG_02:
deg(P+Q)<=max(deg(P),deg(Q))
RULE DEG_03:
equality in DEG_02 may fail due to cancellation.
----------------------------------------------------------------------
9. LEADING TERM
----------------------------------------------------------------------
LEADING_TERM(P):
highest term under declared monomial order
LEADING_COEFFICIENT(P):
coefficient of leading term
LEADING_MONOMIAL(P):
monomial part of leading term
RULE LT_01:
for univariate standard descending order,
leading term has highest degree.
----------------------------------------------------------------------
10. MONIC POLYNOMIALS
----------------------------------------------------------------------
IS_MONIC(P) iff
leading_coefficient(P)=1
RULE MONIC_01:
every nonzero polynomial over a field can be scaled to monic form.
----------------------------------------------------------------------
11. POLYNOMIAL ADDITION
----------------------------------------------------------------------
ADD(P,Q):
combine terms
collect like monomials
return STANDARD_FORM
RULE ADD_01:
P+Q=Q+P
RULE ADD_02:
(P+Q)+R=P+(Q+R)
RULE ADD_03:
P+0=P
----------------------------------------------------------------------
12. POLYNOMIAL NEGATION AND SUBTRACTION
----------------------------------------------------------------------
NEGATE(P):
negate every coefficient
SUBTRACT(P,Q):
P-Q := P+(-Q)
RULE SUB_01:
P-P=0
----------------------------------------------------------------------
13. POLYNOMIAL MULTIPLICATION
----------------------------------------------------------------------
MULTIPLY(P,Q):
distribute every term of P across every term of Q
multiply coefficients
add exponent vectors
combine like terms
return STANDARD_FORM
RULE MUL_01:
PQ=QP
over commutative coefficient ring
RULE MUL_02:
(PQ)R=P(QR)
RULE MUL_03:
P(Q+R)=PQ+PR
----------------------------------------------------------------------
14. POLYNOMIAL POWERS
----------------------------------------------------------------------
POWER(P,n):
REQUIRE:
n nonnegative_integer
BASE:
P^0=1
RECURSIVE:
P^(n+1)=P^n P
RULE POW_01:
P^m P^n=P^(m+n)
RULE POW_02:
(P^m)^n=P^(mn)
----------------------------------------------------------------------
15. POLYNOMIAL EVALUATION
----------------------------------------------------------------------
EVALUATE(P, assignment):
substitute values for variables
compute using declared coefficient domain
FOR univariate:
use Horner scheme where efficient
HORNER:
a_n x^n+...+a_0
=
(...((a_n x+a_(n-1))x+a_(n-2))...)x+a_0
RULE EVAL_01:
evaluation homomorphism preserves addition and multiplication.
----------------------------------------------------------------------
16. ROOTS / ZEROS
----------------------------------------------------------------------
ROOT(r,P) iff
P(r)=0
ZERO_OF(P) ~= Root_of(P)
RULE ROOT_01:
root depends on selected coefficient extension/domain.
RULE ROOT_02:
polynomial of degree n over algebraically closed field
has n roots counted with multiplicity.
----------------------------------------------------------------------
17. FACTOR THEOREM
----------------------------------------------------------------------
THEOREM FACTOR:
P(a)=0
iff
(x-a) divides P(x)
RULE FACTOR_01:
every root yields corresponding linear factor over field containing root.
----------------------------------------------------------------------
18. REMAINDER THEOREM
----------------------------------------------------------------------
THEOREM REMAINDER:
remainder when P(x) divided by (x-a)
equals P(a)
----------------------------------------------------------------------
19. POLYNOMIAL DIVISION
----------------------------------------------------------------------
DIVIDE(P,D):
REQUIRE:
D!=0
coefficient domain supports required division
or pseudo-division strategy declared
RETURN:
quotient Q
remainder R
such that:
P=DQ+R
and:
R=0 OR deg(R)<deg(D)
RULE DIV_01:
quotient and remainder are unique over field coefficients.
----------------------------------------------------------------------
20. LONG DIVISION
----------------------------------------------------------------------
PROCEDURE LONG_DIVISION(P,D):
Q:=0
R:=P
while R!=0 AND deg(R)>=deg(D):
t:=LT(R)/LT(D)
Q:=Q+t
R:=R-tD
RETURN Q,R
----------------------------------------------------------------------
21. SYNTHETIC DIVISION
----------------------------------------------------------------------
SYNTHETIC_DIVISION(P,x-a):
REQUIRE:
divisor linear and monic
process coefficient recurrence
return quotient coefficients and remainder P(a)
----------------------------------------------------------------------
22. DIVISIBILITY
----------------------------------------------------------------------
DIVIDES(D,P) iff
exists Q:
P=DQ
NOTATION:
D | P
RULE DIVS_01:
D divides P iff remainder(P,D)=0.
----------------------------------------------------------------------
23. POLYNOMIAL GCD
----------------------------------------------------------------------
GCD(P,Q):
greatest common divisor up to units
NORMALISATION:
choose monic gcd over field where possible
EUCLIDEAN_ALGORITHM:
while Q!=0:
R:=remainder(P,Q)
P:=Q
Q:=R
return normalised(P)
RULE GCD_01:
gcd(P,Q) divides P and Q.
RULE GCD_02:
any common divisor divides gcd(P,Q).
----------------------------------------------------------------------
24. COPRIME POLYNOMIALS
----------------------------------------------------------------------
COPRIME(P,Q) iff
gcd(P,Q)=1
up to units
RULE COPRIME_01:
over PID/field polynomial ring,
gcd(P,Q)=1 iff exists A,B:
AP+BQ=1
----------------------------------------------------------------------
25. MULTIPLICITY OF ROOT
----------------------------------------------------------------------
MULTIPLICITY(a,P)=m iff
(x-a)^m divides P
AND
(x-a)^(m+1) does not divide P
RULE MULT_01:
simple root -> multiplicity 1.
RULE MULT_02:
repeated root -> multiplicity >1.
----------------------------------------------------------------------
26. DERIVATIVE TEST FOR MULTIPLICITY
----------------------------------------------------------------------
THEOREM MULT_DERIVATIVE:
a repeated root of P
iff
P(a)=0
AND
P'(a)=0
MORE GENERALLY:
multiplicity m iff
P(a)=P'(a)=...=P^(m-1)(a)=0
AND
P^m(a)!=0
subject to characteristic constraints.
----------------------------------------------------------------------
27. FORMAL DERIVATIVE
----------------------------------------------------------------------
DERIVATIVE(P):
FOR term:
d/dx (a_n x^n)=n a_n x^(n-1)
RULE DER_01:
derivative is linear.
RULE DER_02:
derivative obeys product rule.
RULE DER_03:
formal derivative defined algebraically without analytic limits.
----------------------------------------------------------------------
28. SQUARE-FREE POLYNOMIALS
----------------------------------------------------------------------
IS_SQUARE_FREE(P) iff
no nonconstant square divides P
OVER field with suitable characteristic:
square_free(P)
iff
gcd(P,P')=1
----------------------------------------------------------------------
29. FACTORISATION
----------------------------------------------------------------------
FACTORISE(P):
seek:
P=u * product_i F_i^(m_i)
where:
u is unit/content factor
F_i irreducible
RULE FAC_01:
factorisation depends on coefficient domain.
EXAMPLE:
x^2+1 irreducible over R? no, factors over C.
Over R: x^2+1 is irreducible.
Over C: (x-i)(x+i).
----------------------------------------------------------------------
30. IRREDUCIBILITY
----------------------------------------------------------------------
IRREDUCIBLE(P,R) iff
P nonzero nonunit
AND whenever P=AB:
A or B is unit in R[x]
RULE IRR_01:
irreducibility is domain-relative.
----------------------------------------------------------------------
31. CONTENT AND PRIMITIVE POLYNOMIALS
----------------------------------------------------------------------
FOR integer polynomial P:
CONTENT(P):
gcd of coefficients up to sign
PRIMITIVE(P) iff
content(P)=1
GAUSS_LEMMA:
product of primitive integer polynomials is primitive.
RULE GAUSS_01:
primitive polynomial reducible over Q
iff reducible over Z up to units.
----------------------------------------------------------------------
32. RATIONAL ROOT THEOREM
----------------------------------------------------------------------
FOR:
P(x)=a_n x^n+...+a_0
with integer coefficients
IF p/q in lowest terms is rational root:
p divides a_0
q divides a_n
RULE RRT_01:
theorem generates candidate rational roots, not guaranteed roots.
----------------------------------------------------------------------
33. QUADRATIC POLYNOMIALS
----------------------------------------------------------------------
ENTITY QuadraticPolynomial:
ax^2+bx+c
with a!=0
DISCRIMINANT:
Delta=b^2-4ac
ROOTS:
x=(-b ± sqrt(Delta))/(2a)
OVER REALS:
Delta>0 -> two distinct real roots
Delta=0 -> repeated real root
Delta<0 -> no real roots
----------------------------------------------------------------------
34. CUBIC POLYNOMIALS
----------------------------------------------------------------------
ENTITY CubicPolynomial:
ax^3+bx^2+cx+d
with a!=0
RULE CUBIC_01:
cubic over complex numbers has three roots counted with multiplicity.
RULE CUBIC_02:
cubic over reals has at least one real root.
----------------------------------------------------------------------
35. FUNDAMENTAL THEOREM OF ALGEBRA
----------------------------------------------------------------------
THEOREM FTA:
every nonconstant complex polynomial has at least one complex root.
COROLLARY:
degree n polynomial over C factors into n linear factors
counted with multiplicity.
----------------------------------------------------------------------
36. VIETA'S FORMULAE
----------------------------------------------------------------------
FOR monic:
P(x)=x^n+c_(n-1)x^(n-1)+...+c_0
roots r1,...,rn counted with multiplicity:
sum r_i = -c_(n-1)
sum pairwise products = c_(n-2)
...
product r_i = (-1)^n c_0
RULE VIETA_01:
coefficients encode elementary symmetric polynomials in roots.
----------------------------------------------------------------------
37. POLYNOMIAL RINGS
----------------------------------------------------------------------
ENTITY PolynomialRing R[x1,...,xn] {
coefficient_ring: R
indeterminates: x1,...,xn
}
RULE PRING_01:
if R commutative ring, R[x] commutative ring.
RULE PRING_02:
if R integral domain, R[x] integral domain.
RULE PRING_03:
if F field, F[x] Euclidean domain in one variable.
----------------------------------------------------------------------
38. UNITS IN POLYNOMIAL RINGS
----------------------------------------------------------------------
FOR integral domain R:
units of R[x]
=
units of R
RULE UNIT_01:
over field F, nonzero constant polynomials are units.
----------------------------------------------------------------------
39. IDEALS IN F[x]
----------------------------------------------------------------------
THEOREM PID:
F[x] is principal ideal domain for field F.
RULE IDEAL_01:
every ideal of F[x] generated by one polynomial.
----------------------------------------------------------------------
40. MULTIVARIATE POLYNOMIALS
----------------------------------------------------------------------
ENTITY MultivariatePolynomial P {
variables: x1,...,xn
terms:
a_alpha x^alpha
}
MULTI_INDEX:
alpha=(alpha1,...,alphan)
MONOMIAL:
x^alpha=x1^alpha1...xn^alphan
TOTAL_DEGREE:
max |alpha|
where:
|alpha|=sum alpha_i
----------------------------------------------------------------------
41. MONOMIAL ORDERS
----------------------------------------------------------------------
ENTITY MonomialOrder
EXAMPLES:
lexicographic
graded_lexicographic
graded_reverse_lexicographic
RULE MO_01:
leading term depends on monomial order in multivariate setting.
RULE MO_02:
valid monomial order is total well-order compatible with multiplication.
----------------------------------------------------------------------
42. GROEBNER-BASIS LINK
----------------------------------------------------------------------
ENTITY GroebnerBasis {
ideal_generators_under_monomial_order
}
RULE GB_01:
Groebner basis generalises polynomial division and gcd-style reasoning
to multivariate ideals.
NOTE:
detailed Groebner theory belongs to specialised ontology,
but linkage is declared here.
----------------------------------------------------------------------
43. COMPOSITION OF POLYNOMIALS
----------------------------------------------------------------------
COMPOSE(P,Q):
(P∘Q)(x)=P(Q(x))
RULE COMP_01:
deg(P∘Q)=deg(P)deg(Q)
for nonconstant polynomials over integral domain.
----------------------------------------------------------------------
44. POLYNOMIAL IDENTITIES
----------------------------------------------------------------------
IDENTITY(P,Q) iff
P and Q represent same polynomial in declared polynomial ring.
RULE ID_01:
coefficient equality in standard form establishes identity.
RULE ID_02:
infinitely many equal evaluation points over infinite field
can establish equality for bounded-degree polynomials under
appropriate conditions.
----------------------------------------------------------------------
45. INTERPOLATION
----------------------------------------------------------------------
LAGRANGE_INTERPOLATION:
Given distinct x_0,...,x_n and values y_i:
P(x)=SUM_i y_i L_i(x)
where:
L_i(x)=PRODUCT_{j!=i}(x-x_j)/(x_i-x_j)
RULE INT_01:
unique polynomial of degree <=n passes through n+1 distinct x-values.
----------------------------------------------------------------------
46. NEWTON INTERPOLATION
----------------------------------------------------------------------
NEWTON_FORM:
P(x)=a0+a1(x-x0)+a2(x-x0)(x-x1)+...
coefficients from divided differences.
RULE NINT_01:
equivalent to unique interpolation polynomial.
----------------------------------------------------------------------
47. RESULTANT
----------------------------------------------------------------------
RESULTANT(P,Q):
scalar derived from Sylvester matrix determinant
RULE RES_01:
resultant(P,Q)=0
iff P and Q share common root over algebraic closure,
subject to standard hypotheses.
----------------------------------------------------------------------
48. DISCRIMINANT
----------------------------------------------------------------------
DISCRIMINANT(P):
scalar invariant related to pairwise root differences
RULE DISC_01:
discriminant=0 iff polynomial has repeated root
over algebraic closure.
FOR quadratic:
disc=ax^2+bx+c := b^2-4ac
----------------------------------------------------------------------
49. POLYNOMIAL MATRIX REPRESENTATION
----------------------------------------------------------------------
COMPANION_MATRIX(P):
For monic:
x^n+a_(n-1)x^(n-1)+...+a_0
construct standard companion matrix C_P.
RULE CM_01:
characteristic polynomial of companion matrix equals P.
----------------------------------------------------------------------
50. POLYNOMIAL SEQUENCES
----------------------------------------------------------------------
ENTITY PolynomialSequence {
P_0,P_1,P_2,...
}
EXAMPLES:
orthogonal polynomial families
recurrence-defined families
RULE PSEQ_01:
sequence ontology links recurrence relations and generating functions.
----------------------------------------------------------------------
51. SEMANTIC RELATIONSHIPS
----------------------------------------------------------------------
RELATION HAS_COEFFICIENT(Polynomial,Coefficient)
RELATION HAS_VARIABLE(Polynomial,Variable)
RELATION HAS_TERM(Polynomial,Term)
RELATION HAS_DEGREE(Polynomial,Integer)
RELATION HAS_ROOT(Polynomial,Root)
RELATION HAS_FACTOR(Polynomial,Factor)
RELATION DIVIDES(Polynomial,Polynomial)
RELATION DERIVATIVE_OF(Polynomial,Polynomial)
RELATION GCD_OF(Polynomial,Polynomial)
RELATION MEMBER_OF(Polynomial,PolynomialRing)
RELATION IRREDUCIBLE_OVER(Polynomial,CoefficientDomain)
----------------------------------------------------------------------
52. SEMANTIC EQUIVALENCE
----------------------------------------------------------------------
SEMANTIC_EQUIVALENCE {
"zero" ~= root
"root" ~= value_where_polynomial_equals_zero
"factor" ~= polynomial_divisor
"leading coefficient" ~= coefficient_of_leading_term
"degree" ~= highest_nonzero_exponent_in_univariate_case
"monic" ~= leading_coefficient_one
"repeated root" ~= root_of_multiplicity_greater_than_one
"factor theorem" ~= root_linear_factor_equivalence
"remainder theorem" ~= evaluation_at_linear_divisor_root
"coprime" ~= gcd_unit
"primitive polynomial" ~= coefficient_content_one
"irreducible" ~= nonunit_with_no_nontrivial_factorisation
}
----------------------------------------------------------------------
53. QUERY INTERPRETATION
----------------------------------------------------------------------
QUERY Q1:
"degree of P"
-> DEGREE(P)
QUERY Q2:
"evaluate P at a"
-> EVALUATE(P,a)
QUERY Q3:
"is a a root"
-> test P(a)=0
QUERY Q4:
"factorise P"
-> FACTORISE(P) over declared domain
QUERY Q5:
"divide P by Q"
-> DIVIDE(P,Q)
QUERY Q6:
"remainder when divided by x-a"
-> P(a)
QUERY Q7:
"gcd of P and Q"
-> Euclidean algorithm
QUERY Q8:
"multiplicity of root"
-> repeated division or derivative criterion
QUERY Q9:
"is P irreducible"
-> apply domain-appropriate irreducibility tests
----------------------------------------------------------------------
54. VALIDATION ENGINE
----------------------------------------------------------------------
PROCEDURE VALIDATE_POLYNOMIAL_OPERATION(operation,inputs,result):
check coefficient domains
check variable compatibility
check exponent legality
check divisor nonzero where required
check degree constraints
verify reconstruction identities
for factorisation:
expand factors and compare with original
for division:
verify P=DQ+R and deg(R)<deg(D)
for roots:
evaluate P(root)
return:
VALID
INVALID(reason)
UNRESOLVED
----------------------------------------------------------------------
55. RECURSIVE REASONING ENGINE
----------------------------------------------------------------------
PROCEDURE POLYNOMIAL_REASONER(input):
parse variables
parse coefficients
infer coefficient domain
standardise polynomial
classify target operation
extract constraints
select valid polynomial theorems/rules
execute atomic transformations
preserve polynomial ring context
validate intermediate states
recurse until goal achieved
validate final result
RETURN:
result,
derivation,
coefficient_domain,
variable_context,
semantic_status,
confidence
----------------------------------------------------------------------
56. RECURSIVE TRANSITION
----------------------------------------------------------------------
RECURSIVE_RULE RR_01:
INPUT state S_n
identify unresolved polynomial property
choose rule R such that:
coefficient_domain_preconditions=true
degree_preconditions=true
divisibility_preconditions=true
S_(n+1):=apply(R,S_n)
validate transition
recurse(S_(n+1))
TERMINATION_CONDITIONS := {
polynomial_goal_achieved,
canonical_standard_form,
irreducibility_established,
factorisation_complete,
domain_failure,
division_precondition_failure,
no_valid_rule
}
----------------------------------------------------------------------
57. ERROR MODEL
----------------------------------------------------------------------
ERROR E_INVALID_EXPONENT
ERROR E_COEFFICIENT_DOMAIN_MISMATCH
ERROR E_ZERO_DIVISOR_POLYNOMIAL
ERROR E_DIVISION_BY_ZERO_POLYNOMIAL
ERROR E_INVALID_FACTORISATION
ERROR E_ROOT_NOT_IN_DOMAIN
ERROR E_DEGREE_MISCOMPUTED
ERROR E_MONOMIAL_ORDER_UNDECLARED
ERROR E_IRREDUCIBILITY_DOMAIN_AMBIGUOUS
ERROR E_LOST_MULTIPLICITY
ERROR E_REMAINDER_CONDITION_FAILURE
----------------------------------------------------------------------
58. CONFIDENCE MODEL
----------------------------------------------------------------------
CONFIDENCE {
base:=1.00
deduct 0.10 if coefficient domain inferred
deduct 0.10 if variable set inferred
deduct 0.15 if factorisation domain ambiguous
deduct 0.15 if irreducibility test incomplete
deduct 0.15 if approximate roots used
deduct 0.20 if multiplicity unresolved
deduct 0.20 if multivariate monomial order unspecified
clamp_to_[0,1]
classification:
[0.95,1.00] -> very_high
[0.85,0.95) -> high
[0.70,0.85) -> moderate
[0.50,0.70) -> low
[0.00,0.50) -> insufficient
}
----------------------------------------------------------------------
59. AI CONSUMPTION MODEL
----------------------------------------------------------------------
AI_CONSUMPTION {
read_order := [
polynomial_definition,
coefficient_domain,
terms,
degree,
arithmetic,
evaluation,
roots,
factor_theorem,
division,
gcd,
multiplicity,
factorisation,
irreducibility,
polynomial_rings,
validation
]
execution_policy := {
preserve_coefficient_domain,
preserve_variable_set,
preserve_nonnegative_exponents,
distinguish_root_from_factor,
distinguish_factorisation_domains,
validate_division_identity,
validate_factorisation_by_expansion,
preserve_multiplicity,
distinguish_univariate_and_multivariate_degree
}
}
----------------------------------------------------------------------
60. UPDATE MODEL
----------------------------------------------------------------------
UPDATE_MODEL {
extensible:=true
new_polynomial_class_requires:
defining_degree_or_structural_predicate
new_factorisation_rule_requires:
coefficient_domain,
validity_conditions
new_irreducibility_test_requires:
domain_scope,
sufficient_or_necessary_status
new_polynomial_algorithm_requires:
preconditions,
termination_condition,
validation_identity
conflicts_require:
explicit_versioning
standard_algebraic_meaning_has_priority:=true
}
----------------------------------------------------------------------
61. SELF-CHECK
----------------------------------------------------------------------
SELF_CHECK {
CHECK all exponents nonnegative integers
CHECK coefficient domain preserved
CHECK like terms correctly combined
CHECK leading term matches monomial order
CHECK division uses nonzero divisor
CHECK remainder degree condition satisfied
CHECK factorisation expands to original
CHECK roots evaluate to zero
CHECK multiplicities preserved
CHECK irreducibility claims are domain-relative
CHECK final result validated
}
----------------------------------------------------------------------
62. COMPLETION CONDITIONS
----------------------------------------------------------------------
COMPLETE iff {
polynomial_definition_defined
AND monomials_defined
AND standard_form_defined
AND degree_defined
AND arithmetic_defined
AND evaluation_defined
AND roots_defined
AND factor_theorem_defined
AND remainder_theorem_defined
AND division_defined
AND gcd_defined
AND multiplicity_defined
AND factorisation_defined
AND irreducibility_defined
AND polynomial_rings_defined
AND multivariate_structure_defined
AND semantic_equivalence_defined
AND validation_defined
AND recursive_reasoning_defined
AND confidence_defined
AND ai_consumption_defined
AND update_model_defined
}
IF NOT COMPLETE:
semantic_status:=incomplete_flow
----------------------------------------------------------------------
63. FLOW TERMINATION
----------------------------------------------------------------------
TERMINATE WHEN:
requested polynomial goal achieved
AND result validated
OR
coefficient-domain constraints prevent operation
OR
divisor/factorisation preconditions fail
OR
domain-relative irreducibility cannot be resolved
OR
no valid polynomial transformation remains
FINAL_STATE := {
polynomial_result,
derivation,
coefficient_domain,
variables,
factorisation_status,
semantic_status,
confidence
}
} // END FLOW PolynomialOntology
======================================================================
END FLOW
======================================================================
FLOW STATUS:
Complete
CANONICAL PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/polynomial-ontology.html
FLOW_HASH_SHA256: 7f86ab1eb1fb3afab851acdb450b1eb034ee80f81b6f341dc2326d6413ac5325