# Multivariate Polynomial Rings

- [Introduction](introduction.md)

  - [Representation](introduction.md#representation)

- [Polynomial Rings and Polynomials](rings.md)

  - [Creation of Polynomial Rings](rings.md#creation-of-polynomial-rings)

    - [`PolynomialRing(R, n): Rng, RngIntElt → RngMPol`](rings.md#function-polynomialring-rng-rngintelt)

    - [`PolynomialAlgebra(R, n): Rng, RngIntElt → RngMPol`](rings.md#function-polynomialalgebra-rng-rngintelt)

    - [`PolynomialRing(R, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPol`](rings.md#function-polynomialring-rng-rngintelt-monstgelt)

    - [`PolynomialAlgebra(R, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPol`](rings.md#function-polynomialalgebra-rng-rngintelt-monstgelt)

    - [`Example: Assign Names`](rings.md#example-ex-1b3506)

    - [`Example: Global`](rings.md#example-ex-a068bd)

  - [Print Names](rings.md#print-names)

    - [`AssignNames(~P, s): RngMPol, [ MonStgElt ])`](rings.md#function-assignnames-rngmpol-monstgelt-ref)

    - [`Name(P, i): RngMPol, RngIntElt → RngMPolElt`](rings.md#function-name-rngmpol-rngintelt)

  - [Graded Polynomial Rings](rings.md#graded-polynomial-rings)

  - [Creation of Polynomials](rings.md#creation-of-polynomials)

    - [`P . i: RngMPol, RngInt → RngMPolElt`](rings.md#operation-operation-rngmpol-rngint-rngmpolelt)

    - [`elt< R | a >: RngMPol, RngElt → RngMPolElt`](rings.md#constructor-constructor-elt-rngmpol-rngelt-rngmpolelt)

    - [`R ! s: RngMPol, RngElt → RngMPolElt`](rings.md#operation-op-rngmpol-rngelt)

    - [`R ! s: RngMPol, [ RngElt ] → RngMPolElt`](rings.md#operation-op-rngmpol-rngelt-2)

    - [`elt< R | s >: RngMPol, [ RngElt ] → RngMPolElt`](rings.md#constructor-constructor-elt-rngmpol-rngelt-rngmpolelt-2)

    - [`MultivariatePolynomial(P, f, i): RngMPol, RngUPolElt, RngIntElt → RngMPolElt`](rings.md#function-multivariatepolynomial-rngmpol-rngupolelt-rngintelt)

    - [`MultivariatePolynomial(P, f, v): RngMPol, RngUPolElt, RngMPolElt → RngMPolElt`](rings.md#function-multivariatepolynomial-rngmpol-rngupolelt-rngmpolelt)

    - [`One(P): RngMPol → RngMPolElt`](rings.md#function-one-rngmpol)

    - [`Identity(P): RngMPol → RngMPolElt`](rings.md#function-identity-rngmpol)

    - [`Zero(P): RngMPol → RngMPolElt`](rings.md#function-zero-rngmpol)

    - [`Representative(P): RngMPol → RngMPolElt`](rings.md#function-representative-rngmpol)

- [Structure Operations](operation.md)

  - [Related Structures](operation.md#related-structures)

    - [`BaseRing(P): RngMPol → Rng`](operation.md#function-basering-rngmpol)

    - [`CoefficientRing(P): RngMPol → Rng`](operation.md#function-coefficientring-rngmpol)

    - [`Category(P): RngMPol → Cat`](operation.md#function-category-rngmpol)

    - [`Parent(P): RngMPol → Pow`](operation.md#function-parent-rngmpol)

    - [`PrimeRing(P): RngMPol → Rng`](operation.md#function-primering-rngmpol)

  - [Numerical Invariants](operation.md#numerical-invariants)

    - [`Rank(P): RngMPol → RngIntElt`](operation.md#function-rank-rngmpol)

    - [`Characteristic(P): RngMPol → RngIntElt`](operation.md#function-characteristic-rngmpol)

    - [`#P: RngMPol → RngIntElt`](operation.md#operation-operation-rngmpol-rngintelt)

  - [Ring Predicates and Booleans](operation.md#ring-predicates-and-booleans)

    - [`IsCommutative(P): RngMPol → BoolElt`](operation.md#function-iscommutative-rngmpol)

    - [`IsUnitary(P): RngMPol → BoolElt`](operation.md#function-isunitary-rngmpol)

    - [`IsFinite(P): RngMPol → BoolElt`](operation.md#function-isfinite-rngmpol)

    - [`IsOrdered(P): RngMPol → BoolElt`](operation.md#function-isordered-rngmpol)

    - [`IsField(P): RngMPol → BoolElt`](operation.md#function-isfield-rngmpol)

    - [`IsEuclideanDomain(P): RngMPol → BoolElt`](operation.md#function-iseuclideandomain-rngmpol)

    - [`IsPID(P): RngMPol → BoolElt`](operation.md#function-ispid-rngmpol)

    - [`IsUFD(P): RngMPol → BoolElt`](operation.md#function-isufd-rngmpol)

    - [`IsDivisionRing(P): RngMPol → BoolElt`](operation.md#function-isdivisionring-rngmpol)

    - [`IsEuclideanRing(P): RngMPol → BoolElt`](operation.md#function-iseuclideanring-rngmpol)

    - [`IsDomain(P): RngMPol → BoolElt`](operation.md#function-isdomain-rngmpol)

    - [`IsPrincipalIdealRing(P): RngMPol → BoolElt`](operation.md#function-isprincipalidealring-rngmpol)

    - [`P eq Q: RngMPol, RngMPol → BoolElt`](operation.md#operation-op-eq-rngmpol-rngmpol)

    - [`P ne Q: RngMPol, RngMPol → BoolElt`](operation.md#operation-op-ne-rngmpol-rngmpol)

  - [Changing Coefficient Ring](operation.md#changing-coefficient-ring)

    - [`ChangeRing(P, S): RngMPol, Rng → RngMPol`](operation.md#function-changering-rngmpol-rng)

  - [Homomorphisms](operation.md#homomorphisms)

    - [`hom< P -> S | f, y₁, ..., yₙ >: RngMPol, Rng → Map`](operation.md#constructor-constructor-hom-rngmpol-rng-map)

    - [`hom< P -> S | y₁, ..., yₙ >: RngMPol, Rng → Map`](operation.md#constructor-constructor-hom-rngmpol-rng-map-2)

    - [`Example: Homomorphism`](operation.md#example-ex-a59b0c)

- [Element Operations](operation-element.md)

  - [Arithmetic Operators](operation-element.md#arithmetic-operators)

    - [`+ a: RngMPolElt → RngMPolElt`](operation-element.md#operation-operation-rngmpolelt-rngmpolelt)

    - [`- a: RngMPolElt → RngMPolElt`](operation-element.md#operation-operation-rngmpolelt-rngmpolelt-2)

    - [`a + b: RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#operation-op-plus-rngmpolelt-rngmpolelt)

    - [`a - b: RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#operation-op-minus-rngmpolelt-rngmpolelt)

    - [`a * b: RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#operation-op-times-rngmpolelt-rngmpolelt)

    - [`a ^ k: RngMPolElt, RngIntElt → RngMPolElt`](operation-element.md#operation-op-pow-rngmpolelt-rngintelt)

    - [`a / b: RngMPolElt, RngMPolElt → FldFunMElt`](operation-element.md#operation-op-div-rngmpolelt-rngmpolelt)

    - [`a div b: RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#operation-operation-div-rngmpolelt-rngmpolelt-rngmpolelt)

    - [`a +:= b: RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#operation-operation-rngmpolelt-rngmpolelt-rngmpolelt)

    - [`a -:= b: RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#operation-operation-rngmpolelt-rngmpolelt-rngmpolelt-2)

    - [`a *:= b: RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#operation-operation-rngmpolelt-rngmpolelt-rngmpolelt-3)

    - [`a div:= b: RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#operation-operation-div-rngmpolelt-rngmpolelt-rngmpolelt-2)

  - [Equality and Membership](operation-element.md#equality-and-membership)

    - [`a eq b: RngMPolElt, RngMPolElt → BoolElt`](operation-element.md#operation-op-eq-rngmpolelt-rngmpolelt)

    - [`a ne b: RngMPolElt, RngMPolElt → BoolElt`](operation-element.md#operation-op-ne-rngmpolelt-rngmpolelt)

    - [`a in R: RngMPolElt, Rng → BoolElt`](operation-element.md#operation-op-in-rngmpolelt-rng)

    - [`a notin R: RngMPolElt, Rng → BoolElt`](operation-element.md#operation-op-notin-rngmpolelt-rng)

  - [Predicates on Ring Elements](operation-element.md#predicates-on-ring-elements)

    - [`IsDivisibleBy(a, b): RngMPolElt, RngMPolElt → BoolElt, RngMPolElt`](operation-element.md#function-isdivisibleby-rngmpolelt-rngmpolelt)

    - [`IsAlgebraicallyDependent(S): {RngMPolElt} → BoolElt`](operation-element.md#function-isalgebraicallydependent-rngmpolelt)

    - [`IsZero(f): RngMPolElt → BoolElt`](operation-element.md#function-iszero-rngmpolelt)

    - [`IsOne(f): RngMPolElt → BoolElt`](operation-element.md#function-isone-rngmpolelt)

    - [`IsMinusOne(f): RngMPolElt → BoolElt`](operation-element.md#function-isminusone-rngmpolelt)

    - [`IsNilpotent(f): RngMPolElt → BoolElt`](operation-element.md#function-isnilpotent-rngmpolelt)

    - [`IsIdempotent(f): RngMPolElt → BoolElt`](operation-element.md#function-isidempotent-rngmpolelt)

    - [`IsUnit(f): RngMPolElt → BoolElt`](operation-element.md#function-isunit-rngmpolelt)

    - [`IsZeroDivisor(f): RngMPolElt → BoolElt`](operation-element.md#function-iszerodivisor-rngmpolelt)

    - [`IsRegular(f): RngMPolElt → BoolElt`](operation-element.md#function-isregular-rngmpolelt)

    - [`IsIrreducible(f): RngMPolElt → BoolElt`](operation-element.md#function-isirreducible-rngmpolelt)

    - [`IsPrime(f): RngMPolElt → BoolElt`](operation-element.md#function-isprime-rngmpolelt)

  - [Coefficients, Monomials and Terms](operation-element.md#coefficients-monomials-and-terms)

    - [`Coefficients(f): RngMPolElt → [ RngElt ]`](operation-element.md#function-coefficients-rngmpolelt)

    - [`Coefficients(f, i): RngMPolElt, RngIntElt → [ RngElt ]`](operation-element.md#function-coefficients-rngmpolelt-rngintelt)

    - [`Coefficients(f, v): RngMPolElt, RngMPolElt → [ RngElt ]`](operation-element.md#function-coefficients-rngmpolelt-rngmpolelt)

    - [`Coefficient(f, i, k): RngMPolElt, RngIntElt, RngIntElt → RngElt`](operation-element.md#function-coefficient-rngmpolelt-rngintelt-rngintelt)

    - [`Coefficient(f, v, k): RngMPolElt, RngMPolElt, RngIntElt → RngElt`](operation-element.md#function-coefficient-rngmpolelt-rngmpolelt-rngintelt)

    - [`LeadingCoefficient(f): RngMPolElt → RngElt`](operation-element.md#function-leadingcoefficient-rngmpolelt)

    - [`LeadingCoefficient(f, i): RngMPolElt, RngIntElt → RngElt`](operation-element.md#function-leadingcoefficient-rngmpolelt-rngintelt)

    - [`LeadingCoefficient(f, v): RngMPolElt, RngMPolElt → RngElt`](operation-element.md#function-leadingcoefficient-rngmpolelt-rngmpolelt)

    - [`Length(f): RngMPolElt → RngIntElt`](operation-element.md#function-length-rngmpolelt)

    - [`TrailingCoefficient(f): RngMPolElt → RngElt`](operation-element.md#function-trailingcoefficient-rngmpolelt)

    - [`TrailingCoefficient(f, i): RngMPolElt, RngIntElt → RngElt`](operation-element.md#function-trailingcoefficient-rngmpolelt-rngintelt)

    - [`TrailingCoefficient(f, v): RngMPolElt, RngMPolElt → RngElt`](operation-element.md#function-trailingcoefficient-rngmpolelt-rngmpolelt)

    - [`CoefficientDenominator(f): RngMPolElt → RngElt`](operation-element.md#function-coefficientdenominator-rngmpolelt)

    - [`CoefficientNumerator(f): RngMPolElt → RngElt`](operation-element.md#function-coefficientnumerator-rngmpolelt)

    - [`MonomialCoefficient(f, m): RngMPolElt, RngMPolElt → RngElt`](operation-element.md#function-monomialcoefficient-rngmpolelt-rngmpolelt)

    - [`Monomials(f): RngMPolElt → [ RngMPolElt ]`](operation-element.md#function-monomials-rngmpolelt)

    - [`CoefficientsAndMonomials(f): RngMPolElt → [ RngElt ], [ RngMPolElt ]`](operation-element.md#function-coefficientsandmonomials-rngmpolelt)

    - [`LeadingMonomial(f): RngMPolElt → RngMPolElt`](operation-element.md#function-leadingmonomial-rngmpolelt)

    - [`Terms(f): RngMPolElt → [ RngMPolElt ]`](operation-element.md#function-terms-rngmpolelt)

    - [`Terms(f, i): RngMPolElt, RngIntElt → [ RngMPolElt ]`](operation-element.md#function-terms-rngmpolelt-rngintelt)

    - [`Terms(f, v): RngMPolElt, RngMPolElt → [ RngMPolElt ]`](operation-element.md#function-terms-rngmpolelt-rngmpolelt)

    - [`Term(f, i, k): RngMPolElt, RngIntElt, RngIntElt → RngMPolElt`](operation-element.md#function-term-rngmpolelt-rngintelt-rngintelt)

    - [`Term(f, v, k): RngMPolElt, RngMPolElt, RngIntElt → RngMPolElt`](operation-element.md#function-term-rngmpolelt-rngmpolelt-rngintelt)

    - [`LeadingTerm(f): RngMPolElt → RngMPolElt`](operation-element.md#function-leadingterm-rngmpolelt)

    - [`LeadingTerm(f, i): RngMPolElt, RngIntElt → RngMPolElt`](operation-element.md#function-leadingterm-rngmpolelt-rngintelt)

    - [`LeadingTerm(f, v): RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#function-leadingterm-rngmpolelt-rngmpolelt)

    - [`TrailingTerm(f): RngMPolElt → RngElt`](operation-element.md#function-trailingterm-rngmpolelt)

    - [`TrailingTerm(f, i): RngMPolElt, RngIntElt → RngElt`](operation-element.md#function-trailingterm-rngmpolelt-rngintelt)

    - [`TrailingTerm(f, v): RngMPolElt, RngMPolElt → RngElt`](operation-element.md#function-trailingterm-rngmpolelt-rngmpolelt)

    - [`Exponents(f): RngMPolElt → [ RngIntElt ]`](operation-element.md#function-exponents-rngmpolelt)

    - [`Monomial(P, E): RngMPol, [ RngIntElt ] → RngMPolElt`](operation-element.md#function-monomial-rngmpol-rngintelt)

    - [`Polynomial(C, M): [RngElt], [RngMPolElt] → RngMPolElt`](operation-element.md#function-polynomial-rngelt-rngmpolelt)

    - [`Example: Coefficients`](operation-element.md#example-ex-f062e9)

  - [Degrees](operation-element.md#degrees)

    - [`Degree(f, i): RngMPolElt, RngIntElt → RngIntElt`](operation-element.md#function-degree-rngmpolelt-rngintelt)

    - [`Degree(f, v): RngMPolElt, RngMPolElt → RngIntElt`](operation-element.md#function-degree-rngmpolelt-rngmpolelt)

    - [`TotalDegree(f): RngMPolElt → RngIntElt`](operation-element.md#function-totaldegree-rngmpolelt)

    - [`LeadingTotalDegree(f): RngMPolElt → RngIntElt`](operation-element.md#function-leadingtotaldegree-rngmpolelt)

  - [Univariate Polynomials](operation-element.md#univariate-polynomials)

    - [`IsUnivariate(f): RngMPolElt → BoolElt, RngUPolElt, RngIntElt`](operation-element.md#function-isunivariate-rngmpolelt)

    - [`IsUnivariate(f, i): RngMPolElt, RngIntElt → BoolElt, RngUPolElt`](operation-element.md#function-isunivariate-rngmpolelt-rngintelt)

    - [`IsUnivariate(f, v): RngMPolElt, RngMPolElt → BoolElt, RngUPolElt`](operation-element.md#function-isunivariate-rngmpolelt-rngmpolelt)

    - [`UnivariatePolynomial(f): RngMPolElt → RngUPolElt`](operation-element.md#function-univariatepolynomial-rngmpolelt)

    - [`Example: Univariate Polynomial`](operation-element.md#example-ex-f54865)

  - [Derivative, Integral](operation-element.md#derivative-integral)

    - [`Derivative(f, i): RngMPolElt, RngIntElt → RngMPolElt`](operation-element.md#function-derivative-rngmpolelt-rngintelt)

    - [`Derivative(f, v): RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#function-derivative-rngmpolelt-rngmpolelt)

    - [`Derivative(f, k, i): RngMPolElt, RngIntElt → RngMPolElt`](operation-element.md#function-derivative-rngmpolelt-rngintelt-2)

    - [`Derivative(f, k, v): RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#function-derivative-rngmpolelt-rngmpolelt-2)

    - [`Integral(f, i): RngMPolElt, RngIntElt → RngMPolElt`](operation-element.md#function-integral-rngmpolelt-rngintelt)

    - [`Integral(f, v): RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#function-integral-rngmpolelt-rngmpolelt)

    - [`JacobianMatrix( [ f ] ): [ RngMPolElt ] → RngMPol`](operation-element.md#function-jacobianmatrix-rngmpolelt)

  - [Evaluation, Interpolation](operation-element.md#evaluation-interpolation)

    - [`Evaluate(f, s): RngMPolElt, [ RngElt ] → RngElt`](operation-element.md#function-evaluate-rngmpolelt-rngelt)

    - [`Evaluate(f, s): RngMPolElt, < RngElt, ..., RngElt > → RngElt`](operation-element.md#function-evaluate-rngmpolelt-rngelt-rngelt)

    - [`Evaluate(f, i, r): RngMPolElt, RngMPolElt, RngElt → RngMPolElt`](operation-element.md#function-evaluate-rngmpolelt-rngmpolelt-rngelt)

    - [`Evaluate(f, v, r): RngMPolElt, RngMPolElt, RngElt → RngMPolElt`](operation-element.md#function-evaluate-rngmpolelt-rngmpolelt-rngelt-2)

    - [`Interpolation(I, V, i): [ RngElt ], [ RngMPolElt ], RngIntElt → RngMPolElt`](operation-element.md#function-interpolation-rngelt-rngmpolelt-rngintelt)

    - [`Interpolation(I, V, v): [ RngElt ], [ RngMPolElt ], RngIntElt → RngMPolElt`](operation-element.md#function-interpolation-rngelt-rngmpolelt-rngintelt-2)

    - [`Example: Interpolate`](operation-element.md#example-ex-ca9d39)

  - [Quotient and Reductum](operation-element.md#quotient-and-reductum)

    - [`f div g: RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#operation-operation-div-rngmpolelt-rngmpolelt-rngmpolelt-3)

    - [`ExactQuotient(f, g): RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#function-exactquotient-rngmpolelt-rngmpolelt)

    - [`Reductum(f): RngMPolElt → RngMPolElt`](operation-element.md#function-reductum-rngmpolelt)

    - [`Reductum(f, i): RngMPolElt, RngIntElt → RngMPolElt`](operation-element.md#function-reductum-rngmpolelt-rngintelt)

    - [`Reductum(f, v): RngMPolElt, RngMPolElt → RngMPolElt`](operation-element.md#function-reductum-rngmpolelt-rngmpolelt)

  - [Diagonalizing a Polynomial of Degree 2](operation-element.md#diagonalizing-a-polynomial-of-degree-2)

    - [`SymmetricBilinearForm(f): RngMPolElt → ModMatRngElt`](operation-element.md#function-symmetricbilinearform-rngmpolelt)

    - [`DiagonalForm(f): RngMPolElt → RngMPolElt, ModMatRngElt`](operation-element.md#function-diagonalform-rngmpolelt)

    - [`Example: Sym Bi Linear`](operation-element.md#example-ex-649fdb)

- [Greatest Common Divisors](common.md)

  - [Common Divisors and Common Multiples](common.md#common-divisors-and-common-multiples)

    - [`GreatestCommonDivisor(f, g): RngMPolElt, RngMPolElt → RngMPolElt`](common.md#function-greatestcommondivisor-rngmpolelt-rngmpolelt)

    - [`Gcd(f, g): RngMPolElt, RngMPolElt → RngMPolElt`](common.md#function-gcd-rngmpolelt-rngmpolelt)

    - [`GCD(f, g): RngMPolElt, RngMPolElt → RngMPolElt`](common.md#function-gcd-rngmpolelt-rngmpolelt-2)

    - [`GCD(Q): [ RngMPolElt ] → RngMPolElt`](common.md#function-gcd-rngmpolelt)

    - [`LeastCommonMultiple(f, g): RngMPolElt, RngMPolElt → RngMPolElt`](common.md#function-leastcommonmultiple-rngmpolelt-rngmpolelt)

    - [`Lcm(f, g): RngMPolElt, RngMPolElt → RngMPolElt`](common.md#function-lcm-rngmpolelt-rngmpolelt)

    - [`LCM(f, g): RngMPolElt, RngMPolElt → RngMPolElt`](common.md#function-lcm-rngmpolelt-rngmpolelt-2)

    - [`LCM(Q): [ RngMPolElt ] → RngMPolElt`](common.md#function-lcm-rngmpolelt)

    - [`Normalize(f): RngMPolElt → RngMPolElt`](common.md#function-dpoly-normalize)

    - [`ClearDenominators(f): RngMPolElt → RngMPolElt`](common.md#function-cleardenominators-rngmpolelt)

    - [`ClearDenominators(Q): [ RngMPolElt ] → [ RngMPolElt ]`](common.md#function-cleardenominators-rngmpolelt-2)

  - [Content and Primitive Part](common.md#content-and-primitive-part)

    - [`Content(f): RngMPolElt → RngIntElt`](common.md#function-content-rngmpolelt)

    - [`PrimitivePart(f): RngMPolElt → RngMPolElt`](common.md#function-primitivepart-rngmpolelt)

    - [`ContentAndPrimitivePart(f): RngMPolElt → RngIntElt, RngMPolElt`](common.md#function-contentandprimitivepart-rngmpolelt)

    - [`Contpp(f): RngMPolElt → RngIntElt, RngMPolElt`](common.md#function-contpp-rngmpolelt)

- [Factorization and Irreducibility](factorization-irreducibility.md)

  - [`Factorization(f): RngMPolElt → [ < RngMPolElt, RngIntElt >], RngElt`](factorization-irreducibility.md#function-factorization-rngmpolelt)

  - [`SquarefreeFactorization(f): RngMPolElt → [ <RngMPolElt, RngIntElt> ]`](factorization-irreducibility.md#function-squarefreefactorization-rngmpolelt)

  - [`SquarefreePart(f): RngMPolElt → RngMPolElt`](factorization-irreducibility.md#function-squarefreepart-rngmpolelt)

  - [`IsIrreducible(f): RngMPolElt → BoolElt`](factorization-irreducibility.md#function-isirreducible-rngmpolelt-2)

  - [`SetVerbose("PolyFact", v): MonStgElt, RngIntElt`](factorization-irreducibility.md#function-setverbose-monstgelt-rngintelt)

  - [`Example: Trinomials`](factorization-irreducibility.md#example-ex-2f44bb)

  - [`Example: Vandermonde`](factorization-irreducibility.md#example-ex-12897c)

  - [`Example: Heron`](factorization-irreducibility.md#example-ex-f94225)

  - [`Example: Finite Field Factorization`](factorization-irreducibility.md#example-ex-8e8a69)

- [Resultants and Discriminants](resultant-discriminant.md)

  - [`Resultant(f, g, i): RngMPolElt, RngMPolElt, RngIntElt → RngMPolElt`](resultant-discriminant.md#function-resultant-rngmpolelt-rngmpolelt-rngintelt)

  - [`Resultant(f, g, v): RngMPolElt, RngMPolElt, RngMPolElt → RngMPolElt`](resultant-discriminant.md#function-resultant-rngmpolelt-rngmpolelt-rngmpolelt)

  - [`Discriminant(f, i): RngMPolElt, RngIntElt → RngMPolElt`](resultant-discriminant.md#function-discriminant-rngmpolelt-rngintelt)

  - [`Discriminant(f, v): RngMPolElt, RngMPolElt → RngMPolElt`](resultant-discriminant.md#function-discriminant-rngmpolelt-rngmpolelt)

- [Polynomials over the Integers](integer.md)

  - [`Sign(f): RngMPolElt → RngIntElt`](integer.md#function-sign-rngmpolelt)

  - [`AbsoluteValue(f): RngMPolElt → RngMPolElt`](integer.md#function-absolutevalue-rngmpolelt)

  - [`Abs(f): RngMPolElt → RngMPolElt`](integer.md#function-abs-rngmpolelt)

  - [`MaxNorm(f): RngMPolElt → RngIntElt`](integer.md#function-maxnorm-rngmpolelt)

  - [`SumNorm(f): RngMPolElt → RngIntElt`](integer.md#function-sumnorm-rngmpolelt)