# Gröbner Bases - [Introduction](introduction.md) - [Representation and Monomial Orders](order.md) - [Lexicographical: lex](order.md#lexicographical-lex) - [Graded Lexicographical: glex](order.md#graded-lexicographical-glex) - [Graded Reverse Lexicographical: grevlex](order.md#graded-reverse-lexicographical-grevlex) - [Graded Reverse Lexicographical (Weighted): grevlexw](order.md#graded-reverse-lexicographical-weighted-grevlexw) - [Elimination (k): elim](order.md#elimination-k-elim) - [Elimination List: elim](order.md#elimination-list-elim) - [Inverse Block: invblock](order.md#inverse-block-invblock) - [Univariate: univ](order.md#univariate-univ) - [Weight: weight](order.md#weight-weight) - [Polynomial Rings and Ideals](creation.md) - [Creation of Polynomial Rings and Accessing their Monomial Orders](creation.md#creation-of-polynomial-rings-and-accessing-their-monomial-orders) - [`PolynomialRing(R, n): Rng, RngIntElt → RngMPol`](creation.md#function-gb-polynomialring) - [`PolynomialAlgebra(R, n): Rng, RngIntElt → RngMPol`](creation.md#function-gb-polynomialalgebra) - [`PolynomialRing(R, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPol`](creation.md#function-polynomialring-rng-rngintelt-monstgelt) - [`PolynomialAlgebra(R, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPol`](creation.md#function-polynomialalgebra-rng-rngintelt-monstgelt) - [`PolynomialRing(R, n, T): Rng, RngIntElt, Tup → RngMPol`](creation.md#function-gb-polynomialringtup) - [`PolynomialAlgebra(R, n, T): Rng, RngIntElt, Tup → RngMPol`](creation.md#function-polynomialalgebra-rng-rngintelt-tup) - [`MonomialOrder(P): RngMPol → Tup`](creation.md#function-gb-monomialorder) - [`MonomialOrderWeightVectors(P): RngMPol → [ [ FldRatElt ] ]`](creation.md#function-gb-monomialorderweightvectors) - [`SetSparseMonomialMinRank(R): RngIntElt`](creation.md#function-setsparsemonomialminrank-rngintelt) - [`GetSparseMonomialMinRank() → RngIntElt`](creation.md#function-getsparsemonomialminrank) - [`Example: Order`](creation.md#example-ex-7d923a) - [Creation of Graded Polynomial Rings](creation.md#creation-of-graded-polynomial-rings) - [`PolynomialRing(R, Q): Rng, [ RngIntElt ] → RngMPol`](creation.md#function-polynomialring-rng-rngintelt) - [`PolynomialAlgebra(R, Q): Rng, [ RngIntElt ] → RngMPol`](creation.md#function-polynomialalgebra-rng-rngintelt) - [`Grading(P): RngMPol → [ RngIntElt ]`](creation.md#function-grading-rngmpol) - [`VariableWeights(P): RngMPol → [ RngIntElt ]`](creation.md#function-variableweights-rngmpol) - [`HomogeneousWeightsSearch(S): [ RngMPol ] → BoolElt, [ RngIntElt ]`](creation.md#function-homogeneousweightssearch-rngmpol) - [Element Operations Using the Grading](creation.md#element-operations-using-the-grading) - [`Degree(f): RngMPolElt → RngIntElt`](creation.md#function-degree-rngmpolelt) - [`WeightedDegree(f): RngMPolElt → RngIntElt`](creation.md#function-weighteddegree-rngmpolelt) - [`LeadingWeightedDegree(f): RngMPolElt → RngIntElt`](creation.md#function-leadingweighteddegree-rngmpolelt) - [`IsHomogeneous(f): RngMPolElt → BoolElt`](creation.md#function-ishomogeneous-rngmpolelt) - [`HomogeneousComponent(f, d): RngMPolElt, RngIntElt → RngMPolElt`](creation.md#function-homogeneouscomponent-rngmpolelt-rngintelt) - [`HomogeneousComponents(f): RngMPolElt → [ RngMPolElt ]`](creation.md#function-homogeneouscomponents-rngmpolelt) - [`MonomialsOfDegree(P, d): RngMPolElt, RngIntElt → {\@ RngMPolElt \@}`](creation.md#function-gb-monomialsofdegree) - [`MonomialsOfWeightedDegree(P, d): RngMPolElt, RngIntElt → {\@ RngMPolElt \@}`](creation.md#function-gb-monomialsofweighteddegree) - [`Example: Graded`](creation.md#example-ex-fffb08) - [Creation of Ideals and Accessing their Bases](creation.md#creation-of-ideals-and-accessing-their-bases) - [`ideal

: RngMPol, List → RngMPol`](creation.md#constructor-gb-ideal) - [`Ideal(B): [ RngMPolElt ] → RngMPol`](creation.md#function-gb-ideal) - [`Ideal(B): { RngMPolElt } → RngMPol`](creation.md#function-ideal-rngmpolelt) - [`Ideal(f): RngMPolElt → RngMPol`](creation.md#function-gb-idealofelt) - [`IdealWithFixedBasis(B): [ RngMPolElt ] → RngMPol`](creation.md#function-gb-idealwithfixedbasis) - [`Basis(I): RngMPol → [ RngMPolElt ]`](creation.md#function-basis-rngmpol) - [`BasisElement(I, i): RngMPol, RngIntElt → RngMPolElt`](creation.md#function-basiselement-rngmpol-rngintelt) - [Gröbner Bases](groebner.md) - [Gröbner Bases over Fields](groebner.md#grobner-bases-over-fields) - [Gröbner Bases over Euclidean Rings](groebner.md#grobner-bases-over-euclidean-rings) - [Construction of Gröbner Bases](groebner.md#construction-of-grobner-bases) - [`Groebner(I: parameters): RngMPol`](groebner.md#function-gb-groebner) - [`GroebnerBasis(I: parameters): RngMPol → [ RngMPolElt ], [ RngIntElt ]`](groebner.md#function-gb-groebnerbasis) - [`GroebnerBasis(S: parameters): [ RngMPolElt ] → [ RngMPolElt ], [ RngIntElt ], []`](groebner.md#function-groebnerbasis-rngmpolelt) - [`GroebnerBasis(S: parameters): { RngMPolElt } → [ RngMPolElt ]`](groebner.md#function-groebnerbasis-rngmpolelt-2) - [`GroebnerBasisUnreduced(S: parameters): [ RngMPolElt ] → [ RngMPolElt ]`](groebner.md#function-groebnerbasisunreduced-rngmpolelt) - [`GroebnerBasisUnreduced(S: parameters): { RngMPolElt } → [ RngMPolElt ]`](groebner.md#function-groebnerbasisunreduced-rngmpolelt-2) - [`GroebnerBasis(S, d: parameters): [ RngMPol ], RngInt → RngMPolElt`](groebner.md#function-dpoly-gb-deg) - [`SetGBGlobalModular(f): BoolElt`](groebner.md#function-setgbglobalmodular) - [`SetFaugereModular(f): BoolElt`](groebner.md#function-setfaugeremodular) - [The Dense Variant of the $F_4$ algorithm](groebner.md#the-dense-variant-of-the-f-4-algorithm) - [Related Functions](groebner.md#related-functions) - [`HasGroebnerBasis(I): RngMPol → BoolElt`](groebner.md#function-hasgroebnerbasis-rngmpol) - [`EasyIdeal(I): RngMPol → RngMPol, [ RngIntElt ]`](groebner.md#function-easyideal) - [`EasyBasis(I): RngMPol → [ RngMPolElt ]`](groebner.md#function-easybasis) - [`SmallBasis(I): RngMPol → [ RngMPolElt ]`](groebner.md#function-smallbasis-rngmpol) - [`MarkGroebner(I): RngMPol`](groebner.md#function-markgroebner-rngmpol) - [`IsGroebner(S): { RngMPolElt } → BoolElt`](groebner.md#function-isgroebner-rngmpolelt) - [`IsGroebner(S): [ RngMPolElt ] → BoolElt`](groebner.md#function-isgroebner-rngmpolelt-2) - [`Coordinates(I, f): RngMPol, RngMPolElt → [ RngMPolElt ]`](groebner.md#function-gb-coordinates) - [`CoordinateMatrix(I): RngMPol → Matrix`](groebner.md#function-gb-coordinatematrix) - [`NormalForm(f, I): RngMPolElt, RngMPol → RngMPolElt`](groebner.md#function-normalform-rngmpolelt-rngmpol) - [`NormalForm(f, S): RngMPolElt, [ RngMPolElt ] → RngMPolElt, [ RngMPolElt ]`](groebner.md#function-normalform-rngmpolelt-rngmpolelt) - [`SPolynomial(f, g): RngMPolElt, RngMPolElt → RngMPolElt`](groebner.md#function-spolynomial-rngmpolelt-rngmpolelt) - [`Reduce(S): [ RngMPolElt ] → [ RngMPolElt ]`](groebner.md#function-gb-reduce) - [`Reduce(S): { RngMPolElt } → [ RngMPolElt ]`](groebner.md#function-reduce-rngmpolelt) - [`ReduceGroebnerBasis(S): [ RngMPolElt ] → [ RngMPolElt ]`](groebner.md#function-reducegroebnerbasis-rngmpolelt) - [`ReduceGroebnerBasis(S): { RngMPolElt } → [ RngMPolElt ]`](groebner.md#function-reducegroebnerbasis-rngmpolelt-2) - [Gröbner Bases of Boolean Polynomial Rings](groebner.md#grobner-bases-of-boolean-polynomial-rings) - [`BooleanPolynomialRing(n): RngIntElt → RngMPolBool`](groebner.md#function-gb-booleanpolynomialring) - [`BooleanPolynomialRing(n, order): RngIntElt, MonStgElt → RngMPolBool`](groebner.md#function-booleanpolynomialring-rngintelt-monstgelt) - [`BooleanPolynomialRing(B, Q): RngMPolBool, [RngIntElt] → RngMPolBoolElt`](groebner.md#function-booleanpolynomialring-rngmpolbool-rngintelt) - [Construction of Input Systems](groebner.md#construction-of-input-systems) - [`MinRankSystem(K, n, k, r): FldFin, RngIntElt, RngIntElt, RngIntElt → [ RngMPolBool ]`](groebner.md#function-minranksystem-fldfin-rngintelt-rngintelt-rngintelt) - [`HFESystem(q, n, D): RngIntElt, RngIntElt, RngIntElt → [ RngMPolBool ]`](groebner.md#function-hfesystem-rngintelt-rngintelt-rngintelt) - [Verbosity](groebner.md#verbosity) - [`SetVerbose("Groebner", v): MonStgElt, RngIntElt`](groebner.md#function-setverbose-monstgelt-rngintelt) - [`SetVerbose("Buchberger", v): MonStgElt, RngIntElt`](groebner.md#function-setverbose-monstgelt-rngintelt-2) - [`SetVerbose("Faugere", v): MonStgElt, RngIntElt`](groebner.md#function-setverbose-monstgelt-rngintelt-3) - [`SetVerbose("FGLM", v): MonStgElt, RngIntElt`](groebner.md#function-setverbose-monstgelt-rngintelt-4) - [`SetVerbose("GroebnerWalk", v): MonStgElt, RngIntElt`](groebner.md#function-setverbose-monstgelt-rngintelt-5) - [`Example: Cyclic6`](groebner.md#example-ex-a7ca30) - [`Example: Runge Kutta2`](groebner.md#example-ex-a39450) - [`Example: Solve Over GF2`](groebner.md#example-ex-5d686d) - [`Example: G Bover Z`](groebner.md#example-ex-daf567) - [`Example: Finding Primes`](groebner.md#example-ex-9e862d) - [`Example: Quadratic Order GB`](groebner.md#example-ex-6747d7) - [`Example: Coordinates`](groebner.md#example-ex-51a96f) - [`Example: Valuation Ring`](groebner.md#example-ex-eaa5db) - [Degree-$d$ Gröbner Bases](groebner.md#degree-d-grobner-bases) - [`GroebnerBasis(S, d : parameters): [ RngMPolElt ], RngInt → RngMPolElt`](groebner.md#function-groebnerbasis-rngmpolelt-rngint) - [`Example: Degree D`](groebner.md#example-ex-054264) - [Changing Coefficient Ring](change-ring.md) - [`ChangeRing(I, S): RngMPol, Rng → RngMPol`](change-ring.md#function-gb-changering) - [`Example: Change Ring`](change-ring.md#example-ex-d2d66e) - [Changing Monomial Order](change-order.md) - [`ChangeOrder(I, Q): RngMPol, RngMPol → RngMPol, Map`](change-order.md#function-gb-changeorder) - [`ChangeOrder(I, order): RngMPol, ... → RngMPol, Map`](change-order.md#function-changeorder-rngmpol) - [`ChangeOrder(I, T): RngMPol, Tup → RngMPol`](change-order.md#function-gb-changeordertup) - [`Example: Change Order`](change-order.md#example-ex-6f12cc) - [Hilbert-driven Gröbner Basis Construction](hilbert-groebner.md) - [`HilbertGroebnerBasis(S, H): [ RngMPolElt ], FldFunRatUElt → BoolElt, [ RngMPolElt ]`](hilbert-groebner.md#function-hilbertgroebnerbasis-rngmpolelt-fldfunratuelt) - [`HilbertGroebnerBasis(S, N): [ RngMPolElt ], RngUPolElt → BoolElt, [ RngMPolElt ]`](hilbert-groebner.md#function-hilbertgroebnerbasis-rngmpolelt-rngupolelt) - [`SetVerbose("HilbertGroebner", v): MonStgElt, RngIntElt`](hilbert-groebner.md#function-setverbose-monstgelt-rngintelt-6) - [`Example: Hilbert Groebner`](hilbert-groebner.md#example-ex-0965e4) - [SAT solver](sat.md) - [`SAT(B): [ RngMPolBoolElt ] → BoolElt, [ FldFinElt ]`](sat.md#function-sat-rngmpolboolelt) - [`SAT(B): [ RngMPolElt ] → BoolElt, [ FldFinElt ]`](sat.md#function-sat-rngmpolelt) - [`Example: SAT`](sat.md#example-ex-a37e10)