undocumented { installMinprimes } doc /// Node Key MinimalPrimes Headline minimal primes and radical routines for ideals Description Text Find the @TO "minimal primes of an ideal"@ in a polynomial ring over a prime field, or a quotient ring of that. These are the geometric components of the corresponding algebraic set. Multiple strategies are implemented via @TO2 {"Macaulay2Doc :: using hooks", "hooks"}@. In many cases the default @TT "Birational"@ strategy is {\it much} faster, although there are cases when the @TT "Legacy"@ strategy does better. For @ofClass MonomialIdeal@, a more efficient algorithm is used instead. @TT "minprimes"@ and @TT "decompose"@ are synonyms for @TO "minimalPrimes"@. Caveat Only works for ideals in (commutative) polynomial rings or quotients of polynomial rings over a prime field, might have bugs in small characteristic and larger degree (although, many of these cases are caught correctly). Subnodes minimalPrimes (isPrime, Ideal) (radical, Ideal) "minimal primes of an ideal" "radical of an ideal" Node Key "minimal primes of an ideal" Description Text @SUBSECTION "using {\tt minimalPrimes}"@ To obtain a list of the minimal associated primes for an ideal {\tt I} (i.e. the smallest primes containing {\tt I}), use the function @TO minimalPrimes@. Example R = QQ[w,x,y,z]; I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2) minimalPrimes I Text If the ideal given is a prime ideal then @TO minimalPrimes@ will return the ideal given. Example R = ZZ/101[w..z]; I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2); minimalPrimes I Text See @TO "PrimaryDecomposition :: associated primes"@ for information on finding associated prime ideals and @TO "PrimaryDecomposition :: primary decomposition"@ for more information about finding the full primary decomposition of an ideal. Node Key "radical of an ideal" Description Text There are two main ways to find the radical of an ideal. On some large examples the second method is faster. @SUBSECTION {"using ", TO radical}@ Example S = ZZ/101[x,y,z] I = ideal(x^3-y^2,y^2*z^2) radical I Text @SUBSECTION {"using the", TO "minimal primes of an ideal"}@ An alternate way to find the radical of an ideal @TT "I"@ is to take the intersection of its minimal prime ideals. To find the @TO "minimal primes of an ideal"@ @TT "I"@ use the function @TO minimalPrimes@. Then use @TO "intersect"@. Example intersect minimalPrimes I Node Key minimalPrimes (minimalPrimes, Ideal) [minimalPrimes, Verbosity] [minimalPrimes, Strategy] [minimalPrimes, CodimensionLimit] [minimalPrimes, MinimalGenerators] (decompose, Ideal) [(decompose, Ideal), Verbosity] [(decompose, Ideal), Strategy] [(decompose, Ideal), CodimensionLimit] [(decompose, Ideal), MinimalGenerators] Headline minimal primes of an ideal Usage minimalPrimes I minprimes I decompose I Inputs I:Ideal Verbosity => ZZ a larger number will print more information during the computation Strategy => String specifies the computation strategy to use. If it is slow, try @TT "Legacy"@, @TT "Birational"@, or @TT "NoBirational"@. The strategies might change in the future, and there are other undocumented ways to fine-tune the algorithm. CodimensionLimit => ZZ stop after finding primes of codimension less than or equal to this value MinimalGenerators=>Boolean if false, the minimal primes will not be minimalized Outputs :List with entries the minimal associated primes of @TT "I"@ Description Text Given an ideal in a polynomial ring, or a quotient of a polynomial ring whose base ring is either {\tt QQ} or {\tt ZZ/p}, this function computes the minimal associated primes of the ideal @TT "I"@. Geometrically, it decomposes the algebraic set defined by @TT "I"@. Example R = ZZ/32003[a..e] I = ideal"a2b-c3,abd-c2e,ade-ce2" C = minprimes I; netList C Example C2 = minprimes(I, Strategy=>"NoBirational", Verbosity=>2) C1 = minprimes(I, Strategy=>"Birational", Verbosity=>2) Text Example. The homogenized equations of the affine twisted cubic curve define the union of the projective twisted cubic curve and a line at infinity: Example R = QQ[w,x,y,z]; I=ideal(x^2-y*w, x^3-z*w^2) minimalPrimes I Text Note that the ideal is decomposed over the given field of coefficients and not over the extension field where the decomposition into absolutely irreducible factors occurs: Example I = ideal(x^2 + y^2) minimalPrimes I Text For monomial ideals, the method used is essentially what is shown in the example. Example I = monomialIdeal ideal"wxy,xz,yz" minimalPrimes I P = intersect(monomialIdeal(w,x,y), monomialIdeal(x,z), monomialIdeal(y,z)) minI = apply(P_*, monomialIdeal @@ support) Text It is sometimes useful to compute @TT "P"@ instead, where each generator encodes a single minimal prime. This can be obtained directly, as in the following code. Example dual radical I P == oo SeeAlso (isPrime, Ideal) "PrimaryDecomposition :: topComponents" "PrimaryDecomposition :: removeLowestDimension" radical irreducibleCharacteristicSeries "Macaulay2Doc :: dual(MonomialIdeal)" Node Key (isPrime, Ideal) [(isPrime, Ideal), Strategy] [(isPrime, Ideal), Verbosity] Headline whether an ideal is prime Usage isPrime I Inputs I:Ideal in a polynomial ring or quotient of one Strategy => String See @TO [minimalPrimes, Strategy]@ Verbosity => ZZ See @TO [minimalPrimes, Verbosity]@ Outputs :Boolean @TO "true"@ if {\tt I} is a prime ideal, or @TO "false"@ otherwise Description Text This function determines whether an ideal in a polynomial ring is prime. Example R = QQ[a..d]; I = monomialCurveIdeal(R, {1, 5, 8}) isPrime I SeeAlso minimalPrimes --- author(s): Decker Node Key radical (radical, Ideal) [radical, Strategy] [radical, Unmixed] [radical, CompleteIntersection] Headline the radical of an ideal Usage radical I Inputs I:Ideal Unmixed=>Boolean whether it is known that the ideal {\tt I} is unmixed. The ideal $I$ is said to be unmixed if all associated primes of $R/I$ have the same dimension. In this case the algorithm tends to be much faster. Strategy=>Symbol the strategy to use, either @TT "Decompose"@ or @TT "Unmixed"@ CompleteIntersection=>Ideal an ideal @TT "J"@ of the same height as @TT "I"@ whose generators form a maximal regular sequence contained in @TT "I"@. Providing this option as a hint allows a separate, often faster, algorithm to be used to compute the radical. This option should only be used if @TT "J"@ is nice in some way. For example, if @TT "J"@ is randomly generated, but @TT "I"@ is relatively sparse, then this will most likely run slower than just giving the @TO "Unmixed"@ option. Outputs :Ideal the radical of @TT "I"@ Description Text If I is an ideal in an affine ring (i.e. a quotient of a polynomial ring over a field), and if the characteristic of this field is large enough (see below), then this routine yields the radical of the ideal I. The method used is the Eisenbud-Huneke-Vasconcelos algorithm. The algorithms used generally require that the characteristic of the ground field is larger than the degree of each primary component. In practice, this means that if the characteristic is something like 32003, rather than, for example, 5, the methods used will produce the radical of @TT "I"@. Of course, you may do the computation over @TO "QQ"@, but it will often run much slower. In general, this routine still needs to be tuned for speed. Example R = QQ[x, y] I = ideal((x^2+1)^2*y, y+1) radical I Text If @TT "I"@ is @ofClass MonomialIdeal@, a faster, combinatorial algorithm is used. Example R = ZZ/101[a..d] I = intersect(ideal(a^2,b^2,c), ideal(a,b^3,c^2)) elapsedTime radical(ideal I_*, Strategy => Monomial) elapsedTime radical(ideal I_*, Unmixed => true) Text For another example, see @TO "PrimaryDecomposition :: PrimaryDecomposition"@. References Eisenbud, Huneke, Vasconcelos, Invent. Math. 110 207-235 (1992). Caveat The current implementation requires that the characteristic of the ground field is either zero or a large prime (unless @TT "I"@ is @ofClass MonomialIdeal@). SeeAlso minimalPrimes "PrimaryDecomposition :: topComponents" "PrimaryDecomposition :: removeLowestDimension" "Saturation :: saturate" "Saturation :: Ideal : Ideal" Subnodes radicalContainment Node Key radicalContainment (radicalContainment, Ideal, Ideal) (radicalContainment, RingElement, Ideal) [radicalContainment, Strategy] Headline whether an element is contained in the radical of an ideal Usage radicalContainment(g, I) Inputs g:RingElement I:Ideal Outputs :Boolean true if @TT "g"@ is in the radical of @TT "I"@, and false otherwise Description Text This method determines if a given element @TT "g"@ is contained in the radical of a given ideal @TT "I"@. There are 2 algorithms implemented for doing so: the first (default) uses the Rabinowitsch trick in the proof of the Nullstellensatz, and is called with @TT "Strategy => \"Rabinowitsch\""@. The second algorithm, for homogeneous ideals, uses a theorem of Kollar to obtain an effective upper bound on the required power to check containment, together with repeated squaring, and is called with @TT "Strategy => \"Kollar\""@. The latter algorithm is generally quite fast if a Grobner basis of @TT "I"@ has already been computed. A recommended way to do so is to check ordinary containment, i.e. @TT "g % I == 0"@, before calling this function. Example d = (4,5,6,7) n = #d R = QQ[x_0..x_n] I = ideal homogenize(matrix{{x_1^(d#0)} | apply(toList(1..n-2), i -> x_i - x_(i+1)^(d#i)) | {x_(n-1) - x_0^(d#-1)}}, x_n) D = product(I_*/degree/sum) x_0^(D-1) % I != 0 and x_0^D % I == 0 elapsedTime radicalContainment(x_0, I) elapsedTime radicalContainment(x_0, I, Strategy => "Kollar") elapsedTime radicalContainment(x_n, I, Strategy => "Kollar") SeeAlso radical Node Key Hybrid Headline the class of lists encapsulating hybrid strategies Description Text This class is used to encapsulate hybrid strategies for certain method functions. This example demonstrates an example of what is meant by a hybrid strategy. Example debug MinimalPrimes R = ZZ/101[w..z]; I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2); elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2); SeeAlso "PrimaryDecomposition :: primaryDecomposition(...,Strategy=>...)" ///

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