Solving systems of polynomial equations is a main task in Computer Algebra, although the precise meaning of what is an acceptable solution depends on the context. 
In this talk, we interpret it as finding the minimal associated primes of the ideal generated by the polynomials. Geometrically, this is equivalent to decompose the set of solutions into its irreducible components.
We study the existing algorithms, and propose some modifications. 

A common technique used is to reduce the problem to the zero dimensional case. In a paper by Gianni, Trager and Zacharias they use this technique, combined with the splitting tool $I = (I : h^infty) cap langle I, h^m 
angle$ for some specific polynomial $h$ and integer $m$. This splitting introduces a number of redundant components that are not part of the original ideal.

In the algorithm we present here, we use the reduction to the zero dimensional case, but we avoid working with the ideal $langle I, h^m 
angle$. As a result, when the ideal has components of different dimensions, our algorithm is usually more efficient

Laplagne, Santiago

Dagstuhl Research Online Publication Server

Computation of the Minimal Associated PrimesSantiago LaplagneDepartamento de Matema´tica, Universidad de Buenos AiresBuenos Aires, Argentinaslaplagn@dm.uba.arAbstract. We propose a new algorithm for the computation of the mini-mal associated primes of an ideal. In [1] we have introduced modificationsto an algorithm for the computation of the radical by Krick and Logar([2]) (based on ideas by Gianni, Trager and Zacharias ([3])), that madeit more efficient. In this work, we show how these same modifications canbe applied to the algorithm for the computation of the minimal associ-ated primes proposed in [3]. We explain the algorithm, our modificationsand show some benchmarks that confirm that the new algorithm is moreefficient than the original one.Keywords. Minimal associated primes, polynomial ideal, Gro¨bner bases,primary decomposition, radical1 IntroductionSolving systems of polynomial equations is a main task in Computer Algebra,although the precise meaning of what is an acceptable solution depends on thecontext. In this work, we interpret it as finding the minimal associated primesof the ideal generated by the polynomials. Geometrically, this is equivalent todecompose the set of solutions into its irreducible components.A common technique is to reduce the problem to the zero dimensional case.In a paper by Gianni, Trager and Zacharias they use this technique, combinedwith the splitting tool I = (I : h∞)∩ 〈I, hm〉 for some specific polynomial h andinteger m. This splitting introduces a number of redundant components that arenot part of the original ideal.Their ideas can be used to compute the primary decomposition, the radicaland the minimal associated primes. In [2], the authors use these ideas to computethe radical of an ideal. In [1] we propose some modifications to that algorithm.We use the reduction to the zero dimensional case, but we avoid working with theideal 〈I, hm〉 using instead saturations with respect to appropriate polynomials.As a result, when the ideal has components of different dimensions, our algorithmis usually more efficient.In the present work we show how the same modifications can be applied tothe computation of the minimal associated primes of an ideal. We make a briefdescription of GTZ algorithm, we introduce our modifications and we show sometime comparisons using an implementation in Singular [4].Dagstuhl Seminar Proceedings 06271Challenges in Symbolic Computation Softwarehttp://drops.dagstuhl.de/opus/volltexte/2006/7742 Santiago Laplagne2 PreliminariesWe note Vk(I) for the vanishing set of I in kn and k¯ for the algebraic closureof k. An ideal is called zero dimensional if Vk¯(I) has only a finite number ofpoints. This is equivalent to saying that it contains polynomials pure in eachvariable. If I is zero-dimensional, we say that a variable xi separate the pointsof I if the results of evaluating the polynomial xi in each of the points of Vk¯(I)are all different.Following the ideas in [3], the computation of the minimal associated primesof a general ideal can be reduced to the zero dimensional case. For the computa-tion of the minimal associated primes of a zero dimensional ideals, the followingalgorithm, also based in an algorithm proposed in [3], can be used.Proposition 1 Let 〈g〉 = I ∩k[xn] and g = gm11 . . . gmtt , the factorization. ThenI = ∩ti=1〈I, gmii 〉.If xn separate points, then– 〈I, gmii 〉 is primary–√〈I, gmii 〉 = 〈I, gi〉, and these are the minimal associated primes.In [5] [Criterion 4.2.4], an algorithm is given for checking if xn separatesvariables, by looking at the shape that the ideals 〈I, gmii 〉 must have in thatcase.If xn does not separate variables, a random coordinate change must be per-formed. If k is infinite a suitable coordinate change always exist.In [3], the authors use the splitting tool I = (I : h∞) ∩ 〈I, h〉 (for h suchthat I : h = I : h2). They find h such that the minimal associated primes ofI : h can be obtained by reduction to the zero-dimensional case and the onescorresponding to 〈I, h〉 can be obtained by induction.When taking 〈I, h〉 there appear redundant components (that is, componentsthat were not part of the original ideal) that slow down the algorithm perfor-mance.In the algorithm that we proposed in [1] for computing the radical of an ideal,we avoided using 〈I, h〉 and instead we used repeatedly the saturation I : h∞ forappropriate h. This leaded in some cases to a more efficient algorithm.The same ideas can be used for computing the minimal associated primes ofan ideal, obtaining the following algorithmAlgorithm 2 minAssPrimes(I)Input: I ⊂ k[x]Output: P1, . . . , Pt, the minimal assocciated primes of I.1. P˜ ← 〈1〉 (P˜ will be the intersection of the minimal associated primes alreadyobtained).2. RepeatComputation of the Minimal Associated Primes 3(a) Look for g ∈ P˜ r √I. To find it, search over the generators of P˜ andcheck if they are in√I.(b) If there does not exist such g, it means that P˜ ⊂ √I. Since we alwayshave√I ⊂ P˜ , we conclude that P˜ = √I. Exit the cycle.(c) If there exists g ∈ P˜ r √I, this means that there exists at least oneminimal prime P associated to I such that g 6∈ P .J ← I : g∞.(d) Reduction to the zero-dimensional case:Take a maximal independent set u with respect to J and compute P ′1, . . . ,P ′s, the minimal associated primes of the zero-dimensional ideal Jk(u)[xru].(e) Contract the ideals P ′i ⊂ k(u)[xr u] to Pi ⊂ k[x], 1 ≤ i ≤ s.(f) P˜ ← P˜ ∩ P1 ∩ · · · ∩ Ps.(g) P ← P ∪ {P1, . . . , Ps}.3. output = P, the minimal associated primes of I.The correctness and termination of the algorithm can be proven in exactlythe same way as in [1].Remark 1. As in [1], in this algorithm there is no redundancy. All the ideals thatwe add to P are minimal prime ideals associated to I.As an example, we apply the algorithm to the idealI = 〈y + z, xz2w, x2z2〉 ⊂ Q[x, y, z, w].In the first iteration, we take g := 1 and J := I : 1∞ = I. We find that u = {x,w}is a maximal independent set with respect to J . Making the reduction step, weobtain that the only minimal associated primes of J(u)[x r u] is 〈y, z〉, whichcontracted to k[x] is P1 = 〈y, z〉. We take P˜ := P1 and P := {P1}.In the second iteration, we look for g ∈ P˜ such that g 6∈ √I. We obtainthat z 6∈ √I and compute J = I : z∞ = 〈y + z, xw, x2〉. Now u = {z, w} is amaximal independent set with respect to J . The only minimal associated primeof Jk(u)[xru] is 〈y + z, x〉, which contracted to k[x] gives P1 = 〈y + z, x〉. Wetake P˜ := 〈y, z〉 ∩ 〈y + z, x〉 = 〈y + z, xz〉 and P = {〈y, z〉, 〈y + z, x〉}.If we search for g ∈ P˜ such that g 6∈ √I, we obtain that y + z and xz areboth in√I. Therefore, the algorithm terminates. We obtain that the minimalassociated primes of I are 〈y, z〉 and 〈y + z, xz〉.We now apply GTZ algorithm ([3]) to the same ideal, to compare it withours. We start with I = 〈y + z, x z2w, x2z2〉. The first step is the same, weobtain P1 = 〈y, z〉 P˜ := P1 and P = {P1}.The next step is different. We look for h such that I = (I(u)[xru]∩ k[x])∩〈I, h〉. We can take h = xz. Now, √I = 〈y, z〉 ∩ √〈I, xz〉. So it remains tocompute the minimal associated primes of 〈I, xz〉. Carrying on the algorithm,we get that they are 〈y + z, x〉 and 〈w, y, z〉.The last prime is not a minimal associated prime of I (not even an associatedprime of I). It is a new component that appeared when we added xz to I.4 Santiago LaplagneThis is a situation that repeats often in the examples. The polynomials thatthe algorithm adds to I make it more and more complex. The polynomials addedare usually large, since they are the product of coefficients of polynomials in aGro¨bner basis and the size of the Gro¨bner basis of the new ideal can increasedrastically.This does not happen in our proposed algorithm. We compute instead thesaturation with respect to polynomials that are usually simple, and this satu-ration does not increase the complexity of the ideal since it only takes somecomponents away from it. No new components can appear.3 Performance evaluationIn this section, we apply the proposed algorithm to several examples given in[6], [7] and other ideals and evaluate its performance. (We only consider thoseideals that are not zero dimensional.) We implemented the algorithm in Singular([4]). Our routine uses the subroutine for the reduction to the zero dimensionalcase that is already implemented in the library primdec [8] for the computa-tion of the Minimal Associated primes by Gianni-Trager-Zacharias algorithm.We compare the times obtained by our algorithm with the algorithms imple-mented in primdec: Gianni-Trager-Zacharias ([3]) and via Characteristic Sets(proc minAssChar).We created some new examples where the differences are more significant,which we detail below.p1 = a+ c+ d+ e+ f + g+h+ j− 1, p2 = −b+ c+ e+ g+ j, q1 = 59ad+59ah+59dh− 705d− 1199h, q2 = −54acf − 54adf + a+ d, q3 = adfg + a+ dI1 = 〈p1, p2〉 ∩ 〈q1, q2, q3〉 (polynomials taken from DGP25 and DGP28)p1 = x2 + y2 + z2 − t2, p2 = xy + z2 − 1, q1 = w2xy + w2xz + w2z2, q2 =tx2y + x2yz + x2z2, q3 = twy2 + ty2z + y2z2, q4 = t2wx+ t2wz + t2z2I2 = 〈p1, p2〉 ∩ 〈q2, q3, q4〉, I3 = 〈p1, p2〉 ∩ 〈q1, q3, q4〉 (polynomials taken fromDGP31 and DGP32)The results are shown in Table 1. All the computations are done over Q. Theordering of the monomials is always the degree reverse lexicographical orderingwith the underlying ordering of the alphabet.The codes for the examples in the firsts columns are the ones given in [6] and[7]. ”Dim” indicates the dimension of the ideal; ”Prim. comps.”, the total numberof primary components; ”Min. ass.”, the number of minimal associated primes;”Emb. comps.”, the number of embedded components and ”Equidim?” if theideal is equidimensional. The last three columns show the timings. GTZ is thealgorithm of Gianni, Trager and Zacharias ([3]) and Char is an algorithm usingcharacteristic sets implemented in Singular. Timing is measured in hundredthsof seconds. The entry * means that after one day of computations, the algorithmdid not terminate.In the implementation of GTZ in Singular, the original ideal is first decom-posed using factorizing Gro¨bner bases algorithm and then the minimal associatedComputation of the Minimal Associated Primes 5Table 1. Timing resultsSource Code Dim Prim.compsMin.ass.Emb.compsEquidim? thispaperGTZ CharDGP 1 3 4 4 0 Yes 39 37 1037DGP 2 3 16 15 1 No 57 40 86DGP 3 2 11 4 7 No 6 4 2DGP 4 6 4 3 1 No 18 17 14DGP 7 3 6 6 0 Yes 26 20 76DGP 14 1 8 2 6 No 9 7 5DGP 20 4 2 1 1 No 15 14 3185DGP 21 9 9 1 8 No 3 2 1DGP 22 2 9 7 2 No 33 25 370DGP 23 2 18 12 6 No 91 71 22750DGP 24 8 6 5 1 No 14 9 12DGP 25 5 7 5 2 No 101 81 1615DGP 27 4 3 3 0 Yes 13 9 11DGP 28 7 2 2 0 Yes 30 27 18DGP 29 2 12 1 11 No 4 2 9DGP 30 1 14 14 0 Yes 283 259 12145DGP 31 1 1 1 0 Yes 10 10 3DGP 32 2 17 8 9 No 21 15 34DGP 33 2 3 3 0 No 10 8 5CCT M 5 3 3 0 No 58 48 2268CCT 83 5 3 3 0 No 133 603 98CCT O 2 5 5 0 Yes 26 209 3New 1 9 4 4 0 No 281 * 2383New 2 3 11 8 3 No 120 * 32065New 3 3 11 8 3 No 69 * 27088primes of each component are computed. We do the same decomposition in ouralgorithm.We see that for time consuming computations, our proposed algorithm isalways faster than GTZ algorithm.4 AcknowledgementsThe author thanks Teresa Krick for her guidance in this work, and Gert-MartinGreuel, Gerhard Pfister and Hans Scho¨nemann for all their help during my stayat the University of Kaiserslautern.References1. Laplagne, S.: An algorithm for the computation of the radical of an ideal. In: ISSAC’06: Proceedings of the 2006 international symposium on Symbolic and algebraiccomputation, New York, NY, USA, ACM Press (2006) 191–1956 Santiago Laplagne2. Krick, T., Logar, A.: An algorithm for the computation of the radical of an ideal inthe ring of polynomials. AAECC9, Springer LNCS (1991) 195–2053. Gianni, P., Trager, B., Zacharias, G.: Bases and primary decomposition of ideals.J. Symbolic Computation (1988) 149–1674. Greuel, G.M., Pfister, G., Schonemann, H.: Singular 3.0. A Computer AlgebraSystem for Polynomial Computations, Centre for Computer Algebra, University ofKaiserslautern (2005) http://www.singular.uni-kl.de.5. Greuel, G.M., Pfister, G.: A Singular Introduction to Commutative Algebra.Springer (2002)6. Decker, W., Gruel, G.M., Pfister, G.: Primary decomposition: Algorithms andcomparisons. Algorithmic algebra and number theory, Springer Verlag, Heidelberg(1998) 187–2207. Caboara, M., Conti, P., Traverso, C.: Yet another algorithm for ideal decomposition.Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (1997) 39–548. Decker, W., Pfister, G., Schoenemann, H.: primdec.lib. A singular 3.0 libraryfor computing primary decomposition and radical of ideals (2005)

Computation of the Minimal Associated Primes

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Computation of the Minimal Associated Primes

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