Improvements to a Triangulation-Decomposition Algorithmfor Ordina ry Dif ferentia l System s in Highe r Degree Cases

Evelyne HubertINRIA

Sophia Antipolis, France

Evelyne.Hubert@inria.fr

ABSTRACTWe introduce new ideas to improve the efficiency and ra-tionality of a triangulation decomposition algorithm. Onthe one hand we identify and isolate the polynomial remain-der sequences in the triangulation-decomposition algorithm.Subresultant polynomial remainder sequences are then usedto compute them and their specialization properties are ap-plied for the splittings. The gain is two fold: control of ex-pression swell and reduction of the number of splittings. Onthe other hand, we remove the role that initials had in pre-vious triangulation-decomposition algorithms. They are notneeded in theoretical results and it was expected that theyneed not appear in the input and output of the algorithms.This is the case of the algorithm presented. New algorithmsare presented to compute a subsequent characteristic decom-position from the output of the triangulation decompositionalgorithm where the initials need not appear.

Categories and Subject DescriptorsI.1.2H.4 [Symbolic and Algebraic Manipulation]: Com-puting Methodologies—Algorithms

General TermsAlgorithms

KeywordsSystems of differential equations, differemtial elimination, differ-ential ideal theory, triangular sets, subresultant polynomialremainder sequence.

1. INTRODUCTIONTriangulation-decomposition algorithms for differential sys-

tems take their roots in the work of Riquier [33]. Given asystem of partial differential equations they compute a rep-resentation that is equivalent but with which the solutionset can be analyzed. A central problem is to describe the

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.ISSAC’04,July 4–7, 2004, Santander, Spain.Copyright 2004 ACM 1-58113-827-X/04/0007 ...$5.00.

arbitrariness coming into the solution set and consistent setsof initial conditions that ensure existence and uniqueness ofa solution, analytic or as a formal power series [33, 5, 36, 24,20]. Elimination question are in the scope of triangulation-decomposition algorithms. For examples of applications seefor instance [15, 11, 27, 22].

Triangulation-decomposition algorithms were developedwith differential algebra by Ritt and Kolchin [34, 23] andan effective elimination theory was developed by Seidenberg[37]. They were further developed in a computer algebracontext [4, 5, 8, 10, 17, 26, 35, 38, 42]. The elementary op-eration is reduction of a differential polynomial by another.That means pseudo-division by a differential polynomial andits derivatives. More or less obvious polynomial remaindersequences (prs) go around unidentified. Our goal is to iden-tify and isolate those prs and modify the algorithm to takeadvantage of the computational efficiency and specializationproperties of subresultant prs [12, 29, 14, 25]. We base ourwork on the algorithm by Boulier et al. [4, 5]. In this paperwe explore first the simpler case of ordinary differential sys-tems to make clear the basic ideas and check the computa-tional gain in practice. Our implementation of the strategywas tested. We can report a clear benefit (90% time gainon the double pendulum problem!). We shall next move onto partial differential systems, where the algorithm is moreinvolved due to the update of the ∆-polynomials.

The use of subresultant prs so as to reduce the size ofcoefficients appears in the context of triangulation decom-position of polynomial systems in [30, 1] and only in thespecific case of ranking change for differential systems [7].The additional specialization properties are put into actionin the context of triangulation-decomposition for polynomialsystems in [16, 39, 13, 40]. In the former approach the prsis interrupted when an initial is a zero divisor. That is notadequate in our context. In the latter approach the prs iscomputed to the end and cases are considered a posteriori.It is the strategy we develop.

In a nonlinear differential context we cannot expect tohave a finite set that is both generating and allows to decidemembership to a (radical) differential ideal, as it is the caseof Grobner basis in the polynomial case [9, 31]. Geometri-cally speaking, we can only make statements valid outside ofan hypersurface, i.e. valid only where (the product of) a setof differential polynomials are not zero. The key theoreti-cal results to effective algorithms require only the separantsto be non zero [34, 4, 5, 19]. But so far the initials werealso required to be non zero as the full reduction neededin the triangulation-decomposition algorithm assumed pre-

191

multiplication by those. Our algorithm variant remedies thesituation: we use only partial reduction, the algebraic reduc-tion being processed within the prs machinery. We thus donot systematically split on the initials. Actually, for mostcases where there would be two splittings, both on the initialand the separant, only one splitting occurs in our algorithm,either on the initial or the separant. When differential poly-nomials are of degree one in their leader, the separant andthe initial are the same thing. So a gain on the number ofsplittings appears for higher degree differential polynomials.

The algorithm presented here thus gains on two aspects:expression swell and number of splittings.

The prs strategy gives new specifications to the output ofthe triangulation-decomposition algorithm. A triangulation-decomposition is refined in a characteristic decompositionby purely algebraic means [17]. The algorithm of [17] isbased on Grobner bases computations and works withoutany change with the new specifications that initials are notrequired to be non zero. On the contrary, the algorithmsbased on pseudo-gcd, [6, 2] and [19, Algorithm 7.1], requirethat the initials are nonzero. We give a new algorithm basedon pseudo-gcd that no longer requires that hypothesis.

The present approach thus makes clear the predominantrole of separants over initials in a differential context.

We tried our best to make the novel ideas stand out butthe paper relies a lot on previous work. Our basic referenceare the tutorials [18, 19] that give a detailed presentation of aselection of triangulation-decomposition algorithms togetherwith detailed proofs of results about triangular sets as devel-oped by many authors. The notations of the present papercomply with those notes. As restricting to ordinary differ-ential systems though, some difficult technicality disappear.Part of this paper therefore provides a good introduction totriangulation-decomposition algorithms for differential sys-tems.

Section 2 reviews differential algebra and triangulation-decomposition algorithms. Precise definition for the trian-gular sets used in this paper are given in Section 3 with theoutline of a characteristic decomposition algorithm. In Sec-tion 4 we present the former version of the triangulation-decomposition algorithm to be modified. The modifica-tion appears in Section 6 after we formalize the specializa-tion properties of subresultant prs in terms of ideals in Sec-tion 5. In Section 7 we review the final step of the differentialpart of the characteristic decomposition algorithm under theweaker hypothesis that only the separants are assumed tobe nonzero. Section 8 presents the purely algebraic part ofthe characteristic decomposition algorithm under that sameweaker hypothesis. Evidence of the gain of the prs strategyis shown in Section 9 through an experimental comparison.

2. DIFFERENTIAL ALGEBRAWe note F JYK the ring of differential polynomials in the

differential indeterminates Y = y1, . . . , yn. Typically Fis a field of rational functions in the independent variable.Consider a set Σ of differential polynomials in F JYK. Itrepresents a differential system in the dependent variablesy1, . . . , yn. A zero (or solution) of Σ is given by a differentialfield extension, as for instance Laurent series or meromor-phic functions, of F and a n-tuple in this field that makes thedifferential polynomial vanish under substitution. The rad-

ical differential ideal generated by Σ, noted1 JΣK, is shownto be the biggest set that has the same zero set than Σ.Studying JΣK allows to answer typical questions on the zeroset of Σ.

Beside JΣK for the radical differential ideal generated byΣ we write [Σ], (Σ), 〈Σ〉 for the differential ideal, the idealand the radical ideal generated by Σ.

Let H be a subset of F JYK. H∞ denotes the minimalsubset of F JYK that contains F \ 0 and H and is stableby multiplication and taking factors i.e. a, b ∈ H∞ ⇔ ab ∈H∞. Let I be an ideal of F JYK. We define the saturationof I by a subset H of F JYK as I :H∞ = q ∈ F JYK | ∃h ∈H∞ s.t. h q ∈ I.

To complete the picture, JΣK : H∞ is the biggest set ofdifferential polynomials that vanish on the subset of zerosof Σ that are not zeros of any element of H. Some zero ofJΣK : H∞ can make elements of H vanish though. See thedefinition of adherence in [34].

We introduce here a new type of monoid that simplifysome write up. The set h + p |h ∈ H∞, p ∈ I is stableby multiplication. By H∞ + I we shall denote h + p |h ∈H∞, p ∈ I∞. One easily checks the following.

Proposition 2.1 Let I be a (differential) ideal and H asubset of F JYK. We have the following equalities:

- H∞ + I :H∞ = H∞ + I- I : (H∞ + I) = I :H∞.

Informally speaking, precise terminology being given innext section, triangulation-decomposition algorithms com-pute a representation of a radical differential ideal JΣK, whereΣ is a finite set of differential polynomials, as an intersectionof radical differential ideals JAK :H∞ where A is a triangularset and H defines singular points for A. The decompositionof the radical differential ideal translates into a decomposi-tion of the zero set of Σ as a union of the nonsingular zerosof the triangular sets. The zero set of a radical differentialideal defined by a triangular set is easy to analyze.

There are variations on the type of triangular sets that oneobtains and the properties they conceal. The output trian-gular sets can be successively refined at different stages inthe algorithm. In this paper we shall speak of triangulation-decomposition algorithm to mean the first stage that pro-vides an equivalent set of systems with a triangular shape.

We eventually look for a characteristic decomposition ofJΣK, i.e. a representation of JΣK as an intersection of charac-terisable differential ideals. Intrinsically, those are the differ-ential ideals that are properly defined by their characteristicset [17]. Practically, they are defined by regular differentialchains. Then membership to the components and thereforeto the radical differential ideal can be tested. Applicationsof characteristic decomposition are illustrated by examplesin [19, Section 8].

Ritt [34, 23] provides an algorithm to compute charac-teristic decomposition where the components are prime dif-ferential ideals. The algorithm intertwines differential andalgebraic operations, those latter being factorizations in al-gebraic extensions. Factorization free versions are presentedin [8, 17]. On the other hand, Seidenberg established an ef-

1The classical notation, Σ, leads to a contorted write up inan algorithmic context. Note the difference between the nota-tion F JYK for a differential polynomial ring and the conventionalnotation R[[X]] for a power series ring .

192

fective elimination theory for differential systems [37]. Basedon this work, Wang [38] gives a triangulation-decompositionalgorithm for ordinary differential systems according to elim-ination ranking. We base our work on the algorithm byBoulier et al. [5] that is implemented in the Maple diffalgpackage [3]. The splitting strategy of that algorithm owes toSeidenberg’s scheme but the differential polynomials thereare treated on the fly rather than in a top-down elimination.

3. CHARACTERISTIC DECOMPOSITIONWe apply here the strategy of [17]: first obtain some de-

composition in terms of differential triangular sets and thenrefine those towards regular differential chains by purely al-gebraic means. We distinguish three stages in the algorithm.Before we proceed to present it we give the terminology forthe differential triangular sets involved. The terminology ofranking, leader, initial, separant reduction and partial re-duction is consensual and is not repeated here. See [19].

Definition 3.1 A subset A of F JYK is a weak differentialtriangular set if none of its elements belongs to F and lead(a)is not a derivative of lead(b), for any two distinct elementsa and b.

It is a differential triangular set, or d-triangular set forshort, if furthermore any element of A is partially reducedw.r.t. the other ones.

L(A) = lead(A), IA = init(A) and SA = sep(A) denoterespectively the sets of leaders, initials and separants of theelements of A. For a derivative u, A<u (respectively A≤u)denotes the elements of A with leader ranking lower than u(respectively lower or equal to u). ΘA is the set consistingof all the derivatives of the elements of A. When A is theempty set we take then convention that [A] = JAK = [0] andIA = SA = 1.

Definition 3.2 A tuple (G, A, H) is a RG-triple if G andH are finite subsets of F JYK and A is a weak d-triangularset s.t. SA ⊂ H∞ + [G ∪A]. If G is empty, we speak of theRG-pair (A, H).

A pair (A, H) is a regular differential system if A is a d-triangular set, the elements of H are partially reduced w.r.t.A and SA ⊂ H∞ + [A].

Note that the initials no longer appear in the definitionof RG-pairs and RG-triples. Also the condition on the sep-arants is alleviated in the definition of regular differentialsystem, RG-pairs and RG-triples2 compared to [5],[19, Def-inition 4.7] where the requirement was SA ⊂ H∞. In thepresent case we still have [A] : (H ∪ SA)∞ = [A] : H∞ andall the results [19, Theorem 4.8, 4.10-11, 4.12-13, 5.5] holdunder that seemingly more general condition.

Definition 3.3 Let A be a d-triangular set. A is a regulardifferential chain if for each u ∈ L(A)

- init(Au) is not a zero divisor modulo (A<u) :I∞A<u

- sep(Au) is not a zero divisor modulo (A≤u) :I∞A≤u

If A is a regular differential chain then A is a characteristicset of [A] :S∞A so that [A] :S∞A is a characterisable differentialideal [19, Section 5].

2In the partial differential case we deal with quadruples, thefourth set containing the ∆-polynomials

Algorithm 3.4 d-characteristic-decompositionIn: Σ, Ω finite subsets of F JYKOut: A set C of regular differential chains s.t.- C is empty iff 1 ∈ JΣK :Ω∞

- JΣK :Ω∞ =⋂

C∈C [C] :S∞C otherwise

A := d-triangulation-decomposition (Σ, Ω);

B := auto-partial-reduce (A, H) | (A, H) ∈ A ;C := a-characteristic-decomposition (B, K) | (B, K) ∈ B return( C );

The algorithm d-triangulation-decomposition processes RG-triples, starting with (Σ, ∅, Ω), towards RG-pairs. The out-put is a set A of RG-pairs s.t.

JΣK :Ω∞ =⋂

(A,H)∈A

JAK :H∞.

The algorithm auto-partial-reduce transforms a RG-pair (A, H)into a regular differential system (B, K) s.t. [A] :H∞ = [B] :K∞. By [19, Theorem 4.12], [B] : K∞ is radical and so wecan write

JΣK :Ω∞ =⋂

(B,K)∈B

[B] :K∞.

The algorithm a-characteristic-decomposition applied to a reg-ular differential system (B, K) computes an irredundant char-acteristic decomposition of (B) : K∞ in a polynomial ringcontaining all the derivatives appearing in B and K. Theremight be more than one component. By [17], [19, Theorem5.5] that decomposition provides an irredundant character-istic decomposition of [B] :K∞. The output C obtained froma-characteristic-decomposition is therefore such that

JΣK :Ω∞ =⋂

C∈C

[C] :S∞C

is a characteristic decomposition, which needs not be irre-dundant.

The prs strategy comes in d-triangulation-decomposition.This implies a different output to that step: the initials nolonger appear. We thus modify the two subsequent steps tocomply with this new output.

4. TRIANGULATION-DECOMPOSITIONWe reproduce here the Rosenfeld-Grobner algorithm [5]

in the simpler case of ordinary differential systems and ina way we can modify it simply to introduce a prs strategy.The sub-algorithm d-split is the piece to be modified and thesub-algorithm reduce is chosen accordingly.

Algorithm 4.1 d-triangulation-decompositionIn: Σ, Ω finite subsets of F JYKOut: A a set of RG-pairs s.t.- A is empty if it is detected that 1 ∈ JΣK :Ω∞

- JΣK :Ω∞ =⋂

(A,H)∈A JAK :H∞ otherwise

S := (Σ, ∅, Ω) ; A := ∅ ;

while S 6= ∅ do(G, A, H) := an element of S ;

S := S \ (G, A, H) if G = ∅ thenA := A ∪ (A, H) ;

193

elsep := an element of G ;q := reduce (p, A) ;if q = 0 thenS := S ∪ (G \ p, A, H) ;

elif q /∈ F thenS := S ∪ d-split ( G \ p, A, H, q );

fi;

fi;

od;

return( A );

The output of d-split shall consist of RG-triples that arelower than (G, A, H) in the sense of [19, Section 6.2]. So thealgorithm terminates. It is proved correct by showing thatthe following properties are loop invariants.I1 Elements of S and A are respectively RG-triples andRG-pairs

I2 JΣK :Ω∞ =⋂

(G,A,H)∈S

JG ∪AK :H∞ ∩⋂

(A,H)∈A

JAK :H∞

The following d-split algorithm works with a full differen-tial reduction. More precisely, for reduce in d-triangulation-decomposition we take [19, Algorithm 3.13]. The RG-triples(G, A, H) in S and RG-pairs (A, H) in A need then the ad-ditional property that IA ∈ H∞ + [A].

Algorithm 4.2 d-splitIn: a RG-triple (G, A, H) and p reduced w.r.t. A.Out: T a set of RG-triples lower than (G ∪ p, A, H)

s.t. JG ∪ p ∪AK :H∞ =⋂

(F,B,K)∈T

JF ∪BK :K∞

u := lead(p) ;

ip := init(p); pi := p− ip rank(p);

sp := sep(p); ps := deg(p, u) p− u sp;

T := (G ∪ ip, pi, A, H) , (G ∪ sp, ps, A, H ∪ ip) ;if no element of lead(A) is a derivative of u then

T := T ∪ (G, A ∪ p, H ∪ sp, ip);else

a := the element of A with leader a derivative of u;

T := T ∪ (G ∪ a, A \ a ∪ p, H ∪ sp, ip);fi;

return(T );

The correctness of the algorithm comes from the followingequality [19, Proposition 6.7]:

JG ∪ p ∪AK :H∞ = JG ∪ p ∪AK : (H ∪ ip, sp)∞∩ JG ∪ p, sp ∪AK : (H ∪ ip)∞ ∩ JG ∪ p, ip ∪AK :H∞

5. SUBRESULTANTSIn this section we translate in terms of ideals the spe-

cialization properties of subresultants and their links to gcdcomputations. The text reference is [29] and we use the con-vention of [25] that allows flexibility on the degrees of thepolynomials.

We take two differential polynomials a and b that we con-sider as polynomial in a distinguished variable x with coef-ficients in a polynomial ring F [X]. In fact x is the leaderof the differential polynomials and X is the set of the other

derivatives occurring in the polynomials. Let a be of degreee and b be of degree f or lower with f > e.

For i less than e, the ith subresultant of b and a is thedeterminant polynomial of the matrix the rows of which areformed by the coefficients of xe−i−1b, . . ., b, xf−i−1a, . . ., a[29, Section 7.7]. Write it as:

subresi(b, f, a, e) = detpol(xe−i−1b, . . . , b, xf−i−1a, . . . , a)

In particular subrese(b, f, a, e) is equal to a multiplied by apower of its leading coefficient. The degree of subresi(b, f, a, e)in x is i or lower. If it is i, subresi(b, f, a, e) is regular. Thereexist ti, ui s.t. subresi(b, f, a, e) = tia + uib [29, Proposi-tion 7.7.4]. The coefficients of subresi(b, f, a, e) are given byminors of the Sylvester matrix of a and b. They are thuspolynomial functions of the coefficients of a and b dependingon e and f only.

Lemma 5.1 In F [X][x] let a be of degree e and b of degreef or lower, with e < f . For i between 0 and e, let si =subresi(b, f, a, e) and ci be the coefficient of the ith power ofx in si. Then a and b belong to 〈c0, c1, . . . , ci−1, si〉 : cica,where ca is the leading coefficient of a in x.

Proof. Consider a field morphism φ : F(X) → F and,for p ∈ F [X][x], write p for the polynomial obtained by

applying φ on its coefficients. Assume that a and b havea gcd g of degree i. Assume furthermore that ca and ci

are not zero. As there exist ti and ui s.t. tia + uib = si,g must divide si. As g and si now have the same de-gree, si is also a gcd of a and b. It implies that the re-mainder prem (a, si, x) in the pseudo-division of a by si

w.r.t. x is zero [18, Proposition 3.2]. As prem (a, si, x) =detpol(xe−isi, . . . , si, a) [29, Corollary 7.5.4] and ci 6= 0,φ( prem (a, si, x)) = prem (a, si, x).

By [29, Corollary 7.7.9 and 7.8.2], as ca 6= 0, a and b havea gcd g of degree i iff c0 = c1 = . . . = ci−1 = 0 and ci 6= 0.Therefore

φ(c0) = φ(c1) = . . . = φ(ci−1) = 0 and φ(ca ci) 6= 0⇒ φ(prem(a, si, x)) = 0.

By the Hilbert Nullstellensatz we thus have prem(a, si, x)∈〈c0, . . . , ci−1〉 : caci so that a ∈ 〈c0, . . . , ci−1, si〉 : caci sincethere exists q ∈ F [X][x] s.t. prem (a, si, x) = ce−i+1

i a−qsi.Similarly for b.

Proposition 5.2 In F JYK let a be of degree e and b ofdegree f or lower, with e < f , in their common leader x.Let r0, r1, . . . , rκ be the regular subresultants of a and b byincreasing degree: r0 is the resultant and rκ is the product ofa by a power of its leading coefficient, ia. Let now c0, . . . , cκ

be the sequence of the leading coefficients of the ri w.r.t. x.For any S ⊂ F JYK

JS, a, bK : ia =

κ⋂k=1

JS, c0, . . . , ck−1, rkK :ckia

Proof. Consider first for i ≤ e, si = subresi(b, f, a, e)and di = coeff(si, x, i). As si ∈ (b, a) for all 0 ≤ i ≤ e ands0 = d0, using the fact that JSK = JS, hK ∩ JSK : h for all

194

subset S and differential polynomial h we can write

JS, a, bK = JS, s0, a, bK= JS, d0, s1, a, bK :d1 ∩ JS, d0, d1, a, bK= JS, d0, s1, a, bK :d1 ∩ JS, d0, d1, s2, a, bK :d2

∩ JS, d0, d1, d2, a, bK= . . .= JS, d0, s1, a, bK :d1 ∩ . . . ∩ JS, d0, . . . , de−1, a, bK

As de is a positive power of ia, quotienting by ia on all sideand applying Lemma 5.1, we obtain

JS, b, aK : ia =

e⋂k=1

JS, d0, . . . , dk−1, skK :dkia.

For an irregular subresultant si, i > 0, we have di = 0.The component quotiented by di in the decomposition istrivial. We can thus write the above decomposition in termsof the regular subresultants only.

We give the specifications of the algorithm to be used tocompute the sequence of regular subresultants. The originalalgorithm to compute the subresultant polynomial remain-der sequence is presented in textbooks as for instance [29,Definition 7.6.4]. Versions that optimize the intermediatesizes are presented in [14, 25].

Algorithm 5.3 subresultant-prsIn: b and a polynomials in a distinguished variable xOut: a sequence of two lists- the list of the regular subresultants by increasing degree

out of subresi(b, f, a, e)i where e = deg(a, x) and f =max(deg(b, x), e + 1)

- the list of the respective leading coefficients

6. NEW SPLITTING ALGORITHMWe give now the algorithm d-split that implements the

prs strategy. It is coupled with partial reduction. In thealgorithm d-triangulation-decomposition the sub-algorithmreduce shall be that of partial reduction ([19, Algorithm3.12]) or one that allows furthermore algebraic reductions bythe differential polynomials that are of degree one in theirleader. Indeed we no longer systematically split on the ini-tials when introducing a new differential polynomial in theweak d-triangular set.

We compute a complete prs when a single step of algebraicreduction would have been performed in the latter version.We then need to split on initials, but not on the separants.

Algorithm 6.1 d-splitIn: a RG-triple (G, A, H), p partially reduced w.r.t. AOut: T a set of RG-triples lower than (G ∪ p, A, H)

s.t. JG ∪ p ∪AK :H∞ =⋂

(F,B,K)∈T

JF ∪BK :K∞

u := lead(p) ;

if L(A) contains no derivative of u thensp := sep(p); ps := deg(p, u) p− u sep(p);T := (G, A ∪ p, H ∪ sp) , (G ∪ ps, sp, A, H);

elif L(A) contains a proper derivative of u thena := the element of A with leader a derivative of u;

sp := sep(p); ps := deg(p, u) p− u sep(p);T := (G ∪ a, A \ a ∪ p, H ∪ sp) ,

(G ∪ ps, sp, A, H) ;

else # in this case u ∈ L(A)

a := the element of A with leader u;

if deg(a, u) ≤ deg(p, u) thenia := init(a); ai := a− iarank(a);if lead(ai) = u thenT := ( G ∪ ia, A \ a ∪ ai, H ) ;H ′ := H ∪ ia;[r0, . . . , rκ], [c0, . . . , cκ] := subresultant-prs (p, a, u);

fi;

elseip := init(p); pi := p− iprank(p);T := ( G ∪ ip, pi, A, H ) ;H ′ := H ∪ ip;[r0, . . . , rκ], [c0, . . . , cκ] := subresultant-prs (a, p, u);

fi;

G0 := G ∪ c0;for k from 1 to κ (while ck−1 /∈ H∞) do

Gk := Gk−1 ∪ ck−1 ;T := T ∪ (Gk, A \ a ∪ rk, H ′ ∪ ck) ;

od;

fi;

return ( T );

In the case where lead(p) /∈ L(A) the only change com-pared to Section 4 is that we split according to the separantof p only and not according to its initial.

The major difference occurs when lead(p) ∈ L(A). Firstnote that if lead(ai) 6= u then init(a) is a factor of sep(a). Byhypothesis it already belongs to H∞ + [A]. Note also thatin all the RG-triples produced the element a is replaced inthe d-triangular set by an element with the same leader.

By Proposition 5.2 and the basic splitting property [19,Proposition 6.6] the following equality holds:

JG ∪ p ∪AK :H∞ =⋂

(F,B,K)∈T

JF ∪BK :K∞

Also the elements of T are seen to be lower than the inputRG-triple (G ∪ p, A, H).

What needs to be shown is that all the elements of Tare proper RG-triples. More precisely, in the case whereu ∈ L(A) we need to show that

sep(rk) ∈ (H ′ ∪ ck)∞ + JGk ∪A ∪ rk \ aK

Let Ak = A \ a ∪ rk and Hk = H ′ ∪ ck. Notethat Gk = G ∪ c0, . . . , ck−1. By Lemma 5.1, a and p be-long to JGk ∪AkK : H∞

k and there exists q ∈ F [ΘY≤u] s.t.for a certain d ∈ N, cd

k a ≡ q rk mod 〈c0, . . . , ck−1〉. Obvi-ously all the ci involve only derivatives lower than u. Thuscd

k sep(a) ≡ q sep(rk) mod 〈c1, . . . , ck−1, rk〉. As sep(a) ∈H∞+JG ∪ p ∪AK by hypothesis, cd

k sep(a) ∈ H∞k +JGk ∪AkK.

It follows that sep(rk) ∈ H∞k + JGk ∪AkK.

7. TO REGULAR D-SYSTEMSFor a pair (A, H) as output by Algorithm 4.1 and 6.1 we

compute a pair (B, K) that is a regular differential systemwith the property that L(A) = L(B) and [A] : H∞ = [B] :K∞. The proof of algorithm auto-partial-reduce, [19, Algo-rithm 6.8], relies on the initials to belong to H. We remedyto that here. The proof is not essentially different thoughthe presentation is lighten up by using Proposition 2.1.

Algorithm 7.1 auto-partial-reduce

195

In: (A, H) a RG-pairOut: Either- nothing if it is detected that 1 ∈ [A] :H∞.- a regular differential system (B, K) with

L(A) = L(B) and [A] :H∞ = [B] :K∞

B := ∅;for u ∈ L(A) increasingly do

a := pd-red(Au, B); (see [19, Algorithm 3.12])

if lead(a) 6= u then return( ); fi;

B := B ∪ a;od;

K := SB ∪ pd-red(h, B) | h ∈ H \ SA;if 0 ∈ K then return( ); fi;

return((B, K));

We prove first that after the for loop, either it is detectedthat 1 ∈ [A] :S∞A or for any u ∈ L(A) we have:I1 L(A≤u) = L(B≤u)

I2 S∞A≤u+ (ΘA<u) = S∞B≤u

+ (ΘB<u)

I3 (ΘA≤u) :S∞A≤u= (ΘB≤u) :S∞B≤u

The properties are true for the lowest u ∈ L(A) sincethen Au = Bu. Let us assume that the properties hold forall v ∈ L(A) with v < u, that is as we start a new iterationfor u.

Induction hypotheses I3 and I2 imply (ΘA<u) : S∞A<u=

(ΘB<u) : S∞B<u([19, Proposition 3.8]). By induction hy-

pothesis I1, u /∈ ΘL(B<u) since A is a weak d-triangularset. The reduction of Au by B<u is in fact a reductionon the coefficients of Au seen as a polynomial in u. Iflead(a) 6= u it means that all the coefficients of the pos-itive powers of u are reduced to 0. Those are the coeffi-cients of the separant of Au. Therefore sep(Au) belongs to(ΘB<u) :S∞B<u

= (ΘA<u) :S∞A<u. This implies 1 ∈ [A] :S∞A .

Otherwise lead(a) = u so that I1 is preserved.There exists s ∈ S∞B<u

such that s Au ≡ Bu mod (ΘB<u)and so s sep(Au) ≡ sep(Bu) mod (ΘB<u). By the latterequation, we see immediately that I2 is preserved. Then I3follows from the former equation.

So, unless we found that 1 ∈ [A] :S∞A , B is a d-triangularset and (ΘA<v) : S∞A = (ΘB<v) : S∞B for all v ∈ ΘY andtherefore [A] : S∞A = [B] : S∞B . Furthermore S∞A + [A] =S∞B + [B].

Since SA ∈ H∞ + [A] by hypothesis, [A] : H∞ = [A] :(H ∪SA)∞. An element k of K \SB comes from an elementh ∈ H \ SA: there exists an s ∈ S∞B s.t. sh ≡ k mod [B].Thus h ∈ K∞ + [B]. As s ∈ S∞A + [A], it follows thatk ∈ H∞ + [A]. Thus K∞ + [B] = H∞ + [A] and therefore[A] :H∞ = [B] :K∞.

8. TO REGULAR D-CHAINSLet (B, K) be a regular differential system in F JYK. Let

T be the set of derivatives present in B and K outside ofL(B). If (B) : (K ∪SB)∞ =

⋂ri=1(Ci) :I∞Ci

is an irredundantdecomposition in either F [T][L(B)] or F(T)[L(B)] then, asSB ⊂ K∞+[B], [B] :K∞ =

⋂ri=1[Ci] :S

∞Ci

is an irredundantcharacteristic decomposition in F JYK [19, Theorem 5.5 and6.2] [17, Theorem 3.10].

[17, Algorithm 3.8] computes an irredundant character-istic decomposition of a radical ideal of dimension 0, as is

(B) : (K ∪ SB)∞ when considered in F(T)[L(B)] [19, The-orem 7.5]. It can be applied here to compute the algebraiccharacteristic decompositions needed and therefore take theplace of a-characteristic-decomposition in Algorithm 4.1.

In [6, 2] and [19, Algorithm 7.1], variants of pseudo-gcdcomputations are applied to compute the characteristic de-composition of (B) :K∞ in the case the initials of B belongto K. This section presents an algorithm in that spirit with-out the hypothesis on the initials. The algorithm iterates theextension of a squarefree regular chain by a new polynomial.

A triangular set A is a regular chain if init(Au) is nota zero divisor modulo (A<u) : I∞A<u

for all u ∈ L(A). Aregular chain is squarefree if additionally sep(Au) is not azero divisor modulo (A≤u) : I∞A≤u

or, equivalently, modulo

(A) :I∞A [18, Section 7].In the pseudo-gcd computation as proposed by Kalkbrener

[21] there are three main components that are intertwined.We use explicitly the algorithms split and relatively-prime.The first one splits the ideal defined by a regular chain intwo, the components that contain a given polynomial on theone hand, and the components modulo which that polyno-mial is not a zero divisor. The second algorithm computesthe saturation of an ideal defined by a regular chain by agiven polynomial. See [18, Algorithm 8.8 and 8.9] for accu-rate description.

We comply to the notations and definitions of [18]. Inparticular, if A = a1, . . . , ar is a triangular set s.t.lead(ai)<lead(ai+1) we write A = a1 M . . . M ar. For a convenientwrite up we introduce the ideals I(A) = (A) : I∞A andR(A) = 〈A〉 : I∞A . The notation F [X][x] for a polynomialring means that the ranking on X ∪ x is s.t. x > y for ally ∈ X.

Algorithm 8.1 extend-sqrfree-regular-chainIn:- F [X][x] a ring of polynomials- A a squarefree regular chain in F [X]- a a polynomial in F [X][x] with deg(a, x) > 0Out: a set C of squarefree regular chains in K[X][x] s.t.

((A) : I∞A + a) : sep(a)∞ =⋂

C∈C

(C) : I∞C is an irredundant

characteristic decomposition.

S := (A, a); C := ∅;while S 6= ∅ do

(C, c) := an element of S; S := S \ (C, c);if deg(c, x) 6= 0 then

(ZC ,UC) := split (F [X], C, lcoeff(c, x));S := S ∪ (B, tail(c, x)) | B ∈ ZC;C := C ∪ (B, c) | B ∈ UC;

fi;

od;

C :=⋃

(C,c)∈C relatively-prime (F [X][x], C, c, sep(c));

return( C );

The degree of tail(c, x) is lower than the degree c. Thealgorithm thus terminates. The following properties are in-variants of the while loop.I0 C is a squarefree regular chain for all (C, c) ∈ S ∪ C.

I1 a ≡ c mod I(C), for all (C, c) ∈ S ∪ C

I2 C M c is a regular chain for all (C, c) in C.

I3⋂

(C,c)∈S∪C I(C) is an irredundant decomposition.

196

I4 I(A) :s∞a =⋂

(C,c)∈S∪C I(C) :s∞aI0 and I3 come from the output properties of split. For

all B ∈ UC , lcoeff(c, x) is not a zero divisor modulo (B) :I∞B so B M c is a regular chain. Hence I4. When B ∈ ZC

then c ≡ tail(c) mod I(B). Hence I1. Property I1 impliesthat sep(a) ≡ ∂c

∂xmod I(C). So when deg(c, x) = 0 then

sep(a) ∈ I(C) and therefore the related component in thedecomposition can be dropped without affecting I3 nor I4.

Thenrelatively-prime(F [X][x], C, c, sep(c)) computes an ir-redundant characteristic decomposition of 〈I(C) + (c)〉 : ics∞c .Since C M c is a regular chain and SC has no element that di-vides zero modulo I(C M c), (I(C) + (c)) : ics

∞c = I(C M c) :

s∞c = (C M c) : H∞C M c. Thus 〈I(C) + (c)〉 : ics

∞c is radical

[18, Theorem 7.5] and therefore the output of relatively-primeconsists of squarefree regular chains.

From Property I1 and [18, Proposition 9.3], 〈I(A) :sa + (a)〉 :sa =

⋂(C,c)∈C 〈I(C) + (c)〉 : sc. By an argument similar to

[18, Proposition 7.6], (I(C) + (c)) : s∞c = (I(C) + (c)) :(icsc)

∞. Thus 〈I(A) :sa + (a)〉 : sa =⋂

(C,c)∈C(I(C) + (c)) :

(icsc)∞. Thanks to Property I3, that decomposition is irre-

dundant. The conclusion follows from the output propertiesof relatively-prime.

Algorithm 8.2 a-characteristic-decompositionIn:- B a triangular set in a polynomial ring F [X]- K a finite subset of F [X] such that SB ⊂ K∞

Out: a set C of squarefree regular chains such that- C is empty iff (B) :K∞ = (1).- (B) :K∞ =

⋂C∈C (C) :I∞C is an irredundant character-

istic decomposition otherwise.

C := ∅;for x in L(B) increasingly do

C :=⋃

C∈C

extend-sqrfree-regular-chain (F [X<x][x], C, Bx);

od;

for h ∈ K \ SB doC :=

⋃C∈C UC where (ZC ,UC) = split(F [X], C, h);

od;

return(C);

The correctness of the algorithm follows from the outputproperties of extend-sqrfree-regular-chain and split.

9. EXPERIMENTAL COMPARISONWe present the 9 problems on which the experimental

comparison was conducted. For each problem we shall givethe time to compute the characteristic decomposition andthe number of splittings that occur during the triangulation-decomposition whether by the standard algorithm or the oneimplementing the prs strategy. In the number of splittingswe distinguish, in parenthesis, the number of RG-triplesthat are effectively stacked. In the implementation indeedonly the RG-triples that fail trivial inconsistency tests arestacked. We make explicit the gain obtained by the prsstrategy both in terms of time and number of splittings.

The implementations of both algorithms were done withinthe diffalg package [3]. They are identical outside of thesplitting and update procedure of the RG-triples. The com-putations were run with Maple 9 on a Pentium IV 2.2 Ghz.For additionnal information see www-sop.inria.fr/cafe/

Evelyne.Hubert/diffalg/performances.

Problem 1 The prendulum in Cartesian coordinates. Weconsider Σ = m q+λ y+g, y−q, m p+λ x, x−p, x2+y2−1as a subset of Q(m, t) Jλ, x, y, p, qK with an orderly ranking.

Problem 2 We consider Σ = x−x (x+y), y+y (x+y), x2+y2 + z2 − 1 as a subset of Q Jx, y, zK with the purely elimi-nation ranking z > y > x

Problem 3 Lorenz system. We consider Σ = x − σ (y −x), y+x z−ρ x+y, z−x y+b z as a subset of Q(σ, ρ, b) Jx, y, zKwith the purely elimination ranking z > y > x.

Problem 4 Double pendulum. We consider Σ = x−2 x l+

2 X L, y − 2 y l + 2 Y L + 1, X + x− 2 X L, Y + y − 2 Y L +1, x2 + y2− 1, X2 + Y 2− 1 as a subset of Q Jx, y, l, X, Y, LKwith a block elimination ranking l, L x, y, X, Y Problem 5 Identifiability for microbial growth in a batchreactor [28]. Σ = (p2+x2)(x1+p3x1)−p1x2x1, (p2+x2)x2+p4p1x2x1, p1, p2, p3, p4 as a subset of Q Jp1, p2, p3, p4, x1, x2Kwith a block elimination ranking p1, p2, p3, p4 x1, x2.Problem 6 Identifiability in an immunological model formastisis in diary cows [28]. Σ = x1 − p1 x1 − p2 x1 x2, x2 −p3 x2 (1− p4 x2) + p5 x1 x2, p1, . . . , p5 as a subset ofQ Jp1, p2, p3, p4, p5, x1, x2K with a block elimination rankingp1, p2, p3, p4, p5 x1, x2.Problem 7 Non identifiability of parameters [32]. Σ =x1−p1x1

2−p2x1x2−u, x2−p3x12+p4, y−x1, p1, p2, p3, p4

as a subset of Q Jp1, p2, p3, p4, x1, x2, y, uK with a block elim-ination ranking x1, x2 p1, p2, p3, p4 y, u.Problem 8 Non observability of state variables. The samesystem as above but with a block elimination rankingp1, p2, p3, p4 x1, x2 y, u.Problem 9 Higher index DAE. See www-sop.inria.fr/cafe/Evelyne.Hubert/diffalg/performances.

standard algo. prs strategy gain# time split time split time split1 0.2 4(4) 0.1 2(2) 18% 50(50)%2 0.3 14(11) 0.2 9(6) 32% 36(45)%3 2.8 0(0) 1.4 5(1) 48%4 7.3 64(61) 0.8 20(12) 88% 69(80)%5 0.5 13(12) 0.3 9(5) 33% 31(58)%6 1.1 14(13) 0.4 4(3) 63% 71(77)%7 >240 >17(14) 0.5 3(3) >99%8 >240 >34(26) 0.8 4(2) >99%9 >900 >6(5) 97. 2(1) huge

10. REFERENCES[1] P. Aubry. Ensembles triangulaires de polynomes et

resolution de systemes algebriques. Implantation enAxiom. PhD thesis, Universite de Paris 6, 1999.

[2] P. Aubry and D. Wang. Reasonning about surfacesusing differential zero and ideal decomposition. InADG 2000, number 2061 in LNAI, 2001.

[3] F. Boulier and E. Hubert. diffalg: description andexamples of use. U. of Waterloo, 1998,www.inria.fr/cafe/Evelyne.Hubert/diffalg.

[4] F. Boulier, D. Lazard, F. Ollivier, and M. Petitot.Representation for the radical of a finitely generateddifferential ideal. In ISSAC’95. ACM, 1995.

[5] F. Boulier, D. Lazard, F. Ollivier, and M. Petitot.Computing representations for radicals of finitelygenerated differential ideals. Technical Report IT-306,LIFL, 1997.

197

[6] F. Boulier and F. Lemaire. Computing canonicalrepresentatives of regular differential ideals. In ISSAC2000. ACM, 2000.

[7] F. Boulier, F. Lemaire, and M. Moreno-Maza. Pardi!In ISSAC 2001. ACM, 2001.

[8] D. Bouziane, A. Kandri Rody, and H. Maarouf.Unmixed-dimensional decomposition of a finitelygenerated perfect differential ideal. Journal ofSymbolic Computation, 31(6):631–649, 2001.

[9] G. Carra Ferro. Grobner bases and differentialalgebra. In AAECC, volume 356 of Lecture Notes inComputer Science. Springer-Verlag, 1987.

[10] S-C. Chou and X-S. Gao. Automated reasonning indifferential geometry and mechanics using thecharacteristic set method. Part II. Mechanicaltheorem proving. Journal of Automated Reasonning,10:173–189, 1993.

[11] P. A. Clarkson and E. L. Mansfield. Symmetryreductions and exact solutions of a class of non-linearheat equations. Physica, D70:250–288, 1994.

[12] G. E. Collins. Subresultants and reduced polynomialremainder sequences. J. Assoc. Comput. Mach.,14:128–142, 1967.

[13] S. Delliere. Triangularisation de systemesconstructibles. Application a l’evaluation dynamique.PhD thesis, Univ. de Limoges, 1999.

[14] L. Ducos. Optimizations of the subresultantalgorithm. Journal of Pure and Applied Algebra,145(2):149–163, 2000.

[15] M. Fliess and S.T. Glad. An algebraic approach tolinear and nonlinear control. In Essays on control:Perspectives in the theory and its applications,volume 14. Birkhauser, Boston, 1993.

[16] T. Gomez-Dıaz. Applications de l’evaluationdynamique. PhD thesis, Univ. de Limoges, 1994.

[17] E. Hubert. Factorisation free decompositionalgorithms in differential algebra. Journal of SymbolicComputation, 29(4-5):641–662, 2000.

[18] E. Hubert. Notes on triangular sets andtriangulation-decomposition algorithms I: Polynomialsystems. In Winkler and Langer [41].

[19] E. Hubert. Notes on triangular sets andtriangulation-decomposition algorithms II: Differentialsystems. In Winkler and Langer [41].

[20] E. Hubert and N. Le Roux. Computing power seriessolutions of a nonlinear pde system. In ISSAC 2003.ACM, 2003.

[21] M. Kalkbrener. A generalized Euclidean algorithm forcomputing triangular representations of algebraicvarieties. Journal of Symbolic Computation,15(2):143–167, 1993.

[22] A. A. Kapaev and E. Hubert. A note on the Lax pairsfor Painleve equations. Journal of Physics. A.Mathematical and General, 32(46), 1999.

[23] E. R. Kolchin. Differential Algebra and AlgebraicGroups, volume 54 of Pure and Applied Mathematics.Academic Press, 1973.

[24] Francois Lemaire. Les classements les plus generauxassurant l’analycite des solutions des systemesorthonomes pour des conditions initiales analytiques.In CASC 2002. Technische Universitat Munchen,

Germany, 2002.

[25] H. Lombardi, M-F. Roy, and M. S. El Din. Newstructure theorem for subresultants. Journal ofSymbolic Computation, 29(4-5):663–690, 2000.

[26] E. L. Mansfield. Differential Grobner Bases. PhDthesis, University of Sydney, 1991.

[27] E. L. Mansfield, G. J. Reid, and P. A. Clarkson.Nonclassical reductions of a 3+1-cubic nonlinearSchrodinger system. Computer PhysicsCommunications, 115:460–488, 1998.

[28] G. Margaria, E. Riccomagno, M. J. Chappell, andH. P. Wynn. Differential algebra methods for thestudy of the structural identifiability of rationalfunction state-space models in the biosciences.Mathematical Biosciences, 174(1):1–26, 2001.

[29] B. Mishra. Algorithmic Algebra, volume XIV of Textsand Monographs in Computer Science. Springer-VerlagNew York, 1993.

[30] M. Moreno-Maza. Calculs de pgcd au-dessus des toursd’extensions simples et resolution des systemesd’equations algebriques. PhD thesis, Universite Paris6, 1997.

[31] F. Ollivier. Standard bases of differential ideals. InAAECC’90, pages 304–321. Springer, 1991.

[32] F. Ollivier and A. Sedoglavic. Algorithmes efficacxespour tester l’identifiabilite locale. In ConferenceInternationale Francophone d’Automatique. IEEE,2002.

[33] C. Riquier. Les systemes d’equations aux deriveespartielles. Gauthier-Villars, Paris, 1910.

[34] J. F. Ritt. Differential Algebra, volume XXXIII ofColloquium publications. American MathematicalSociety, 1950. http://www.ams.org/online bks.

[35] C. Rust. Rankings of derivatives for eliminationalgorithms and formal solvability of analytic partialdifferential equations. PhD thesis, University ofChicago, 1998.

[36] C. J. Rust, G. J. Reid, and A. D. Wittkopf. Existenceand uniqueness theorems for formal power seriessolutions of analytic differential systems. In ISSAC’99.ACM, 1999.

[37] A. Seidenberg. An elimination theory for differentialalgebra. University of California Publications inMathematics, 3(2):31–66, 1956.

[38] D. Wang. An elimination method for differentialpolynomial systems. I. Systems Science andMathematical Sciences, 9(3):216–228, 1996.

[39] D. Wang. Decomposing polynomial systems intosimple systems. Journal of Symbolic Computation,25(3):295–314, 1998.

[40] D. Wang. Computing triangular systems and regularsystems. Journal of Symbolic Computation,30(2):221–236, 2000.

[41] F. Winkler and U. Langer, editors. Symbolic andNumerical Scientific Computing, volume 2630 ofLNCS. Springer Verlag, 2003.

[42] W. T. Wu. On the foundation of algebraic differentialgeometry. Systems Science and MathematicalSciences, 2(4):289–312, 1989.

198