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A mod- of engines based on Gr¨obner bases, in terms of output size. ular algorithm for computing these GCDs appears in [14]: if sat(T ) is known to be radical, the performance (both in theory and practice) of this algorithm are very satisfactory 2. REGULAR CHAINS whereas if sat(T ) is not radical, the complexity of the algo- We review hereafter the notion of a regular chain and its rithm increases substantially w.r.t. the radical case. In this related concepts. Then we state basic properties (Proposi- paper, the ring A will be of the form k[x]/ sat(T ) while our tions 1, 2, 3, 4, and Corollaries 1, 2) of regular chains, which p algorithms will not need to compute a basis nor a character- are at the core of the proofs of the algorithms of Section 4. istic set of sat(T ). For the purpose of polynomial system p solving (when retaining the multiplicities of zeros is not re- Throughout this paper, k is a field, K is the algebraic closure quired) this weaker notion of a polynomial GCD is clearly of k and k[x] denotes the ring of polynomials over k, with sufficient. In addition, this leads us to a very simple pro- ordered variables x = x1 < · · · monomial, the leading term Extracting common work from similar computations. Up and the reductum of p, regarded as a univariate polynomial to technical details, if T consists of a single polynomial t in mvar(p), are called respectively the initial, the main de- whose main variable is the same as p, say v, computing gree, the rank, the head and the tail of p; they are denoted by Intersect(p,T,R) can be achieved by successively computing init(p), mdeg(p), rank(p), head(p) and tail(p) respectively. Let q be another polynomial of k[x]. If q is not constant, then we denote by prem(p, q) and pquo(p, q) the pseudo- (s1) the resultant r of p and t w.r.t. v, remainder and the pseudo-quotient of p by q as univariate polynomials in mvar(q). We say that p is less than q and (s2) a regular GCD of p and t modulo the squarefree part k k of r. write p ≺ q if either p ∈ and q∈ / or both are non- constant polynomials such that mvar(p) < mvar(q) holds, or mvar(p) = mvar(q) and mdeg(p) < mdeg(q) both hold. Observe that Steps (s1) and (s2) reduce essentially to com- We write p ∼ q if neither p ≺ q nor q ≺ p hold. puting the subresultant chain of p and t w.r.t. v. The algo- rithms of Section 4 extend this simple observation for com- Notations for polynomial sets. Let F ⊂ k[x]. We denote by puting Intersect(p,T,R) with an arbitrary regular chain. In hF i the ideal generated by F in k[x]. For an ideal I ⊂ k[x], broad terms, the intermediate polynomials computed during we denote by dim(I) its dimension. A polynomial is regular the “elimination phasis” of Intersect(p,T,R) are recycled for modulo I if it is neither zero, nor a zerodivisor modulo I. performing the “extension phasis” at essentially no cost. Denote by V (F ) the zero set (or algebraic variety) of F in Kn. Let h ∈ k[x]. The saturated ideal of I w.r.t. h, denoted The techniques developed for Intersect(p,T,R) are applied by I : h∞, is the ideal {q ∈ k[x] |∃m ∈ N s.t. hmq ∈ I}. to other key sub-algorithms, such as:

Triangular set. Let T ⊂ k[x] be a triangular set, that is, a • the regularity test of a polynomial modulo the satu- set of non-constant polynomials with pairwise distinct main rated of a regular chain, see Section 4, variables. The set of main variables and the set of ranks of the polynomials in T are denoted by mvar(T ) and rank(T ), • the squarefree part of a regular chain, see Appendix B. respectively. A variable in x is called algebraic w.r.t. T if it belongs to mvar(T ), otherwise it is said free w.r.t. T . The primary application of the operation Intersect is to ob- For v ∈ mvar(T ), denote by Tv the polynomial in T with tain triangular decomposition encoding all the points of the main variable v. For v ∈ x, we denote by Tcommutative ring with unity. where Tmax is the polynomial in T with maximum rank, and Let p,t,g ∈ A[y] with t 6= 0 and g 6= 0. We say that g ∈ A[y] the initial of Tmax is regular w.r.t. sat(T \ {Tmax}). The is a regular GCD of p,t if: empty regular chain is simply denoted by ∅.

(R1) the leading coefficient of g in y is a regular element; Triangular decomposition. Let F ⊂ k[x] be finite. Let T := {T1,...,Te} be a finite set of regular chains of k[x]. We (R2) g belongs to the ideal generated by p and t in A[y]; call T a Kalkbrener triangular decomposition of V (F ) if we e (R3) if deg(g,y) > 0, then g pseudo-divides both p and t, have V (F )= ∪i=1W (Ti). We call T a Lazard-Wu triangular e that is, prem(p,g)= prem(t,g) = 0. decomposition of V (F ) if we have V (F )= ∪i=1W (Ti).

Proposition 1 (Th. 6.1. in [1]). Let p and T be re- Proposition 5. For 1 ≤ k ≤ n, let T ⊂ k[x1,...,xk−1] spectively a polynomial and a regular chain of k[x]. Then, be a regular chain, possibly empty. Let p,t,g ∈ k[x1,...,xk] prem(p,T ) = 0 holds if and only if p ∈ sat(T ) holds. be polynomials with main variable xk. Assume T ∪ {t} is a regular chain and g is a regular GCD of p and t modulo sat(T ). We have: Proposition 2 (Prop. 5 in [16]). Let T and T ′ be two p regular chains of k[x] such that sat(T ) ⊆ sat(T ′) and ′ p p dim(sat(T )) = dim(sat(T )) hold. Let p ∈ k[x] such that p (i) if mdeg(g)= mdeg(t), then sat(T ∪ t)= sat(T ∪ g) ′ p p is regular w.r.t. sat(T ). Then p is also regular w.r.t. sat(T ). and W (T ∪ t) ⊆ Z(hg,T ∪ t) ∪ W (T ∪ g) both hold, (ii) if mdeg(g) < mdeg(t), let q = pquo(t,g), then T ∪ q is Proposition 3 (Prop. 4.4 in [1]). Let p ∈ k[x] and a regular chain and the following two relations hold: T ⊂ k[x] be a regular chain. Let v = mvar(p) and r = (ii.a) sat(T ∪ t) = sat(T ∪ g) ∩ sat(T ∪ q), prem(p,T≥v) such that r ∈ sat(Tprime ideal p associated with sat(T ), we have Since mdeg(t) = mdeg(g) holds, we have q ∈ k[x1,...,xk−1], e0 p 6∈ p, and thus we have hg ht = q hg . Since ht and hg are reg- ular modulo sat(T ), the same property holds for q. To- (iii) the iterated resultant res(p,T ) is not zero. gether with (2), we obtain g ∈ sat(T ∪ t). Therefore p sat(T ∪ t) = sat(T ∪ g). The inclusion relation in (i) p p Corollary 2. Let p ∈ k[x] and T ⊂ k[x] be a regular follows from (1). chain. Let v := mvar(p) and r := res(p,T≥v). We have: We prove (ii). Assume mdeg(t) > mdeg(g). With (1) and (2), this hypothesis implies that T ∪ q is a regular chain (1) the polynomial p is regular w.r.t. sat(T ) if and only if and t ∈ sat(T ∪ q) holds. Since t ∈ sat(T ∪ g) also r is regular w.r.t. sat(T

Let p,t be two polynomials of k[x1,...,xk], for k ≥ 1. Let (L1) sat(T ) ⊆ sat(Ti) m = deg(p,xk), n = mdeg(t,xk). Assume that m, n ≥ 1. p p

Let λ = min(m, n). Let T be a regular chain of k[x1,...,xk−1]. (L2) W (Ti) ⊆ V (p) (or equivalently p ∈ sat(Ti)) p Let B = k[x1,...,xk−1] and A = B/ sat(T ). e p (L3) V (p) ∩ W (T ) ⊆ ∪i=1W (Ti)

Let S0,...,Sλ−1 be the subresulant polynomials [15, 8] of B i p and t w.r.t. xk in [xk]. Let si = coeff(Si,xk) be the We write as (p,T ) −→ T1,...,Te. Observe that the above principle subresultant coefficient of Si, for 0 ≤ i ≤ λ − 1. three conditions are equivalent to the following relation. If m ≥ n, we define Sλ = t, Sλ+1 = p, sλ = init(t) and sλ+1 = init(p). If m < n, we define Sλ = p, Sλ+1 = t, V (p) ∩ W (T ) ⊆ W (T1) ∪···∪ W (Te) ⊆ V (p) ∩ W (T ). sλ = init(p) and sλ+1 = init(t). Geometrically, this means that we may compute a little more than V (p)∩W (T ); however, W (T1)∪···∪W (Te) is a “sharp” The following theorem provides sufficient conditions for Sj approximation of the intersection of V (p) and W (T ). (with 1 ≤ j ≤ λ + 1) to be a regular GCD of p and t in A[xk]. Next we list the specifications of our triangular decompo- sition algorithm and its subroutines. We denote by R the Theorem 1. Let j be an integer, with 1 ≤ j ≤ λ + 1, polynomial ring k[x], where x = x1 < · · ·

Proof. By Definition 1, it suffices to prove that both • Input: F , a finite set of polynomials of R prem(p,Sj ,x ) = 0 and prem(t,Sj ,x ) = 0 hold in A. By k k • Output: A Lazard-Wu triangular decomposition of symmetry we only prove the former equality. V (F ). Let p be any prime ideal associated with sat(T ). Define D = k[x1,...,xk−1]/p and let L be the fraction field of the Intersect(p,T,R) integral domain D. Let φ be the homomorphism from B to L. By Theorem 4 of Appendix A, we know that φ(Sj ) • Input: p, a polynomial of R; T , a regular chain of R is a GCD of φ(p) and φ(t) in L[xk]. Therefore there ex- ists a polynomial q of L[xk] such that p = qSj in L[xk], • Output: a set of regular chains {T1,...,Te} such chat which implies that there exists a nonzero element a of D ′ ′ (p,T ) −→ T1,...,Te. and a polynomial q of D[xk] such that ap = q Sj in D[xk]. Therefore prem(ap,Sj ) = 0in D[xk], which implies that prem(p,Sj )=0 in D[xk]. Therefore prem(p,Sj ) belongs to Regularize(p,T,R) p and thus to sat(T ). So prem(p,Sj ,x )=0 in A. p k • Input: p, a polynomial of R; T , a regular chain of R. 4. THE INCREMENTAL ALGORITHM In this section, we present an algorithm to compute Lazard- • Output: a set of pairs {[p1,T1],..., [pe,Te]} such that Wu triangular decompositions in an incremental manner. for each i, 1 ≤ i ≤ e: (1) Ti is a regular chain; (2) p = pi mod sat(Ti); (3) if pi = 0, then pi ∈ sat(Ti) We recall the concepts of a process and a regular (delayed) p p split, which were introduced as Definitions 9 and 11 in [16]. otherwise pi is regular modulo sat(Ti); moreover we p To serve our purpose, we modify the definitions as below. have T −→ T1,...,Te.

Definition 2. A process of k[x] is a pair (p,T ), where SubresultantChain(p,q,v,R) p ∈ k[x] is a polynomial and T ⊂ k[x] is a regular chain. The process (0,T ) is also written as T for short. Given ′ ′ ′ • Input: v, a variable of {x1,...,xn}; p and q, polyno- two processes (p,T ) and (p ,T ), let v and v be respectively mials of R, whose main variables are both v. the greatest variable appearing in (p,T ) and (p′,T ′). We ′ ′ ′ ′ say (p,T ) ≺ (p ,T ) if: (i) either v < v ; (ii) or v = v • Output: a list of polynomials (S0,...,Sλ), where ′ ′ ′ and dim T < dim T ; (iii) or v = v , dim T = dim T and λ = min(mdeg(p), mdeg(q)), such that Si is the i-th T ≺ T ′; (iv) or v = v′, dim T = dim T ′, T ∼ T ′ and subresultant of p and q w.r.t. v. sys Input size Output size #v #e deg dim GL GS GD TL TK 1 4corps-1parameter-homog 4 3 8 1 - - 21863 - 30738 2 8-3-config-Li 12 7 2 7 67965 - 72698 7538 1384 3 Alonso-Li 7 4 4 3 1270 - 614 2050 374 4 Bezier 5 3 6 2 - - 32054 - 114109 5 Cheaters-homotopy-1 7 3 7 4 26387452 - 17297 - 285 7 childDraw-2 10 10 2 0 938846 - 157765 - - 8 Cinquin-Demongeot-3-3 4 3 4 1 1652062 - 680 2065 895 9 Cinquin-Demongeot-3-4 4 3 5 1 - - 690 - 2322 10 collins-jsc02 5 4 3 1 - - 28720 2770 1290 11 f-744 12 12 3 1 102082 - 83559 4509 4510 12 Haas5 4 2 10 2 - - 28 - 548 14 Lichtblau 3 2 11 1 6600095 - 224647 110332 5243 16 Liu-Lorenz 5 4 2 1 47688 123965 712 2339 938 17 Mehta2 11 8 3 3 - - 1374931 5347 5097 18 Mehta3 13 10 3 3 - - - 25951 25537 19 Mehta4 15 12 3 3 - - - 71675 71239 21 p3p-isosceles 7 3 3 4 56701 - 1453 9253 840 22 p3p 8 3 3 5 160567 - 1768 - 1712 23 Pavelle 8 4 2 4 17990 - 1552 3351 1086 24 Solotareff-4b 5 4 3 1 2903124 - 14810 2438 872 25 Wang93 5 4 3 1 2772 56383 1377 1016 391 26 Xia 6 3 4 3 63083 2711 672 1647 441 27 xy-5-7-2 6 3 3 3 12750 - 599 - 3267 Table 1 The input and output sizes of systems

RegularGcd(p,q,v,S,T,R) CleanChain(C,T,xi, R)

• Input: v, a variable of {x1,...,xn}, • Input: T , a regular chain of R; C, a regular chain of

k[x1,...,xi−1] such that sat(T

• Output: a set of pairs {[g1,T1],..., [ge,Te]} such that Extend(C,T,xi, R) T −→ T1,...,Te and for each Ti: if dim T = dim Ti, then gi is a regular GCD of p and q modulo sat(Ti); p otherwise gi = 0, which means undefined. • Input: C, is a regular chain of k[x1,...,xi−1]. T , a regular chain of R such that sat(T

• Output: a set of regular chains {T1,...,Te} such that Algorithm SubresultantChain is standard, see [8]. The algo- (p,C) −→ (T1,...,Te). rithm Triangularize is a principle algorithm which was first presented in [16]. We use the following conventions in our pseudo-code: the keyword return yields a result and ter- IntersectAlgebraic(p,T,xi,S,C,R) minates the current function call while the keyword output yields a result and keeps executing the current function call.

• Input: p, a polynomial of R with main variable xi, 5. PROOF OF THE ALGORITHMS – T , a regular chain of R, where xi ∈ mvar(T ), Theorem 2. All the algorithms in Fig. 1 terminate.

– S, the subresultant chain of p and Txi w.r.t. xi,

– C, a regular chain of k[x1,...,xi−1], such that: Proof. The key observation is that the flow graph of Fig.

init(Txi ) is regular modulo sat(C); the resul- 1 can be transformed into an equivalent flow graph satisfying p the following properties: (1) the algorithms Intersect and tant of p and Txi , which is S0, belongs to sat(C). p Regularize only call each other or themselves; (2) all the other • Output: a set of regular chains T1,...,Te such that algorithms only call either Intersect or Regularize. Therefore,

(p,C ∪ Txi ) −→ T1,...,Te. it suffices to show that Intersect and Regularize terminate. Algorithm 1: Intersect(p,T,R) Algorithm 5: Regularize(p,T,R) 1 if prem(p,T ) = 0 then return {T }; 2 if p ∈ k then return { }; 1 if p ∈ k or T = ∅ then return [p,T ]; 3 r := p; P := {r}; S := { }; 2 v := mvar(p); 4 while mvar(r) ∈ mvar(T ) do 3 if v∈ / mvar(T ) then 4 5 v := mvar(r); src := SubresultantChain(r, T v,v,R); for [f,C] ∈ Regularize(init(p),T,R) do 6 S := S ∪ {src}; r := resultant(src); 5 if f = 0 then output Regularize(tail(p),C,R); 7 if r = 0 then break; 6 ; 8 if r ∈ k then return { }; 7 else output [p,C]; 9 P := P ∪ {r} 8 else ′ 10 T := {∅}; T := { }; i := 1; 9 src := SubresultantChain(p,Tv,v,R); r := resultant(src); 11 while i ≤ n do 10 for [f,C] ∈ Regularize(r, Tv]; 24 T return 23 q := pquo(Tv,g); 24 output Regularize(p, D ∪ q ∪ T>v, R); 25 for E ∈ Intersect(hg ,D,R) do Algorithm 2 : RegularGcd(p,q,v,S,T,R) 26 for F ∈ Extend(E,T,v,R) do 1 T := {(T, 1)}; 27 output Regularize(p,F,R) 2 while T 6= ∅ do 3 let (C, i) ∈ T; T := T \ {(C, i)}; 4 for [f, D] ∈ Regularize(si,C,R) do 5 if dim D < dim C then output [0, D] ; 6 else if f = 0 then T := T ∪ {(D, i + 1)} ; 7 else output [Si, D] Algorithm 6: Extend(C,T,xi, R)

1 if T≥xi = ∅ then return C; 2 let p ∈ T with greatest main variable; T ′ := T \ {p}; Algorithm 3: IntersectFree(p,xi,C,R) ′ 3 for D ∈ Extend(C,T ,xi, R) do 1 for [f, D] ∈ Regularize(init(p),C,R) do 4 for [f,E] ∈ Regularize(init(p), D) do 2 if f = 0 then output Intersect(tail(p),D,R) ; 5 if f 6= 0 then output E ∪ p; 3 else 4 output D ∪ p; 5 for E ∈ Intersect(init(p),D,R) do 6 output Intersect(tail(p),E,R)

Algorithm 7: CleanChain(C,T,xi, R)

1 if xi ∈/ mvar(T ) or dim C = dim T

g) ⊆ V (p) holds. Since g is a regular GCD of p and Txi Regularize IntersectAlgebraic modulo sat(D), the conclusion follows from point (iii) of p Proposition 5.

e Extend RegularGcd Finally we prove (L3), that is Z(p,C ∪ Txi ) ⊆ W (Tj). Sj=1 Let D1,...,Ds be the regular chains returned from Algo- rithm RegularGcd. We have C −→ D1,...,Ds, which im- e Figure 1: Flow graph of the Algorithms plies Z(p,C ∪ Txi ) ⊆ ∪j=1Z(p, Dj ∪ Txi ). Next since g is a regular GCD of p and Tx modulo sat(Dj ), the conclusion i p follows from point (iv) of Proposition 5. Note that the input of both functions is a process, say (p,T ). One can check that, while executing a call with (p,T ) as Proposition 8. Intersect satisfies its specification. input, any subsequent call to either functions Intersect or Regularize will take a process (p′,T ′) as input such that (p′,T ′) ≺ (p,T ) holds. Since a descending chain of processes Proof. The first while loop can be seen as a projection is necessarily finite, both algorithms terminate. process. We claim that it produces a nonempty triangular set P such that V (p) ∩ W (T ) = V (P ) ∩ W (T ). The claim holds before staring the while loop. For each iteration, let P ′ Since all algorithms terminate, and following the flow graph be the set of polynomials obtained at the previous iteration. of Fig. 1, each call to one of our algorithms unfold to a finite We then compute a polynomial r, which is the resultant of dynamic acyclic graph (DAG) where each vertex is a call to a polynomial in P ′ and a polynomial in T . So r ∈ hP ′,T i. one of our algorithms. Therefore, proving the correctness of By induction, we have hp,T i = hP,T i. So the claim holds. these algorithms reduces to prove the following two points. Next, we claim that the elements in T satisfy the follow- ing invariants: at the beginning of the i-th iteration of the • Base: each algorithm call, which makes no subsequent second while loop, we have calls to another algorithm or to itself, is correct.

• Induction: each algorithm call, which makes subse- (1) each C ∈ T is a regular chain; if Txi exists, then

quent calls to another algorithm or to itself, is correct, init(Txi ) is regular modulo sat(C), as soon as all subsequent calls are themselves correct. (2) for each C ∈ T, we have sat(T

For all algorithms in Fig. 1, proving the base cases is straight- (3) for each C ∈ T, we have W (C) ⊆ V (P

Proposition 6. IntersectFree satisfies its specification. When i = n+1, we then have sat(T ) ⊆ sat(C), W (C) ⊆ p p V (P ) ⊆ V (p) for each C ∈ T and V (p)∩W (T ) ⊆ ∪C∈TW (C). Proof. We have the following two key observations: So (L1), (L2), (L3) of Definition 3 all hold. This concludes the correctness of the algorithm.

• C −→ D1,...,Ds, where Di are the regular chains in Now we prove the above claims (1), (2), (3), (4) by induc- the output of Regularize. tion. The claims clearly hold when i = 1 since C = ∅ and V (p) ∩ W (T ) = V (P ) ∩ W (T ). Now assume that the • V (p) ∩ W (D)= W (D,p) ∪ V (init(p), tail(p)) ∩ W (D). loop invariants hold at the beginning of the i-th iteration. We need to prove that it still holds at the beginning of the Then it is not hard to conclude that (p,C) −→ T1,...,Te. (i + 1)-th iteration. Let C ∈ T be an element picked up at the beginning of i-th iteration and let L be the set of the new elements of T′ generated from C. Proposition 7. IntersectAlgebraic is correct. Then for any C′ ∈ L, claim (1) clearly holds by specification of CleanChain. Next we prove (2). Proof. We need to prove: (p,C ∪ Txi ) −→ T1,...,Te. Let us prove (L1) now, that is, for each regular chain Tj in the output, we have sat(C ∪ Tx ) ⊆ sat(Tj ). First by • if xi ∈/ mvar(T ), then T

(2) if p′ = 0, then p is zero modulo sat(T ′), otherwise p Theorem 3. Let F ⊂ k[x] be finite, with cardinality #(F ). p is regular modulo sat(T ), Assume F generates a proper ideal of k[x]. Then, for any minimal prime ideal p associated with hF i, the height of p is (3) we have sat(T ) ⊆ sat(T ′), p p less than or equal to #(F ). ′ (4) we have W (T ) ⊆ ∪T ′∈L W (T ). Corollary 3. Let T be a Kalkbrener triangular decom- Statement (1) is due to Proposition 2. Next we prove (2). position of V (F ). Let T be a regular chain of T, the height First, when there are recursive calls, the conclusion is obvi- of which is greater than #(F ). Then T \ {T } is also a Kalk- ous. Let [f,C] be a pair in theoutputof Regularize(r, Tv) together to CleanChain, IntersectFree and IntersectAlgebraic. p p p with sat(T ) ⊆ sat(D ∪ q ∪ T>v) hold. Hence (3) holds. p p In IntersectFree, we pass its input height A to every function Finally by point (ii.b) of Proposition 5, we have W (D∪Tv) ⊆ call. Besides, Lines 5 to 6 are executed only if the height of Z(hg , D ∪ Tv) ∪ W (D ∪ g) ∪ W (D ∪ q). So (4) holds. D is strictly less than A, since otherwise we would obtain regular chains of height greater than A. In other algorithms, we apply similar strategies as in Intersect and IntersectFree. Proposition 10. Extend satisfies its specification. 7. EXPERIMENTATION

Proof. It clearly holds when T≥xi = ∅, which is the Part of the algorithms presented in this paper are imple- base case. By induction and the specification of Regularize, mented in Maple14 while all of them are present in the cur- we know that sat(T ′) ⊆ sat(E). Since init(p) is reg- rent development version of Maple. Tables 1 and 2 report p ′ p ular modulo both sat(T ) and sat(E), by Corollary 1, we on our comparison between Triangularize and other Maple sys Triangularize Triangularize versus other solvers TK13 TK14 TK TL13 TL14 TL STK STL GL GS WS TL TK 1 - 241.7 36.9 - - - 62.8 - - - - - 36.9 2 8.7 5.3 5.9 29.7 24.1 25.8 6.0 26.6 108.7 - 27.8 25.8 5.9 3 0.3 0.3 0.4 14.0 2.4 2.1 0.4 2.2 3.4 - 7.9 2.1 0.4 4 - - 88.2 ------88.2 5 0.4 0.5 0.7 - - - 451.8 - 2609.5 - - - 0.7 7 ------1326.8 1437.1 19.3 - - - - 8 3.2 0.7 0.6 - 55.9 7.1 0.7 8.8 63.6 - - 7.1 0.6 9 166.1 5.0 3.1 - - - 3.3 - - - - - 3.1 10 5.8 0.4 0.4 - 1.5 1.5 0.4 1.5 - - 0.8 1.5 0.4 11 - 29.1 12.7 - 27.7 14.8 12.9 15.1 30.8 - - 14.8 12.7 12 452.3 454.1 0.3 - - - 0.3 - - - - - 0.3 14 0.7 0.7 0.3 801.7 226.5 143.5 0.3 531.3 125.9 - - 143.5 0.3 16 0.4 0.4 0.4 4.7 2.6 2.3 0.4 4.4 3.2 2160.1 40.2 2.3 0.4 17 - 2.1 2.2 - 4.5 4.5 2.2 6.2 - - 5.7 4.5 2.2 18 - 15.6 14.4 - 126.2 51.1 14.5 63.1 - - - 51.1 14.4 19 - 871.1 859.4 - 1987.5 1756.3 859.2 1761.8 - - - 1756.3 859.4 21 1.2 0.6 0.3 - 1303.1 352.5 0.3 - 6.2 - 792.8 352.5 0.3 22 168.8 5.5 0.3 - - - 0.3 - 33.6 - - - 0.3 23 0.8 0.9 0.5 - 10.3 7.0 0.4 12.6 1.8 - - 7.0 0.5 24 1.5 0.7 0.8 - 1.9 1.9 0.9 2.0 35.2 - 9.1 1.9 0.8 25 0.5 0.6 0.7 0.6 0.8 0.8 0.8 0.9 0.2 1580.0 0.8 0.8 0.7 26 0.2 0.3 0.4 4.0 1.9 1.9 0.5 2.7 4.7 0.1 12.5 1.9 0.4 27 3.3 0.9 0.6 - - - 0.7 - 0.3 - - - 0.6 Table 2 Timings of Triangularize versus other solvers solvers. The notations used in these tables are defined be- To be precise, let “dec” and “gb” be respectively the out- low. put of the Triangularize and Groebner functions. The Maple command we use are length(convert(map(Equations, dec, R), string)) and length(convert(gb, string)). From Table 1, it is Notation for Triangularize. We denote by TK and TL the clear that Triangularize produces much smaller output than latest implementation of Triangularize for computing, respec- commands based on Gr¨obner basis computations. tively, Kalkbrener and Lazard-Wu decompositions, in the current version of Maple. Denote by TK14 and TL14 the TK, TL, GS, WS (and, to some extent, GL) can all be seen corresponding implementation in Maple14. Denote by TK13, as polynomial system solvers in the sense of that they pro- TL13 the implementation based on the algorithm of [16] in vide equidimensional decompositions where components are Maple13. Finally, STK and STL are versions of TK and represented by triangular sets. Moreover, they are imple- TL respectively, enforcing that all computed regular chains mented in Maple (with the support of efficient C code in are squarefree, by means of the algorithms in Appendix B. the case of GS and GL). The specification of TK are close to those of GS while TL is related to WS, though the triangular Notation for the other solvers. Denote by GL, GS, GD, re- sets returned by WS are not necessarily regular chains. spectively the function Groebner:-Basis (plex order), Groebner:- Solve, Groebner:-Basis (tdeg order) in current beta version of In Table 2, we provide the timings of different versions of Maple. Denote by WS the function wsolve of the package Triangularize and other solvers. From this table, it is clear Wsolve [19], which decomposes a variety as a union of quasi- that the implementations of Triangularize, based on the al- components of Wu Characteristic Sets. gorithms presented in this paper (that is TK14, TL14, TK, TL) outperform the previous versions (TK13, TL13), based The tests were launched on a machine with Intel Core 2 on [16], by several orders of magnitude. We observe also that Quad CPU (2.40GHz) and 3.0Gb total memory. The time- TK outperforms GS and GL while TL outperforms WS. out is set as 3600 seconds. The memory usage is limited to 60% of total memory. In both Table 1 and 2, the symbol Acknowledgments. The authors would like to thank the “-” means either time or memory exceeds the limit we set. support of Maplesoft, Mitacs and Nserc of Canada.

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We denote by dpol(M) the element of A[x], called the [10] E.´ Hubert. Factorization free decomposition determinant polynomial of M, given by ℓ−k ℓ−k−1 algorithms in differential algebra. J. Symb. Comp., det Mkx + det Mk+1x + · · · + det Mℓ. 29(4-5):641–662, 2000. Let f1(x),...,fk(x) be a set of polynomials of A[x]. Let [11] M. Kalkbrener. A generalized euclidean algorithm for ℓ = 1 + max(deg f1(x),..., deg fk(x)). The matrix M of ℓ−j computing triangular representations of algebraic f1,...,fk is defined by Mij = coeff(fi,x ). varieties. J. Symb. Comp., 15:143–167, 1993. m n Let f = amx + · · · + a0, g = bnx + · · · + b0 be two [12] D. Lazard. A new method for solving algebraic polynomials of A[x] with positive degrees m and n. Let systems of positive dimension. Discr. App. Math, λ = min(m, n). For any 0 ≤ i<λ, let M be the matrix of 33:147–160, 1991. the polynomials xn−1−if,...,xf,f,xm−1−ig,...,xg,g. We [13] G. Lecerf. 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For any integer 0 ≤ k <λ, if φ(sk) 6= 0, [17] T. Shimoyama and K. Yokoyama. Localization and then φ(am) and φ(bn) does not vanish at the same time. primary decomposition of polynomial ideals. J. Symb. Moreover, we have both deg φ(f) ≥ k and deg φ(g) ≥ k. Comput., 22(3):247–277, 1996. Proof. Observe that [18] A.J. Sommese, J. Verschelde, and C. W. Wampler. Solving polynomial systems equation by equation. In am am−1 · · · a0

Algorithms in Algebraic Geometry, pages 133–152. ··· ···

Springer-Verlag, 2008. am am−1 · · · ak sk = . bn bn−1 · · · b0 [19] D. K. Wang. The Wsolve package. ··· ··· http://www.mmrc.iss.ac.cn/∼dwang/wsolve.txt. bn bn−1 · · · bk

Therefore there exists i ≥ k, j ≥ k such that φ(ai) 6= 0 and Done. φ(bj ) 6= 0. The conclusion follows.

Theorem 4. We have the following relations between the Lemma 2. Assume that φ(s0) = · · · = φ(s 1) = 0. λ− subresultants and the GCD of φ(f) and φ(g): Then, if m ≤ n, we have

1. Let k, 0 ≤ k<λ, be an integer such that φ(sk) 6= 0 and (1) if φ(am) 6= 0 and φ(bn)= · · · = φ(bm) = 0, then φ(g)= 0 for any i, 0 ≤ i n, we have (a) if m ≤ n and φ(am) 6= 0, then gcd(φ(f),φ(g)) = φ(f); symmetrically, if m > n and φ(bn) 6= 0, then (3) if φ(bn) 6= 0 and φ(am)= · · · = φ(an) = 0, then φ(f)= 0 we have gcd(φ(f),φ(g)) = φ(g) (b) if m ≤ n and φ(am) = 0 but φ(bn) 6= 0, then we (4) if φ(bn) = 0 and φ(am) 6= 0, then φ(g) = 0 have gcd(φ(f),φ(g)) = φ(g); symmetrically, if m ≥ n and φ(bn) = 0 but φ(am) 6= 0, then we have Proof. We prove (1) and (2), whose correctness implies gcd(φ(f),φ(g)) = φ(f) (3) and (4) by symmetry. Let i = λ − 1 = m − 1, then we have (c) if φ(am)= φ(bn) = 0, then n−m Sm−1 = dpol(x f,...,xf,f,g). gcd(φ(f),φ(g)) = gcd(φ(red(f)),φ(red(g))) Therefore

am · · · a0 Proof. Let us first prove (1). W.l.o.g, we assume φ(am) 6= ′ . . 0. From Lemma 1, we know that k ≤ n . Therefore for i ≤ k, .. .. ′ sm−1 = . we have i ≤ n . By Lemma 3, am am−1 ′ ′ n−n n −1−i bn · · · bm bm−1 φ(Si) = φ(am) dpol(x φ(f),...,xφ(f),φ(f), m−1−i x φ(g),...,xφ(g),φ(g)) So from φ(bn) = · · · = φ(bm) = 0 and φ(sm−1) = 0, we ′ conclude that φ(bm−1) = 0. On theother hand, if φ(am) = 0 ′ n−n If i < n , we have φ(Si) = φ(am) Si(φ(f),φ(g)). If and φ(bn) 6= 0, then φ(am−1) = 0. i = n′, since i

′ n−n m−1−i Now let consider Sm−2. We have φ(Si) = φ(am) dpol(x φ(g),...,xφ(g),φ(g)) ′ 1 n−n ′ m− −i am am−1 · · · a0 = φ(am) φ(bn ) φ(g).

. . ′ .. .. So for all i

bn · · · bm−1 bm−2 φ(g)= φ(Sk).

From φ(bm−1) = 0, we conclude that φ(bm−2) = 0. From Next we prove (2a). By symmetry, we prove it when m ≤ n. φ(am−1) = 0, we conclude that φ(am−2) = 0. If φ(bn)= · · · = φ(bm) = 0, it follows directy from Lemma 2. Otherwise, we have n′ ≥ m. By Lemma 3, for all i

19 else 20 for [f,E] ∈ Regularize(init(p),D,R) do 21 if f 6= 0 then T := T ∪ {E};

Algorithm 11: Squarefree(T, R)

Input: a polynomial ring R = k[x1,...,xn], a regular chain T of R. Output: a set of squarefree regular chains T1,...,Te such that T −→ T1,...,Te. 1 T := {SquarefreePart(p) | p ∈ T }; 2 S := { }; 3 for p ∈ T do 4 if mdeg(p) > 1 then 5 S := S ∪ {SubresultantChain(p,p′, mvar(p), R)}; 6 T := {∅}; T′ := { }; i := 1; 7 while i ≤ n do 8 for C ∈ T do 9 if xi ∈/ mvar(T ) then ′ ′ 10 T := T ∪ CleanChain(C,T,xi+1, R) 11 else

12 if mdeg(Txi ) = 1 then ′ ′ 13 T := T ∪ CleanChain(C ∪ {Txi },T,xi+1, R) 14 else

15 for D ∈ Squarefree(Txi ,xi,Sxi ,C,R) do ′ ′ 16 T := T ∪ CleanChain(D,T,xi+1, R)

17 T := T′; T′ := { }; i := i + 1; 18 return T