Affine solution sets of sparse polynomial systems

Affine solution s ets of sparse p olynomia l syst ems ∗ Mar ´ ıa Isab el Herrero ♯ , Gabriela Jeronimo ♯, ⋄ , Juan Sabia † , ⋄ ♯ Departamento de Matem´ atica, F acultad de Ciencias Exacta s y Natura le s, Univ ersidad de Buenos Aires, Ciudad Universitaria, (14 2 8) Buenos Aires, Arge ntina † Departamento de Ciencias Exa c tas, Ciclo B´ asico Com ´ un, Univ ersidad de Buenos Aires, Ciudad Universitaria, (14 2 8) Buenos Aires, Arge ntina ⋄ IMAS, CONICET, Argentina Abstract This pap er fo cuses on the equidimensional decomp osition of affine v a rieties de- fined by sparse p olynomia l sy stems. F or generic systems with fixed suppor ts, we give combinatorial conditions f or the existence of p ositive dimensional comp onents whic h characterize the eq uidimensio nal decomp osition of the asso cia ted affine v ariet y . This result is applied to design an equidimensional deco mp ositio n algorithm for generic sparse systems. F or arbitrary sparse systems of n p olynomials in n v ariables with fixed supp orts, we o btain an upp er b ound for the degr ee o f the a ffine v ariety defined and we pr esent a n a lg orithm which computes finite sets of p oints repr esent ing its equidimensional comp onents. Keyw ords: Spars e p olynomial systems, equidimen s ional decomp osition of algebraic v a- rieties, degree of affine v arieties, algorithms and complexit y 1 In tro duction The aim of this pap er is to describ e the affine solution set of a p olynomial system tak- ing into account the sets of exp onents of the mon omials with nonzero coefficient s in the p olynomials inv olv ed, that is, their supp ort sets. Bernstein ([1]), Ku s hnirenko ([23]) and Kho v ans kii ([21]) prov ed that the n um b er of isolated s olutions in ( C ∗ ) n of a p olynomial system with n equ ations in n v ariables is b ound ed by a com b inatorial inv arian t (the mixed volume) asso ciated with their supp orts. This result, whic h ma y b e considered the basis for the current study of sparse p olynomial systems, hints at the fact that the algorithms solving these systems should h a ve sh orter computing time than th e general ones. There are sev eral algorithms to compute e ither n um er ically or symbolically the isolated ro ots of sparse p olynomial systems in ( C ∗ ) n (see, for example,[37, 16, 28, 19]). The effi- ciency of some of these algorithms r elies on the use of p olyhedral deformations pr eserving the monomial stru ctur e of the p olynomial system un der consideration. ∗ P artially supp orted by the A rgen tinian researc h grants CONICET PIP 099/11 and U BAC YT 2002009 010006 9 (2010-2013). 1 The first step to wards the study of the solutions of sparse systems in the affine case w as to obtain upp er b ound s for the num b er of isolated solutions in C n in terms of the structure of their su pp orts and to design numerica l algorithms to compute them (see [27, 25, 29, 17, 7, 8 ]). Sym b olic algorithms p erforming this task w ere giv en in [19] and [15]. The next natural s tep is to c h aracterize th e comp onen ts of higher dimension of th e affine v ariet y defin ed b y a system of sparse p olynomial equations taking int o accoun t their supp orts. I n this context , in [36], certificates for the existence of cur ves are giv en in th e n umerical framew ork. There are different sy mb olic algorithms describing the equidim en sional decomp osition of a v ariet y which only take into consideration the degrees of the p olynomials d efining it and not their particular mon omial stru ctur e. The earliest deterministic ones can b e found in [4] a nd [11] (see a lso [10], where the more ge neral problem of the pr imary decomp osition of ideals is consider ed ). P r obabilistic algorithms with shorter runnin g ti me are given in [6] and [20]. Th e complexities of these probab ilistic algorithms are p olynomial in the B´ ezout n um b er of the system, whic h, in th e generic case, coincides w ith the degree of the v ariet y the sys tem defines. Other p r obabilistic algorithms are p resen ted in [24] and [18] with complexities dep end ing on a new inv arian t related to the system (the ge ometric de gr e e ) whic h r efines the B ´ ezout b oun d . Some of these algorithms can b e d erandomized easily via the Sch w artz-Zipp el lemma ([32, 38]) pro vided upp er b oun ds for the degrees of th e p olynomials c haracterizing exceptional instances are kno wn. Algorithms dealing with th e problem f rom th e numerical p oint of view can b e traced bac k to [33]. A series of pap ers by Sommese, V ersc h elde and W ampler p r esen t successiv e impro v ements to this pro cedure, leading to th e irredu cible decomp osition algorithm based on homotop y cont in uation describ ed in [34] (see references therein). In this pap er we an alyze, b oth from the theoretic and algorithmic p oint s of view, the equidimensional decomp osition of the affine v ariet y defined by a sparse p olynomial system. First, w e co nsider the ca se of generic sparse systems. In this con text, there exists a m a- jor differen ce with th e case of d ense p olynomials. The set of solutions of a generic system of n p olynomials in n v ariables with fixed degrees co nsists o nly o f isolated p oin ts. How ev er , fixing the set of s upp orts of the n p olynomials in n v ariables inv olv ed in a sparse system, for generic c hoices of its co efficien ts, there may app ear affine comp onents of p ositiv e di- mension (see, for instance, Examples 3 and 8 b elo w). W e sh o w th at the existence of these generic comp onent s of p ositiv e dimension d ep ends only on the combinato rial stru cture of the supp orts: in Prop osition 6 b elo w, we give conditions that yield these comp onents. Suc h conditions provide not only a theoretic descrip tion of the equid imensional decom- p osition of the affine v ariet y V ( f ) defined by a ge neric sp arse system f in terms of the solution sets in the torus of smaller systems f I asso ciated to sub s ets I ⊂ { 1 , . . . , n } b ut also a formula f or the degree of V ( f ) in this generic case (see T heorem 7 b elo w). Previous results on this sub ject can b e found in [3]. There, usin g also a com binatorial approac h , the authors analyze thoroughly the problem of d eciding wh ether a system of n binomials in n v ariables has a fin ite n um b er of affine s olutions and, in th is case, the computational complexit y of the corresp onding coun ting problem. Our result is used to desig n a probabilistic algorithm whic h, f or a generic sparse system f , computes the equidimens ional d ecomp osition of V ( f ) w ith a complexit y d ep ending on its degree and com b inatorial in v ariants asso ciated w ith th e system (see Theorem 14). The idea of the algorithm is to compute firs t a family of subs ets I ⊂ { 1 , . . . , n } wh ic h ma y lead to comp onen ts of V ( f ) and solv e the corresp on d ing p olynomial systems f I b y apply- 2 ing symb olic p olyhedr al d eformations ([19]) and a Newton-Hensel based p ro cedure ([13]). The output of the algorithm is, f or eac h k = 0 , . . . , n − 1, a list of ge ometric r esolutions represent ing the equidimensional comp onen t of dimension k of V ( f ). A geometric resolu- tion of an equ idimensional v ariet y is a parametric t yp e description of the v ariet y w hic h is widely used in s ym b olic compu tations (see, for instance, [12, 30, 13, 31]); in Section 2.1 b elo w we giv e the pr ecise d efinition w e use. The next step is to consider the equidimensional decomp osition of the affine v ariet y defined by an arbitr ary system of sp arse p olynomials. A question to answ er b eforehand is whic h parameter should b e in v olv ed in th e alge braic complexit y of an al gorithm solving this task. F rom previous exp erience, a n atural inv arian t exp ected to app ear in the complexit y b ound s is the d egree of the v ariet y , which is, in particular, an upp er b ou n d for the n um b er of its irreducible comp onents. Unlik e the B ´ ezout b ound for dense p olynomials, in th e spars e setting, the degree of the affine v ariet y defined by a generic square system is not an up p er b oun d for the d egree of the v ariet y defi ned by any system w ith th e same supp orts (see Example 15). In [22], a b ound for t he degree o f the affine v ariet y defined b y a n arbitrary sparse polynomial system dep endin g on a mixed v olume related to the u nion of the supp orts of the p olynomials is present ed (see also [28, Theorem 1] for a related result). Here, we obtain a sharp er b oun d for this degree also giv en by a mixed vo lume asso ciated to the sup p orts but not inv olving their un ion: Theorem. L e t f = ( f 1 , . . . , f n ) b e n p olynomials in C [ X 1 , . . . , X n ] supp orte d on A = ( A 1 , . . . , A n ) and let V ( f ) = { x ∈ C n | f i ( x ) = 0 for e v ery 1 ≤ i ≤ n } . Then deg( V ( f )) ≤ M V n ( A 1 ∪ ∆ , . . . , A n ∪ ∆) , wher e ∆ = { 0 , e 1 , . . . , e n } with e i the i th ve ctor of the c anonic al b asis of R n and M V n stands for the n -dimensional mixe d volume. Finally , w e obtain an algorithm wh ic h, using a p olyhedral deformation, describ es p oin ts in ev ery irredu cible comp onent of the affine v ariet y defin ed by an arbitrary squ are sp arse system with complexit y d ep ending on the degree b ound pr eviously stated. The idea of the algorithm relie s on the fact that cutting the v ariet y with a g eneric a ffine linear v ariet y of co dimension k , sufficientl y many p oints in eac h irredu cible comp onen t of d imension k can b e obtained. T o ke ep the complexit y within the desired b ound s, instead of computing this intersectio n, we pr o ceed in a particular wa y whic h enables us to compute a finite sup erset of the inte rsection included in th e v ariet y . Represen ting a p ositiv e dimensional v ariet y by means of a finite set of p oints is a well-kno wn appr oac h in n umerical algebraic geometry (see th e notion of witness p oint sup ersets in [34]). Theorem. L e t f = ( f 1 , . . . , f n ) b e n p olynomials in Q [ X 1 , . . . , X n ] supp orte d on A = ( A 1 , . . . , A n ) . Ther e is a pr ob abilistic algorithm which, taking as input the sp arse r ep- r esentation of f c omputes a family of n ge ometric r esolutions ( R (0) , R (1) , . . . , R ( n − 1) ) such that, for eve ry 0 ≤ k ≤ n − 1 , R ( k ) r epr esents a finite set c ontaining deg V k ( f ) p oints in the e quidimensional c omp onent V k ( f ) of dimension k of V ( f ) . The numb er of arithmetic op er ations over Q p erforme d by the algorithm is of or der O e ( n 4 dN D 2 ) , wher e d = max 1 ≤ j ≤ n { deg( f j ) } , N = P n j = 1 #( A j ∪ ∆) and D = M V n ( A 1 ∪ ∆ , . . . , A n ∪ ∆) . Here O e r efers to the standard soft-oh notation whic h do es not take in to accoun t logarithmic factors. F urthermore, w e ha ve ignored factors dep endin g p olynomially on 3 the size of certain combinato rial ob jects asso ciated to the p olyhedral deformation. F or a precise complexit y statement , see Theorem 21, and for error probabilit y consid er ations, see Remark 23. The pap er is organized as follo ws. In S ection 2 , the b asic d efinitions and notations used throughout the p ap er are introdu ced. S ection 3 is dev oted to the equidimens ional decomp osition of affine v arieties defi ned by generic sparse systems: first, we consider the solution sets in ( C ∗ ) n of und erdetermined systems (see S ubsection 3.1); th en, we pro v e our main theoretic result on equid im en sional d ecomp osition and present our algorithm to compute it (see Subs ection 3.2). Finally , in S ection 4, we consider th e case of arbitrary sparse systems: w e pro v e the upp er b oun d for the degree o f affine v arieties defined by these systems in Su bsection 4.1 and, in Subsection 4.2, w e describ e our algorithm to compu te represent ativ e p oint s of the equ id imensional comp onen ts. 2 Preliminaries 2.1 Basic definitions and notation Throughout this pap er, unless otherwise explicitly stated, we d eal with p olynomials in Q [ X 1 , . . . , X n ], that is to say p olynomials with rational coefficient s in n v ariables X = ( X 1 , . . . , X n ). If f = ( f 1 , . . . , f s ) is a family of such p olynomials, V ( f ) will d enote the algebraic v ariet y of their common zero es in C n , the n -dim en sional affine s p ace o v er the complex num b ers. The algebraic v ariet y V ( f ) ⊂ C n can b e decomp osed uniquely as a finite union of irre- ducible v arieties in a non-r edundant wa y . This leads to the e quidimensional de c omp osition of the v ariet y: V ( f ) = n [ k =0 V k ( f ) , where, for every 0 ≤ k ≤ n , V k ( f ) is the (p ossibly empt y) u nion of all the irredu cible comp onen ts of dimension k of V ( f ). The degree of eac h equidimensional comp onent V k ( f ) is the num b er of p oints in its in tersectio n with a generic affine linear v ariet y of co d im en sion k , and the d egree of V ( f ), whic h we denote b y deg( V ( f )), is the sum of the d egrees of its equidimensional co mp onents (see [14]). A common wa y to describ e zero-dimensional affine v arieties d efined b y p olynomials o ver Q is a ge ometric r esolution (see, for instance, [13] and the references th erein). The precise defin ition we are going to use is the follo wing: Let V = { ξ (1) , . . . , ξ ( D ) } ⊂ C n b e a zero-dimensional v ariet y defi ned b y r ational p olyno- mials. Giv en a linear form ℓ = ℓ 1 X 1 + · · · + ℓ n X n in Q [ X 1 , . . . , X n ] suc h that ℓ ( ξ ( i ) ) 6 = ℓ ( ξ ( j ) ) if i 6 = j , the f ollo wing p olynomials completely c haracterize V : • the minimal p olynomial q = Q 1 ≤ i ≤ D ( u − ℓ ( ξ ( i ) )) ∈ Q [ u ] of ℓ o ver the v ariet y V (where u is a n ew v ariable), • p olynomials v 1 , . . . , v n ∈ Q [ u ] with deg ( v j ) < D for ev ery 1 ≤ j ≤ n satisfying V = { ( v 1 ( η ) , . . . , v n ( η )) ∈ C n | η ∈ C , q ( η ) = 0 } . The family of univ ariate p olynomials ( q , v 1 , . . . , v n ) ∈ Q [ u ] n +1 is called a geometric reso- lution of V (asso ciated with the linear form ℓ ). 4 An equiv alent description of V can b e giv en th rough the so-called Kr one cker r epr esen- tation (see [13]), whic h consists of the minimal polynomial q and p olynomials w 1 , . . . , w n ∈ Q [ u ] suc h th at V = { ( w 1 q ′ ( η ) , . . . , w n q ′ ( η )) ∈ C n | η ∈ C , q ( η ) = 0 } , wh ere q ′ is the deriv ativ e of q . Either r epresen tatio n can b e obtained from th e other one in p olynomial time. The notion of geometric resolution can b e extended to an y equ idimensional v ariet y: Let V ⊂ C n b e an equidimensional v ariet y of dimension r d efi ned b y p olynomials f 1 , . . . , f n − r ∈ Q [ X 1 , . . . , X n ]. Assume that for eac h irreducible co mp onent C of V , the id entit y I ( C ) ∩ Q [ X 1 , . . . , X r ] = { 0 } holds, where I ( C ) is the id eal of all p olyno- mials in Q [ X 1 , . . . , X n ] v anish ing iden tical ly o ver C. Let ℓ b e a nonzero linear form in Q [ X r +1 , . . . , X n ] and π ℓ : V → C r +1 the morp hism defi n ed b y π ℓ ( x ) = ( x 1 , . . . , x r , ℓ ( x )). Then, there exists a unique (up to scaling by nonzero elemen ts of Q ) p olynomial Q ℓ ∈ Q [ X 1 , . . . , X r , u ] of minimal degree defining π ℓ ( V ). Let q ℓ ∈ Q ( X 1 , . . . , X r )[ u ] denote the (unique) monic m ultiple of Q ℓ with deg u ( q ℓ ) = deg u ( Q ℓ ). In these terms, if ℓ is a generic linear form, a ge ometric r esolution of V is ( q ℓ , v r +1 , . . . , v n ) ∈ ( Q ( X 1 , . . . , X r )[ u ]) n − r +1 , where, for r + 1 ≤ i ≤ n , v i satisfies ∂ q ℓ ∂ u ( ℓ ) X i = v i ( ℓ ) in Q ( X 1 , . . . , X r ) ⊗ Q [ V ] and deg u ( v i ) < deg u ( q ℓ ). 2.2 Algorithms and co dification Although w e work w ith p olynomials, our algorithms only deal with elemen ts in Q . The notion of c omplexity of an algorithm w e consider is the num b er of op erations and com- parisons in Q it has to p erform . W e w ill enco de multiv ariate p olynomials in d ifferen t w a ys: • in sparse form, th at is, by means of the list of p airs ( a, c a ) where a run s ov er the set of exp onen ts of the monomials app earing in the p olynomial with nonzero coefficien ts and c a is the corresp ond ing co efficient , • in the stand ard dense form, whic h enco des a p olynomial as the vec tor of its co effi- cien ts includin g zero es (w e use this enco ding only f or u niv ariate p olynomials), • in the str aight-line pr o gr am (slp for sh ort) enco ding. A str aight-l ine program is an algorithm w ithout br anc hin gs wh ich allo ws th e ev aluation of the p olynomial at a generic v alue (for a precise defin ition and pr op erties of slp’s, see [2]). In our complexity estimates, we will use th e usual O notation: for f , g : Z ≥ 0 → R , f ( d ) = O ( g ( d )) if | f ( d ) | ≤ c | g ( d ) | for a p ositiv e constant c . W e will also use the notation M ( d ) = d log 2 ( d ) log (log ( d )), where log denotes logarithm to base 2. W e recall that multi- p oin t ev aluation and in terp olation of u niv ariate p olynomials of degree d with co efficien ts in a c haracteristic-0 commutativ e r in g R can b e p erf ormed with O ( M ( d )) op erations and that m ultiplication and division with remainder of suc h p olynomials can b e done with O ( M ( d ) / log ( d )) arithm etic op erations in R . W e denote b y Ω the exp onent in th e complexit y estimate O ( d Ω ) for the m ultiplication of t wo d × d matrices with rati onal coefficien ts. It is kno wn that Ω < 2 . 376 (se e [9, Chapter 12]). Finally , w e write Ω for the exp onen t (Ω < 4) in the complexit y O ( d Ω ) of op erations on d × d matrices with en tries in a commuta tiv e r ing R . 5 Our algorithms are probabilistic in the sen s e that th ey mak e random c hoices of p oints whic h lead to a correct computation pr o vided the p oints lie outside certain p rop er Z ariski closed sets of suitable affine spaces. Then, u s ing the Scwhartz-Zipp el lemma ([32, 38]), the error probability of our algorithms can b e control led by making these r an d om c hoices within su fficien tly large sets of intege r n um b ers w h ose size d ep end on th e degrees of the p olynomials defin ing the p reviously men tioned Zariski closed sets. 2.3 Sparse systems Giv en a family A = ( A 1 , . . . , A s ) of finite subs ets of ( Z ≥ 0 ) n , a sp arse p olynomial sys- tem supp orte d on A is giv en b y p olynomials f j = P a ∈A j c j,a X a in the v ariables X = ( X 1 , . . . , X n ), with c j,a ∈ C \ { 0 } for eac h a ∈ A j and 1 ≤ j ≤ s . W e write f = ( f 1 , . . . , f s ) for this system. Assume s = n . W e denote by M V n ( A ) th e mixe d volume of the conv ex hulls of A 1 , . . . , A n in R n (see, for example, [5, Chapter 7] for th e defin ition), wh ic h is an upp er b ound for th e num b er of isolate d ro ots in ( C ∗ ) n of a sp arse system sup p orted on A (see [1]). The mixed v olume M V n ( A ) can b e co mputed as the sum of the n -dimensional v olumes of the con ve x hulls of all the mixe d c e l ls in a fin e mixed sub division of A . S u c h a sub division can b e obtained b y means of a standard lifting p ro cess (see [16 , Section 2]): let ω = ( ω 1 , . . . , ω n ) b e a n -tuple of generic fun ctions ω j : A j → R and consider th e p olytop e P in R n +1 obtained by taking the p oint w ise sum of the conv ex h ulls of the graphs of ω j for 1 ≤ j ≤ n . Then, the pro jection of th e lo wer facets of P (that is, the n -dimen sional faces with inner normal vecto r with a p ositiv e last co ordinate) induces a fine m ixed su b division of A . Th e dyn amic enumeration p ro cedure describ ed in [26] app ears to b e the f astest algorithm known up u n til now to ac hieve this computation of mixed cells. The stable mixe d volume of A , which is d enoted b y S M n ( A ), is in tro d uced in [17] as an u pp er b ound for the n um b er of isolated ro ots in C n of a sparse system sup p orted on A . Consider A 0 = ( A 0 1 , . . . , A 0 n ) the family with A 0 j := A j ∪ { 0 } for eve ry 1 ≤ j ≤ n , and let ω 0 = ( ω 0 1 , . . . , ω 0 n ) b e a lifting for A 0 defined by ω 0 j ( q ) = 0 if q ∈ A j and ω 0 j (0) = 1 if 0 / ∈ A j . The stable mixed v olume of A is d efined as th e sum of the mixed v olumes of all the cells in the sub division of A 0 induced by ω 0 corresp onding to facets ha vin g inner normal v ecto rs with non-negativ e en tries. 3 Generic sparse systems 3.1 T or ic comp onents Let n and m b e p ositiv e in tegers and let A = ( A 1 , . . . , A m ) b e a family of fi nite subsets of ( Z ≥ 0 ) n . F or 1 ≤ j ≤ m , let F j ( C j , X ) = X a ∈A j C j,a X a where X = ( X 1 , . . . , X n ); for a = ( a 1 , . . . , a n ), X a = Q 1 ≤ j ≤ n X a j j , and C j = ( C j,a ) a ∈A j are N j = # A j indeterminate co efficien ts. F ollo win g [35], consider the inciden ce v ariety { ( x, c ) ∈ ( C ∗ ) n × ( P N 1 − 1 × · · · × P N m − 1 ) | F j ( c j , x ) = 0 for every 1 ≤ j ≤ m } 6 and its pr o jection to the s econd factor Z = { c ∈ P N 1 − 1 × · · · × P N m − 1 | ∃ x ∈ ( C ∗ ) n with F j ( c j , x ) = 0 for ev ery 1 ≤ j ≤ m } . Note that th e elemen ts in Z corresp ond essentia lly to co efficien ts of systems su pp orted on A whic h ha v e a solution in ( C ∗ ) n . Lemma 1 The Zariski closur e of Z e quals P N 1 − 1 × · · · × P N m − 1 if and o nly if, for ev e ry J ⊆ { 1 , . . . , m } , dim  P j ∈ J A j  ≥ # J . In p articular, if m > n , a ge ne ric system su pp orte d on A has no solutions in ( C ∗ ) n . Mor e over, if m ≤ n and d im  P j ∈ J A j  ≥ # J for every J ⊆ { 1 , . . . , m } , the solution set in ( C ∗ ) n of a gene ric system supp orte d on A is an e quidimensional variety of dimension n − m and de gr e e M V n ( A 1 , . . . , A m , ∆ ( n − m ) ) , wher e ∆ = { 0 , e 1 , . . . , e n } with e i the i th v e ctor of the c anonic al b asis of R n and the sup erscript ( n − m ) indic ates that it i s r ep e ate d n − m times. Pr o of: The fir st statemen t of th e Lemma follo ws as in [35, Theorem 1.1]. Assume that m ≤ n and for ev ery J ⊆ { 1 , . . . , m } , d im  P j ∈ J A j  ≥ # J . Then, if e A = ( A 1 , . . . , A m , ∆ ( n − m ) ), we ha ve that for ev ery e J ⊆ { 1 , . . . , n } , the inequalit y dim  P j ∈ e J e A j  ≥ # e J h olds, since dim(∆) = n . T herefore, M V n ( e A ) > 0, whic h im- plies that a generic system s upp orted on e A has finitely many solutions in ( C ∗ ) n (as many as M V n ( e A )). No w, the solution set of a generic sy s tem supp orted on e A is the intersect ion of the solution set of a generic system of m equations s u pp orted on A , which is a v ariet y of dimension at least n − m in ( C ∗ ) n , and n − m generic h yp erplanes. W e conclude that the solution set in ( C ∗ ) n of a generic system supp orted on A is an equidimens ional v ariety of dimension n − m and d egree M V n ( e A ).  Assume now that f = ( f 1 , . . . , f m ) are generic p olynomials in the v ariables X = ( X 1 , . . . , X n ) supp orted on A = ( A 1 , . . . , A m ) ⊂ ( Z n ≥ 0 ) m , with m ≤ n . The pr evious lemma states that the affine v ariet y V ∗ ( f ) ⊂ C n consisting of th e union of all the irr e- ducible comp onent s of V ( f ) = { x ∈ C n | f j ( x ) = 0 for ev ery 1 ≤ j ≤ s } th at h a ve a non-empt y intersect ion with ( C ∗ ) n is either the emp t y set or an equidimensional v ariety of dimension n − m . In what follo ws, we extend the sym b olic algorithm f r om [19, Section 5], whic h deals with the case m = n , to a pro cedure for the computation o f a geometric resolution of V ∗ ( f ) for arb itrary m ≤ n . As in [19], our algo rithm assumes th at a fine mixed sub division of ( A , ∆ ( n − m ) ) indu ced by a generic lifting function ω = ( ω 1 , . . . , ω n ) is giv en b y a pre- pro cessing. Algorithm Generi cToricSol ve INPUT: A sparse represen tation of a generic system f = ( f 1 , . . . , f m ) in the v ariables X = ( X 1 , . . . , X n ) supp orted on A = ( A 1 , . . . , A m ), a lifting function ω = ( ω 1 , . . . , ω n ) and the mixed cells in the sub division of ( A , ∆ ( n − m ) ) indu ced by ω . 1. If the fi ne mixed sub division of ( A , ∆ ( n − m ) ) do es not con tain an y mixed cell, return R = ∅ . Otherw ise, con tinue to Step 2. 7 2. Cho ose randomly the entrie s of a matrix A = ( a hl ) ∈ Q n × n and a ve ctor b = ( b 1 , . . . , b n ) ∈ Q n . 3. F or 1 ≤ h ≤ n − m , consider th e affine linear forms L h = P n l =1 a hl X l − b h . 4. Apply [19 , Algorithm 5.1] to obtain a geometric resolution ( q ( u ) , v 1 ( u ) , . . . , v n ( u )) of the isolated common zero es of f , L 1 , . . . , L n − m in ( C ∗ ) n . 5. Obtain an slp for the p olynomials g := f ( A − 1 Y ) in th e new v ariables Y = ( Y 1 , . . . , Y n ). 6. Compute ( w 1 ( u ) , . . . , w n ( u )) t := A ( v 1 ( u ) , . . . , v n ( u )) t . 7. Apply the Global Newton Iterator from [13, Algorithm 1] to the p olynomials g ( Y ), the geometric r esolution ( q ( u ) , w n − m +1 ( u ) , . . . , w n ( u )) of V ( g ( b, Y n − m +1 , . . . , Y n )), and p r ecision κ = M V n ( A , ∆ ( n − m ) ) to obtain a ge ometric r esolution R Y of an equidi- mensional v ariet y of dim en sion n − m . 8. Obtain the geometric resolution R := A − 1 R Y of V ∗ ( f ). OUTPUT: The geometric resolution R of V ∗ ( f ). In the sequel w e will justify the correctness of the ab ov e p r o cedure and estimate its complexit y . Since f is a generic sp arse system of m equations in n v ariables, as a consequence of Lemma 1, if L 1 , . . . , L n − m are generic linear forms, V ∗ ( f ) ∩ V ( L 1 , . . . , L n − m ) is either th e empt y set (wh en V ∗ ( f ) is the empty set) or a finite set consisting of deg ( V ∗ ( f )) p oin ts (when V ∗ ( f ) is not the emp t y set). Step 1 decides wh ether V ∗ ( f ) ∩ V ( L 1 , . . . , L n − m ) is empt y or not, since the mixed v olume of the su pp orts of these p olynomials is the s u m of the volumes of the mixed cells in a fi ne mixed su b division ([16]). If it is not emp t y , this finite set of p oint s can b e regarded as a generic fi b er of a generic linear su rjectiv e pro jection and therefore, it enab les us to reco ver the v ariet y V ∗ ( f ) by deformation tec h niques. Th us, the idea of the algorithm is to c h o ose n − m linear forms at random, then compute a geometric resolution of the set V ∗ ( f ) ∩ V ( L 1 , . . . , L n − m ) and fin ally , apply a Newton-Hensel lifting to the finite s et obtained in order to get a geometric resolution of V ∗ ( f ). Step 2 d eals with the random c hoice of the entries of a m atrix and a v ector. This random c hoice do es n ot affect the o verall complexit y of the p r o cedure (see Remark 23 b elo w ). I n Step 3, the sp arse e nco ding of n − m linear forms constructed from the previous data is obtained. The idea of Step 4 is to obtain a geometric resolution of V ∗ ( f ) ∩ V ( L 1 , . . . , L n − m ). In order to do this, the algorithm computes the isolated common zero es in ( C ∗ ) n of the generic system f , L 1 , . . . , L n − m supp orted on ( A , ∆ ( n − m ) ). Note th at, if L 1 , . . . , L n − m are generic, this set of p oin ts m eets only the irreducible comp onen ts of V ∗ ( f ), that is, it con tains no p oint in the irreducible comp onen ts of V ( f ) with v anish ing co ordinates. By applying the result in [19, Prop osition 5.13], it follo ws that the complexit y of this step is O  ( n 3 ( N + ( n − m ) n ) log d + n 1+Ω ) M ( D ) M ( M )( M ( D ) + M ( E ))  , where • N := P 1 ≤ j ≤ m # A j ; • d := max 1 ≤ j ≤ m { deg( f j ) } ; 8 • D := M V n ( A , ∆ ( n − m ) ); • M := max {k µ k} , where the maxim um ran ges o ver all p rimitiv e normal vecto rs to the mixed cells in the fine mixed su b division of ( A , ∆ ( n − m ) ) given by ω ; • E := M V n +1 (∆ × { 0 } , A 1 ( ω 1 ) , . . . , A m ( ω m ) , ∆( ω m +1 ) , . . . , ∆( ω n )), where A j ( ω j ) (1 ≤ j ≤ m ) and ∆( ω l ) ( m + 1 ≤ l ≤ n ) are, resp ectiv ely , the s upp orts of f , L 1 , . . . , L n − m lifted by ω . No w the algorithm lifts the geo metric resolution of the zero-dimensional subset of V ( f ) obtained so far to a geometric resolution of th e union of the irreducible comp onents of t his v ariet y ha vin g a n on -emp t y inte rsection with ( C ∗ ) n . In order to do this, we consider the c han ge of v ariables g iv en by Y = A.X and mak e this c hange of v ariables in the p olynomials f and the geometric resolution already o btained (Steps 5 and 6). A p ossib le w a y of making this c han ge of v ariables is by fir st compu ting A − 1 with O ( n Ω ) op erations and using it to obtain an slp of length L = O ( n 2 + n log ( d ) N ) f or the p olynomials in g (n ote that the length of this slp dep end s only on the cost O ( n 2 ) of computing the pro d u ct of A − 1 times a vect or, and not on the cost of in v erting A ). T aking int o account that the d egrees of th e p olynomials v 1 , . . . , v n are b ound ed b y D , to wr ite th e geometric resolution in the new v ariables, w e p erform O ( n 2 D ) op erations. Note that ( w 1 ( u ) , . . . , w n − m ( u )) = b and ( q ( u ) , w n − m +1 ( u ) , . . . , w n ( u )) is a geometric resolution of the isolated p oints in V ( g ( b, Y n − m +1 , . . . , Y n )) corresp onding to the isolated p oin ts in ( C ∗ ) n of V ( f , L 1 , . . . , L n − m ). No w, the geometric resolution of V ∗ ( f ) with resp ect to the linear pro jection giv en by Y consists of p olynomials in Q [ Y 1 , . . . , Y n − m , u ] ha ving total degrees b ound ed b y D . T h erefore, it s uffices to compute the represent ativ es of th ese p olynomials in ( Q [ Y 1 , . . . , Y n − m ] / h Y 1 − b 1 , . . . , Y n − m − b n − m i D +1 )[ u ]. T o this end, in Step 7 w e app ly successiv ely the Global Newton Iterator from [13] to the p olynomials g , starting with the geometric resolution ( q ( u ) , w n − m +1 ( u ) , . . . , w n ( u )) obtained in Step 6, wh ic h can b e regarded as a represen tativ e in ( Q [ Y 1 , . . . , Y n − m ] / h Y 1 − b 1 , . . . , Y n − m − b n − m i )[ u ], up to th e required precision D = M V n ( A , ∆ ( n − m ) ). Using [13, L emma 2] and enco din g the elemen ts of Q [ Y 1 , . . . , Y n − m ] / h Y 1 − b 1 , . . . , Y n − m − b n − m i k as ( k + 1)-tuples of slp’s (one slp for eac h homogeneous comp onen t), the complexit y of Step 7 is of order O (( mL + m Ω ) M ( D ) D 2 ). Finally , the algorithm changes v ariables bac k in order to obtain the desired geometric resolution of V ∗ ( f ), whic h adds O ( n 2 D ) to the complexit y . T aking in to accoun t the pr evious complexit y estimates, w e ha v e the follo win g result: Prop osition 2 L et f = ( f 1 , . . . , f m ) b e a system of m ≤ n generic p olynomials in Q [ X 1 , . . . , X n ] supp orte d on A = ( A 1 , . . . , A m ) . Gener icToricSol ve i s a pr ob abilistic algorithm that c omputes a ge ometric r esolution of the affine variety V ∗ ( f ) c onsisting of the union of al l the irr e ducible c omp onents of V ( f ) that have a non-empty interse ction with ( C ∗ ) n . U sing the pr evious notation, the c omplexity of this algorithm is of or der O  n 3 ( N + ( n − m ) n ) log( d ) M ( D )( M ( M )( M ( D ) + M ( E )) + D 2 )  . 3.2 Affine comp onen t s 3.2.1 Theoretic results This section is dev oted to sho wing a com binatorial descrip tion of the equid imensional de- comp osition of the affine v ariet y d efined by a generic sparse p olynomial system. More 9 precisely , we pr o ve com binatorial conditions on the s upp orts of the p olynomials that d e- termine the existence of irred ucible comp onen ts of the different p ossible d imensions not in tersecting ( C ∗ ) n and giv e a com b inatorial c haracterization of the set of linear s ubspaces where these comp onents lie. This c haracterization enables us to giv e a com binatorial form ula for the degree of th ese v arieties. The follo win g example shows that generic square sparse systems ma y defin e affine v arieties co n taining p ositiv e dimensional co mp onent s. It also sh ows th at n either the mixed v olum e nor the stable mixed volume of the system are u pp er b ounds for the d egree of the affine v ariet y defined : Example 3 Consider a generic sparse system supp orted on A = ( A 1 , A 2 , A 3 ) wh ere A 1 = { (1 , 1 , 2) , (1 , 1 , 1) } , A 2 = { (2 , 0 , 1) , (1 , 0 , 1) } and A 3 = { (0 , 2 , 1) , (0 , 1 , 1 ) } :      aX 1 X 2 X 2 3 + bX 1 X 2 X 3 = 0 cX 2 1 X 3 + dX 1 X 3 = 0 eX 2 2 X 3 + f X 2 X 3 = 0 with a, b, c, d, e, f nonzero complex num b ers. The affin e v ariet y defined by th e system has 5 irreducible comp onen ts of degree 1: { x 3 = 0 } , { x 1 = 0 , x 2 = − f e } , { x 1 = − d c , x 2 = 0 } , { x 1 = 0 , x 2 = 0 } and { ( − d c , − f e , − b a ) } . How ev er, we h av e that M V 3 ( A 1 , A 2 , A 3 ) = 1 and S M 3 ( A 1 , A 2 , A 3 ) ≤ M V 3 ( A 1 ∪ { 0 } , A 2 ∪ { 0 } , A 3 ∪ { 0 } ) = 4. Let f = ( f 1 , . . . , f s ) b e generic p olynomials in th e v ariables X = ( X 1 , . . . , X n ) sup- p orted on A = ( A 1 , . . . , A s ) ⊂ ( Z n ≥ 0 ) s . F or I ⊂ { 1 , . . . , n } , w e define J I = { j ∈ { 1 , . . . , s } | f j | T i ∈ I { x i =0 } 6≡ 0 } , that is, the set of ind ices of the p olynomials in f that do not v anish iden tically un der the sp ecializatio n X i = 0 for ev ery i ∈ I , and f I = (( f j ) I ) j ∈ J I where, for a p olynomial f ∈ C [ X 1 , . . . , X n ], f I denotes the p olynomial in C [( X i ) i / ∈ I ] ob- tained from f by sp ecializing X i = 0 for ev ery i ∈ I . Namely , f I is the set of p olynomials obtained by sp ecializing th e v ariables indexed by I to 0 in the p olynomials in f and discarding the ones that v anish iden ticall y . W e d enote by A I j the supp ort of ( f j ) I , by π I : C n → C n − # I the pro jection π I ( x 1 , . . . , x n ) = ( x i ) i / ∈ I on to the co ordinates not in I and by ϕ I : C n − # I → C n the map that inserts zero es in the co ordinates ind exed by I . F or an irr educible subv ariet y W of V ( f ) = { x ∈ C n | f j ( x ) = 0 for every 1 ≤ j ≤ s } , let I W = { i ∈ { 1 , . . . , n } | W ⊂ { x i = 0 }} . Lemma 4 Under the pr ev ious assumptions, let W b e an irr e ducible c omp onent of V ( f ) . Then dim W = n − # I W − # J I W . Mor e over π I W ( W ) is an irr e ducible c omp onent of V ( f I W ) ⊂ C n − # I W interse c ting ( C ∗ ) n − # I W . 10 Pr o of: Without loss of generalit y , we ma y assum e that I W = { r + 1 , . . . , n } and J I W = { 1 , . . . , m } for some r > 0 and m ≤ n . Then, π := π I W : C n → C r is the pro jection to the first r co ordinates and f I W = ( f 1 ( x 1 , . . . , x r , 0 ) , . . . , f m ( x 1 , . . . , x r , 0 )), where 0 is the origin of C n − r . Note that W = π ( W ) × { 0 } and π ( W ) ⊂ V ( f I W ). If π ( W ) ⊂ C ⊂ V ( f I W ) for an irreducible comp onen t C of V ( f I W ), it follo w s that W ⊂ C × { 0 } ⊂ V ( f ), with C × { 0 } an irreducible v ariety . Since W is an irreducible comp onent of V ( f ), the equalit y W = C × { 0 } holds. This implies that π ( W ) = C is an irreducible comp onent of V ( f I W ). In addition, by the d efi nition of I W , w e h a ve th at W ∩ ( T r i =1 { x i 6 = 0 } ) 6 = ∅ : otherwise, W ⊂ S r i =1 { x i = 0 } , which imp lies that W ⊂ { x i = 0 } for some 1 ≤ i ≤ r since W is an irreducible v ariet y . Therefore, π ( W ) ∩ ( C ∗ ) r 6 = ∅ . By L emm a 1, we conclude that π ( W ) ∩ ( C ∗ ) r has dim en sion r − m and so, d im( W ) = dim( π ( W )) = n − # I W − # J I W .  The previous lemma allo ws us to p ro ve that a resu lt established for bin omials in [3, Theorem 2.6] also holds f or arb itrary p olynomials. Prop osition 5 With our pr evious notatio n, assuming tha t s = n and 0 ∈ V ( f ) , we have that V ( f ) c onsists only of isolate d p oints if and only if for eve ry I ⊂ { 1 , . . . , n } , # I + # J I ≥ n . Pr o of: Assume that there exists I ⊂ { 1 , . . . , n } s u c h that # I + # J I < n . S in ce 0 ∈ V ( f I ) ⊂ C n − # I and this v ariet y is d efined b y # J I p olynomials in n − # I v ariables, it follo ws that dim( V ( f I )) ≥ n − # I − # J I > 0. T aking into accoun t that V ( f ) ⊇ ϕ I ( V ( f I )), w e conclude that dim( V ( f )) > 0. Con v ersely , if dim( V ( f )) > 0 and W is a p ositiv e dimensional ir reducible comp onent of V ( f ), b y L emm a 4, n − # I W − # J I W > 0.  No w we will c haracterize the sets I ⊂ { 1 , . . . , n } suc h that the irr ed ucible comp onen ts of V ( f I ) int ersecting ( C ∗ ) n − # I yield irredu cible comp onents of V ( f ). Prop osition 6 Under the pr ev ious assumptio ns, let I ⊂ { 1 , . . . , n } . Then V ( f I ) ∩ ( C ∗ ) n − # I is not empty if and only if for every J ⊂ J I , dim( P j ∈ J A I j ) ≥ # J and, in this c ase, V ( f I ) ∩ ( C ∗ ) n − # I is an e quidimensional variety of dimension n − # I − # J I . In addition, if W i s an irr e ducible c omp onent of V ( f I ) ∩ ( C ∗ ) n − # I , we have that ϕ I ( W ) is an irr e ducib le c omp onent of V ( f ) ∩ T i / ∈ I { x i 6 = 0 } if and only if for every e I ⊂ I , # e I + # J e I ≥ # I + # J I . Pr o of: The fir st statemen t of th e Prop osition follo ws from Lemma 1. Without loss of generalit y , assume that I = { r + 1 , . . . , n } . Let W b e an irreducible comp onen t of V ( f I ) ∩ ( C ∗ ) r . Supp ose that for a s ubset e I ⊂ I the inequ alit y # e I + # J e I < # I + # J I holds. Assume e I = { ˜ r + 1 , . . . , n } for ˜ r > r . Note that if ξ = ( ξ 1 , . . . , ξ r ) ∈ V ( f I ), then ( ξ , 0 n − r ) ∈ V ( f ) and , therefore, ( ξ , 0 ˜ r − r ) ∈ V ( f e I ). Thus, we ma y c onsider W × { 0 ˜ r − r } ⊂ V ( f e I ), whic h w ill b e included in an irreducible comp onen t f W of V ( f e I ). T aking in to acco unt that f e I consists of # J e I p olynomials in n − # e I v ariables and applying Lemma 1 to W and f I , it follo ws that dim( f W ) ≥ n − # e I − # J e I > n − # I − # J I = dim( W ) . 11 W e conclude that ϕ I ( W ) = W × { 0 n − r } ( f W × { 0 n − ˜ r } ⊂ V ( f ) and, therefore, ϕ I ( W ) is not an irredu cible comp onent of V ( f ) ∩ T i / ∈ I { x i 6 = 0 } . Con v ersely , if ϕ I ( W ) = W × { 0 n − r } is not an irr ed ucible comp onen t of V ( f ), there is an irredu cible comp onen t f W of this v ariet y su c h that W × { 0 n − r } ( f W . The p revious inclusion implies that I f W ⊂ I . Assume I f W = { ˜ r + 1 , . . . , n } for ˜ r > r . Due to Lemm a 4, π I f W ( f W ) is an irr educible comp on ent of V ( f I f W ) ha ving a non-empty intersec tion with ( C ∗ ) ˜ r . Therefore, n − # I f W − # J I f W = dim( π I f W ( f W )) = dim( f W ) > dim W = n − # I − # J I , and so, # I + # J I > # I f W + # J I f W .  As a consequence of Prop osition 6, w e hav e that the irreducible comp onen ts of V ( f ) ⊂ C n are con tained in the linear subsp aces T i ∈ I { x i = 0 } asso ciated to th e subsets I ⊂ { 1 , . . . , n } in Γ = n I ⊂ { 1 , . . . , n } | ∀ J ⊂ J I , dim( X j ∈ J A I j ) ≥ # J ; ∀ e I ⊂ I , # J e I + # e I ≥ # J I + # I o . Note that th ere ma y b e sets I 1 ( I 2 ⊂ { 1 , . . . , n } b oth in Γ, as it can b e seen in Example 3, where the three sets { 1 } , { 2 } and { 1 , 2 } giv e irreducible comp onen ts of the v ariet y . If we write V ∗ ( f I ) to denote the union of all the irr educible comp onent s of V ( f I ) ha vin g a n on -emp t y in tersection with ( C ∗ ) n − # I , fr om Lemma 4 and Prop osition 6, w e deduce that, for ev ery I ∈ Γ, ϕ I ( V ∗ ( f I )) = [ W i rred. comp. of V ( f ) such th at I W = I W . W e obtain the follo wing c h aracterizat ion of the equidimensional decomp osition of V ( f ) and, using Lemma 1, a combinatoria l expression for the degree of V ( f ): Theorem 7 L et f = ( f 1 , . . . , f s ) b e generic p olynomials in n variables supp orte d on A = ( A 1 , . . . , A s ) ⊂ ( Z n ≥ 0 ) s . F or k = 0 , . . . , n , let V k ( f ) b e the e qu idimensional c omp onent of dimension k of V ( f ) . Then, using the pr e v ious notations: V k ( f ) = [ I ∈ Γ , # I +# J I = n − k ϕ I ( V ∗ ( f I )) . Mor e over, deg( V ( f )) = P I ∈ Γ M V n − # I ( A I , ∆ ( n − # I − # J I ) ) . Example 8 Consider the follo wing system of generic p olynomials in Q [ X 1 , X 2 , X 3 , X 4 ]:            a 1 X 1 X 4 + a 2 X 2 1 X 2 4 + a 3 X 1 X 2 X 3 + a 4 X 2 X 3 = 0 b 1 X 1 X 2 + b 2 X 1 X 2 2 + b 3 X 1 X 3 X 4 + b 4 X 3 X 4 + b 5 X 3 X 2 4 = 0 c 1 X 1 X 2 X 4 + c 2 X 1 X 3 X 4 + c 3 X 2 X 3 + c 4 X 2 X 3 X 4 = 0 d 1 X 1 + d 2 X 2 1 + d 3 X 1 X 2 + d 4 X 2 3 + d 5 X 3 X 4 = 0 Then, with the pr evious n otatio n, Γ = {∅ , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 }} and then, 12 • V 0 ( f ) = V ∗ ( f ) ∪ { (0 , 0 , b 4 d 5 b 5 d 4 , − b 4 b 5 ) , ( − d 1 d 2 , 0 , 0 , a 1 d 2 a 2 d 1 ) , ( − d 1 d 2 + b 1 d 3 b 2 d 2 , − b 1 b 2 , 0 , 0) } : – F or I = ∅ w e h av e 19 isolated solutions in ( C ∗ ) 4 (this qu an tity is giv en b y the mixed vol ume of the f amily of supp orts asso ciated to th e system) – F or I = { 1 , 2 } we h a ve : f { 1 , 2 } = ( b 4 X 3 X 4 + b 5 X 3 X 2 4 d 4 X 2 3 + d 5 X 3 X 4 ; V ∗ ( f { 1 , 2 } ) = n b 4 d 5 b 5 d 4 , − b 4 b 5 o , whic h giv es the p oin t (0 , 0 , b 4 d 5 b 5 d 4 , − b 4 b 5 ). – F or I = { 2 , 3 } we h a ve : f { 2 , 3 } = ( a 1 X 1 X 4 + a 2 X 2 1 X 2 4 d 1 X 1 + d 2 X 2 1 ; V ∗ ( f { 2 , 3 } ) = n − d 1 d 2 , a 1 d 2 a 2 d 1 o , whic h giv es the p oin t ( − d 1 d 2 , 0 , 0 , a 1 d 2 a 2 d 1 ). – F or I = { 3 , 4 } we h a ve : f { 3 , 4 } = ( b 1 X 1 X 2 + b 2 X 1 X 2 2 d 1 X 1 + d 2 X 2 1 + d 3 X 1 X 2 ; V ∗ ( f { 3 , 4 } ) = n − d 1 d 2 + b 1 d 3 b 2 d 2 , − b 1 b 2 o , whic h giv es the p oin t ( − d 1 d 2 + b 1 d 3 b 2 d 2 , − b 1 b 2 , 0 , 0). • V 1 ( f ) = { x ∈ C 4 | x 2 = 0 , x 4 = 0 , d 1 x 1 + d 2 x 2 1 + d 4 x 2 3 = 0 } : – F or I = { 2 , 4 } we h a ve : f { 2 , 4 } = { d 1 X 1 + d 2 X 2 1 + d 4 X 2 3 ; V ∗ ( f { 2 , 4 } ) = { ( x 1 , x 3 ) | d 1 x 1 + d 2 x 2 1 + d 4 x 2 3 = 0 } , whic h giv es the cur v e { x 2 = 0 , x 4 = 0 , d 1 x 1 + d 2 x 2 1 + d 4 x 2 3 = 0 } . • V 2 ( f ) = { x ∈ C 4 | x 1 = 0 , x 3 = 0 } : – F or I = { 1 , 3 } we h a ve : f { 1 , 3 } = ∅ ; V ∗ ( f { 1 , 3 } ) = C 2 , whic h giv es the p lane { x 1 = 0 , x 3 = 0 } . Remark 9 In the case of a generic system f = ( f 1 , . . . , f s ) in n v ariables such that A 1 = · · · = A s , we hav e that the sets I ∈ Γ , I 6 = ∅ , are all I ⊂ { 1 , . . . , n } such that # J I = 0 and for every e I ( I , # J e I > 0; and ∅ ∈ Γ if and only if dim( A 1 ) ≥ s . Moreo v er, f or ev ery I ∈ Γ, I 6 = ∅ , ϕ I ( V ∗ ( f I )) = { x i = 0 ∀ i ∈ I } and so, apart fr om the comp onen ts int ersecting ( C ∗ ) n that corresp ond to I = ∅ (if ∅ ∈ Γ), the only irreducible comp onen ts of V ( f ) are linear sub spaces of C n . 13 3.2.2 Algorithmic result s According to Prop osition 6, the subsets I ⊂ { 1 , . . . , n } wh ic h yield irr ed ucible co mp onents of the v ariet y V ( f ) are the ones satisfying simulta neously 1. ∀ J ⊂ J I , d im( P j ∈ J A I j ) ≥ # J , 2. ∀ I ′ ⊂ I , # J I ′ + # I ′ ≥ # J I + # I . No w we pr esen t an algorithm to obtain the sets I satisfying condition (2) and the inequalit y # I + # J I ≤ n , whic h is a n ecessary condition for the system f I to ha v e zero es in ( C ∗ ) n − # I , weak er bu t easier to c hec k than condition (1). O ur algorithm to find all the affine comp onents of V ( f ) (see Algorithm G enericAffi neComps b elo w) c hec ks only among these sets whether cond ition (1) is fulfilled or not by means of a mixed v olume computation. Algorithm Specia lSets INPUT: A family of su pp orts A = ( A 1 , . . . , A s ) ⊂ ( Z n ≥ 0 ) s . 1. P ∅ := min { n, s } . 2. If P ∅ = s , add ( ∅ , { 1 , . . . , s } ) to an empt y list e Γ. 3. F or k = 1 , . . . , n : F or ev ery I suc h that # I = k : (a) P I := min { n, { P I ′ } I ′ ⊂ I , # I ′ = k − 1 , k + # J I } . (b) If P I = k + # J I , add ( I , J I ) to the list e Γ. OUTPUT: The list e Γ of all pairs of su bsets ( I , J I ), with I ⊂ { 1 , . . . , n } suc h th at # I + # J I ≤ n and for eve ry e I ⊂ I , # e I + # J e I ≥ # I + # J I . First, let u s pr o ve the correctness of this algorithm: Lemma 10 F or every I ⊂ { 1 , . . . , n } , P I = min e I ⊂ I { n, # e I + # J e I } . Pr o of: By indu ction on # I . F or # I = 0, since # J ∅ = s , w e ha v e that P ∅ = min { n, # ∅ + # J ∅ } . Assuming the identit y holds for ev ery su bset of cardinalit y k − 1, let I ⊂ { 1 , . . . , n } with # I = k . Consid er a prop er subset e I 0 ( I . T hen, there exists I ′ ⊂ I with # I ′ = k − 1 suc h that e I 0 ⊂ I ′ . By the ind uctiv e assu mption, P I ′ = min e I ⊂ I ′ { n, # e I + # J e I } and so, P I ′ ≤ # e I 0 + # J e I 0 . On the other hand , by th e definition of P I , w e h a ve that P I ≤ P I ′ . Thus, P I ≤ # e I 0 + # J e I 0 .  14 No w w e estimate the complexit y of the algorithm. Let N = P s j = 1 # A j . F or eac h I of cardinalit y k ≥ 1, it tak es k N + # J I + 1 op erations to compu te k + # J I . T a king the minim um among k + 2 n um b ers take s k + 1 comparisons. Thus, the complexit y of Step 3a is k ( N + 1) + # J I + 2. In Step 3b, w e add one comparison. As we hav e to d o this for eac h subset of { 1 , . . . , n } of cardinalit y k ≥ 1 and for the empty set, the complexity is b oun d ed b y 2 + P n k =1  n k  ( k ( N + 1) + s + 3) = 2 + n 2 n − 1 ( N + 1) + (2 n − 1)( s + 3) w hic h is of order O ( nN 2 n ). Unfortunately , the exp onential complexit y of the algorithm cannot b e a voided, as the follo wing examples sh o w: Example 11 Let L 1 , . . . , L n ∈ Q [ X 1 , . . . , X n ] b e generic affine linear forms. Consider the set of generic p olynomials f 1 = X 1 .L 1 , . . . , f n = X n .L n . In this case, if A = ( A 1 , . . . , A n ) is their family of su pp orts, Γ = { I | I ⊆ { 1 , . . . , n }} and therefore #Γ = 2 n . One ma y think the exp onen tialit y of the cardinal of th e set Γ in this example arises from the fact that the v ariables are factors of the p olynomials. How ev er, this is not alw ays the case as the follo wing example sh o ws. Th is example also shows how subroutine SpecialS ets is useful to discard s ubsets wh ic h do not lead to affine comp onent s: in this case, the only element in e Γ whic h do es not corresp ond to a set in Γ is ( ∅ , { 1 , . . . , 2 n } ). Example 12 Consider generic p olynomials f 1 , . . . , f 2 n ∈ Q [ X 1 , . . . , X 2 n ], f j = X 1 ≤ k ≤ n c j k X 2 k − 1 X 2 k , j = 1 , . . . , 2 n, all supp orted on the set A = { e 2 k − 1 + e 2 k | 1 ≤ k ≤ n } wh er e e i denotes the i th v ector of the canonical b asis of R 2 n . Then, it is easy to see that, for any subset S ⊂ { 1 , . . . , n } , the set I S = { 2 k − 1 | k ∈ S } ∪ { 2 k | k ∈ { 1 , . . . , n } \ S } is in Γ, and that the sets I S are the only ones (toge ther with the empt y set) obtained as first co ordinate of elemen ts of e Γ by the p r evious algorithm. T herefore, in this case, w e hav e that th e num b er of elemen ts of the list e Γ is 2 n + 1. In the follo win g example, the usefulness of su broutine Sp ecialSets is more eviden t: Example 13 Let A = ( A 1 , . . . , A n ) b e a family of finite sets o f ( Z ≥ 0 ) n suc h that, for eve ry 1 ≤ j ≤ n , ther e exists a non n egativ e inte ger d j i suc h that d j i .e i ∈ A j for every 1 ≤ i ≤ n . Then the output of subroutine Specia lSets in this case is e Γ = { ( ∅ , { 1 , . . . , n } ) } . The follo win g algorithm compu tes a family of geometric resolutions describing the affine v ariet y defined by a generic system f with giv en supp orts A . Algorithm Generi cAffineSo lve INPUT: A sp ars e represent ation of the generic system f = ( f 1 , . . . , f s ) of p olynomials in Q [ X 1 , . . . , X n ]. 15 1. Apply algorithm Sp ecialSets to the family of the supp orts A = ( A 1 , . . . , A s ) of f to obtain th e list e Γ of pairs of s ets ( I , J I ) with I ⊂ { 1 , . . . , n } suc h that # I + # J I ≤ n and ∀ e I ⊂ I , # e I + # J e I ≥ # I + # J I . 2. F or ev ery ( I , J I ) ∈ e Γ: (a) F o r j ∈ J I , compu te A I j := { π I ( a ) | a ∈ A j suc h that a i = 0 ∀ i ∈ I } , and obtain th e sp arse representat ion of the sys tem f I supp orted on A I = ( A I j ) j ∈ J I . (b) Apply algorithm Gen ericToric Solve to th e sparse system f I to obtain a ge- ometric resolution R I (p ossibly empt y) of the affine comp onents of the set of solutions of f I in tersecting the torus ( C ∗ ) n − # I . (c) If R I 6 = ∅ : i. Obtain the geometric r esolution ϕ I ( R I ) of the u nion of all irreducible com- p onents W of V ( f ) suc h that I W = I b y add ing zero es to R I in the co or- dinates indexed by I . ii. If n − (# I + # J I ) = k , add ϕ I ( R I ) to the list V k . OUTPUT: A family of lists V k , 0 ≤ k ≤ n − 1, of geometric r esolutions, eac h list either empt y or describing the equidimensional comp onent of V ( f ) of dimens ion k . The correctness of this algorithm is straigh tforward from Prop osition 6 and Theorem 7. When applyin g algorithm Gen ericToric Solve in Step 2b, we n eed a p re-pro cessing obtaining a fi ne mixed su b division. T o do so, w e m a y apply the d ynamic en umeration pro cedure from [26]. This pro cedu r e pro v ed to b e very effici en t ev en for large systems, but there are no explicit complexit y b ou n ds; for th is r eason, we do n ot includ e its cost in our complexit y estimates. This pr e-pro cessing, in particular, decides wh ether a set I satisfies dim( P j ∈ J A I j ) ≥ # J for ev ery J ⊂ J I and, therefore, it discards the sets I ∈ e Γ \ Γ. F or this reason, we will only consider the complexit y of the compu tations for the sets I ∈ Γ. This complexit y can b e estimated from the complexities of the sub routines applied at the in termediate steps (see Pr op osition 2 ) and is of order O  X I ∈ Γ ( n − # I ) 3  N I +( n − # I − # J I )( n − # I )  log( d I ) M ( D I )  M ( M I ) ( M ( D I ) + M ( E I )) + D 2 I   where, for eve ry I ∈ Γ, N I , d I , D I , M I and E I are the parameters defined in the complexit y of Algorithm Gen ericToricS olve , asso ciated to the system f I . In order to estimate the o v erall complexit y of the algorithm, note that N I ≤ N := P s j = 1 # A j and d I ≤ d := max 1 ≤ j ≤ s { deg( f j ) } for ev ery I ∈ Γ. In add ition, D := P I ∈ Γ D I = deg V ( f ). Note that, if ω max is th e maxim um v alue of the lifting fu nctions applied to the su pp orts A I , for ev ery I ∈ Γ we hav e E I ≤ M V n − # I +1 (∆ × { 0 } , ( A I j × { 0 , ω max } ) j ∈ J I , (∆ × { 0 , ω max } ) ( n − # I − # J I ) ) ≤ ≤ ω max  ( n − # I − # J I ) M V n − # I ( A I , ∆ ( n − # I − # J I ) )+ X ℓ ∈ J I M V n − # I (( A I j ) j 6 = ℓ , ∆ ( n − # I − # J I +1) )  . Then, if E max := max I ∈ Γ { ( n − # I − # J I ) M V n − # I ( A I , ∆ ( n − # I − # J I ) ) + P ℓ ∈ J I M V n − # I (( A I j ) j 6 = ℓ , ∆ ( n − # I − # J I +1) ) } and M max := max I ∈ Γ { M I } , taking int o ac- coun t the complexit y of Step 1 , we ha ve: 16 Theorem 14 L e t f = ( f 1 , . . . , f s ) b e a system of generic p olynomials in Q [ X 1 , . . . , X n ] supp orte d on A = ( A 1 , . . . , A s ) . Gen ericAffine Solve is a pr ob abilistic algorithm that c omputes a family of lists V k , 0 ≤ k ≤ n − 1 , of ge ometric r esolutions, e ach list e i ther empty or describi ng the e quidimensional c omp onent of V ( f ) of dimension k . Usi ng the pr evious notation, the c omplexity of this algorithm is of or der O  n 2 n N + n 3 ( N + n 2 ) log ( d ) M ( D )  M ( M max ) ( M ( D ) + M ( ω max E max )) + D 2  . 4 Arbitrary sparse sy stems 4.1 An upp er b ound for the degree The aim of this section is to sho w a b ound for the degree of the affine v ariet y defined by a square system of sp arse p olynomials which tak es into accoun t its sp ars it y . The follo wing example shows that the degree of an affine v ariet y defined by a generic sparse system with giv en s u pp orts is not an upp er b oun d for the degree of the v ariety defined by a particular system with the same supp orts. On e ma y think the p roblem arises from th e presence of irr educible comp onen ts not in tersecting the torus either for the generic or the particular systems. Ho w ever, this is not the case: Example 15 Consider the follo wing sys tem:      X 1 X 2 − X 1 − X 2 + 1 = ( X 1 − 1)( X 2 − 1) = 0 X 1 X 3 − X 1 − X 3 + 1 = ( X 1 − 1)( X 3 − 1) = 0 X 2 X 3 − X 2 − X 3 + 1 = ( X 2 − 1)( X 3 − 1) = 0 The v ariety defin ed by the system consists of 3 lines, eac h ha vin g a non-empty inter- section with ( C ∗ ) 3 . Ho wev er, if A = ( A 1 , A 2 , A 3 ) is the family of the supp orts of the p olynomials, the d egree of the v ariety d efined by a generic s ys tem with the same su p p orts is M V 3 ( A 1 , A 2 , A 3 ) = 2. Our b ound for the d egree, whic h i s stated in the follo wing theorem, is the mixed v olume of a family of s ets obtained b y enlarging the sup p orts of th e p olynomials inv olved. Theorem 16 L e t f = ( f 1 , . . . , f n ) b e n p olynomials in C [ X 1 , . . . , X n ] supp orte d on A = ( A 1 , . . . , A n ) and let V ( f ) = { x ∈ C n | f i ( x ) = 0 for every 1 ≤ i ≤ n } . L et ∆ = { 0 , e 1 , . . . , e n } wher e e i the i th ve ctor of the c anonic al b asis of R n . Then deg( V ( f )) ≤ M V n ( A 1 ∪ ∆ , . . . , A n ∪ ∆) . Before pr oving the theorem, w e will fix some notation and d efinitions. Let r j = #( A j ∪ ∆) − 1 for 1 ≤ j ≤ n . Consider the morp hism of v arieties ϕ : C n → P n × P r 1 × · · · × P r n defined by ϕ ( x ) = ((1 : x ) , ( x a ) a ∈A 1 ∪ ∆ , . . . , ( x a ) a ∈A n ∪ ∆ ) . (1) Let X = ϕ ( C n ). F o r 1 ≤ j ≤ n , w e d enote by L j the lin ear form in P r j giv en by the co efficien ts of the p olynomial f j , that is to say , if f j = P a ∈A j c j,a X a , then L j = P a ∈A j c j,a X j,a . F or eac h in teger k (0 ≤ k ≤ n ) and eac h sub set S ⊂ { 1 , . . . , k } w e defin e the v ariet y X k ,S recursiv ely in the follo wing wa y: 17 1. X 0 , ∅ = X . 2. Pro vid ed X k ,S is defined for eve ry S ⊂ { 1 , . . . , k } , we define X k +1 ,T with T ⊂ { 1 , . . . , k + 1 } as follo ws: • If k + 1 / ∈ T , X k +1 ,T is the union of the ir reducible comp onents of X k ,T included in { L k +1 = 0 } . • If k + 1 ∈ T , X k +1 ,T is the intersecti on of { L k +1 = 0 } with the un ion of the irreducible comp onents of X k ,T \{ k +1 } not included in { L k +1 = 0 } . Note that, fr om this definition, eac h X k ,S is an equidimensional v ariety of dimension n − # S . Moreo ver, if π : P n × P r 1 × · · · × P r n → P n is the pr o jection on to the fir st factor, it is easy to see inductive ly that, for eve ry 1 ≤ k ≤ n , [ S ⊂{ 1 ,.. .,k } π ( X k ,S ) = V ( f 1 , . . . , f k ) ⊂ P n . F or an equidimensional subv ariet y W of X , w e define its multidegrees deg ( r , 0 k , 1 n − k ) ( W ), for r , k ∈ Z ≥ 0 suc h that n − k + r = d im( W ), as deg ( r , 0 k , 1 n − k ) ( W ) = max    #  W ∩ r \ j = 1 { ℓ 0 ,j = 0 } ∩ n \ j = k +1 { ℓ j = 0 }     where the maxim u m is taken o ver all ( ℓ 0 , 1 , . . . , ℓ 0 ,r , ℓ k +1 , . . . , ℓ n ) such that eac h ℓ 0 ,j is a linear form in P n and eac h ℓ j is a linear form in P r j and the inte rsection is fin ite. Note that the 1 subscrip t indicates ho w many p ro jectiv e spaces are cut by a linear f orm and the 0 sub s cript shows h o w man y remain uncut. As in the case of the standard degree of affine or pro jectiv e v arieties (see [14]), the maximum is attained generically . In particular, it is clear that deg (0 , 0 0 , 1 n ) ( X ) = M V n ( A 1 ∪ ∆ , . . . , A n ∪ ∆). Lemma 17 Under the pr evious assumptions, definitions and notations, deg ( k − # S, 0 k , 1 n − k ) ( X k ,S ) ≥ deg ( k +1 − # S , 0 k +1 , 1 n − k − 1 ) ( X k +1 ,S )+deg ( k − # S, 0 k +1 , 1 n − k − 1 ) ( X k +1 ,S ∪{ k +1 } ) Pr o of: As the v ariet y X k ,S = X k +1 ,S ∪ g X k ,S , wh ere g X k ,S is the u nion of the irreducible comp onen ts of X k ,S not con tained in { L k +1 = 0 } , usin g genericit y in the definition of m ultidegrees, we hav e that deg ( k − # S, 0 k , 1 n − k ) ( X k ,S ) = deg ( k − # S, 0 k , 1 n − k ) ( X k +1 ,S ) + deg ( k − # S, 0 k , 1 n − k ) ( g X k ,S ) . Note that, b y adding the simp lex ∆ to the supp orts A 1 , . . . , A n , the p oin ts of the v arieties in P n × P r 1 × · · · × P r n w e are considerin g ha v e a cop y of their co ordinate in P n in eac h co ordinate in P r j for j = 1 , . . . , n (see Equation (1)). Because of this, if ℓ 0 is a linear form i n P n and ℓ k +1 is exac tly the same linear form in volving on ly the corresp onding co ordinates in P r k +1 , w e h a ve th at X k +1 ,S ∩ { ℓ 0 = 0 } = X k +1 ,S ∩ { ℓ k +1 = 0 } an d th er efore, deg ( k − # S, 0 k , 1 n − k ) ( X k +1 ,S ) ≥ deg ( k +1 − # S , 0 k +1 , 1 n − k − 1 ) ( X k +1 ,S ) and deg ( k − # S, 0 k , 1 n − k ) ( g X k ,S ) ≥ deg ( k − # S, 0 k +1 , 1 n − k − 1 )  g X k ,S ∩ { L k +1 = 0 }  = = deg ( k − # S, 0 k +1 , 1 n − k − 1 ) ( X k +1 ,S ∪{ k +1 } ) .  18 No w, w e can prov e Theorem 16: Pr o of: Consider the p ro jection π : P n × P r 1 × · · · × P r n → P n on to the first factor. As [ S ⊂{ 1 ,.. .,n } π ( X n,S ) = V ( f ) ⊂ P n , w e ha ve that deg( V ( f )) = deg  [ S ⊂{ 1 ,.. .,n } π ( X n,S )  ≤ X S ⊂{ 1 ,.. .,n } deg π ( X n,S ) = = X S ⊂{ 1 ,.. .,n } deg ( n − # S, 0 n , 1 0 ) ( X n,S ) (this last equalit y is n othin g but our definition of multidegree). Applying indu ctivel y Lemma 17, we get that, for eac h 0 ≤ k ≤ n , X S ⊂{ 1 ,.. .,k } deg ( k − # S, 0 k , 1 n − k ) ( X k ,S ) ≤ deg (0 , 0 0 , 1 n ) X 0 , ∅ and therefore we obtain that X S ⊂{ 1 ,.. .,n } deg ( n − # S, 0 n , 1 0 ) ( X n,S ) ≤ deg (0 , 0 0 , 1 n ) ( X 0 , ∅ ) = M V n ( A 1 ∪ ∆ , . . . , A n ∪ ∆) .  In the follo win g examples, we sh o w that this b ound may b e attained: Example 18 Let S ( { 1 , . . . , n } and let f 1 , . . . , f n ∈ Q [ X 1 , . . . , X n ] b e p olynomials of degree d in the v ariables ( X i ) i ∈ S with no zero co efficient s (that is to say , the monomials app earing in f 1 , . . . , f n are those of degree less or equal to d in the v ariables ( X i ) i ∈ S ). If A is their common supp ort, it is eviden t that M V n ( A ( n ) ) = 0 and that S M n ( A ( n ) ) = 0. Ho wev er, M V n (( A ∪ ∆) ( n ) ) = d # S and this degree can be attai ned for sp ecial choice s of the p olynomials: If f 1 , . . . , f # S ∈ Q [( X i ) i ∈ S ] is a family with d # S isolated solutions in C # S , tak e linear com b in ations f # S +1 , . . . , f n of f 1 , . . . , f # S . Th en, the family of p olynomials f 1 , . . . , f # S , f # S +1 , . . . , f n ∈ Q [ X 1 , . . . , X n ] d efines a v ariet y formed b y d # S affine linear spaces of dimension n − # S . Our b ound can b e also attained for generic systems: Example 19 Consider the family of generic p olynomials f 1 , . . . , f n defined in Example 11 an d let their sup p orts A = ( A 1 , . . . , A n ). Th en, the affine v ariet y they define consists of 2 n p oin ts, and therefore, its d egree is 2 n = M V n ( A 1 ∪ ∆ , . . . , A n ∪ ∆). 4.2 An algorithm in the non-generic case In the sequ el we will describ e an algorithm that, giv en an arbitr ary square sys tem of sparse p olynomials, pro vides a fin ite s et of p oin ts in eac h irreducible comp onent of th e affine v ariet y the system d efines. T he complexit y of this algorithm dep end s on the b ound for the degree of the v ariet y obtained in th e p r evious s ection. 19 Let f = ( f 1 , . . . , f n ) b e n p olynomials in Q [ X 1 , . . . , X n ] su pp orted on A = ( A 1 , . . . , A n ) and, for a fixed k , 1 ≤ k ≤ n − 1, let L 1 , . . . , L k b e ge neric affine linear forms in Q [ X 1 , . . . , X n ]. Then , if V k ( f ) is the equidimen s ional comp onen t of d imension k of V ( f ), the isolated common zero es of f 1 , . . . , f n , L 1 , . . . , L k are deg( V k ( f )) p oin ts in V k ( f ). The idea is to repr esent eac h equidimensional comp onent b y means of the corresp onding set of p oin ts (c.f. the notion of witness p oint set in [34]). Example 20 Consider the follo wing p olynomial system: f =            X 3 1 X 2 X 3 − X 1 X 2 X 3 3 − X 2 1 + X 2 3 = ( X 1 X 2 X 3 − 1)( X 1 − X 3 )( X 1 + X 3 ) X 2 1 X 2 2 X 3 − X 2 1 X 2 X 3 − X 1 X 2 2 X 2 3 + X 1 X 2 X 2 3 − X 1 X 2 + X 1 + X 3 X 2 − X 3 = = ( X 1 X 2 X 3 − 1)( X 1 − X 3 )( X 2 − 1) X 1 X 3 2 X 3 − X 1 X 2 X 2 3 − X 2 2 + X 3 = ( X 1 X 2 X 3 − 1)( X 2 2 − X 3 ) The equidimensional comp onen ts of V ( f ) are V 0 ( f ) = { ( − 1 , 1 , 1) } , V 1 ( f ) = { x 1 − x 3 = 0 , x 2 2 − x 3 = 0 } , V 2 ( f ) = { x 1 x 2 x 3 − 1 = 0 } . T aking L 1 = X 1 − X 2 and L 2 = 6 X 2 − X 3 + 7, we h a ve: • the set of isolated p oin ts of V ( f ) ∩ { x 1 − x 2 = 0 } is { (1 , 1 , 1) , (0 , 0 , 0) } , whic h is a set with 2 = deg ( V 1 ( f )) p oint s in V 1 ( f ). • V ( f ) ∩ { x 1 − x 2 = 0 , 6 x 2 − x 3 + 7 = 0 } = { ( − 1 , − 1 , 1) , ( − 1 2 , − 1 2 , 4) , ( 1 3 , 1 3 , 9) } , wh ich is a set with 3 = deg( V 2 ( f )) p oint s in V 2 ( f ). F or a fixed k , 0 ≤ k ≤ n − 1, in order to compu te the isolated common zero es of f 1 , . . . , f n , L 1 , . . . , L k , by taking n generic linear com binations of these p olynomials, w e obtain a system of n p olynomials in n v ariables h a ving these p oin ts among its isolated zero es (see [14]). Note that, in order to ac hieve th is, it suffi ces to take linear com binations of the form f i ( X ) + k X j = 1 b ij L j ( X ) , i = 1 , . . . , n for generic b ij (1 ≤ i ≤ n, 1 ≤ j ≤ k ). Pro cedure Point sInEquidC omps describ ed b elow compu tes a family of n geometric resolutions R ( k ) , for 0 ≤ k ≤ n − 1, enco ding a fin ite s et of p oints and suc h th at R ( k ) represent s at least deg( W ) p oin ts in eac h irreducible comp onent W of dimension k of V ( f ). The in termediate subroutine CleanGR tak es as input a geometric resolution ( q ( u ) , v 1 ( u ) , . . . , v n ( u )) ⊂ ( Q [ u ]) n +1 of a finite set of p oints P ⊂ C n and a list f = ( f 1 , . . . , f n ) of p olynomials in Q [ X 1 , . . . , X n ], and computes a ge ometric resolution ( Q ( u ) , V 1 ( u ) , . . . , V n ( u )) of P ∩ V ( f ): Q ( u ) = gcd( q ( u ) , f 1 ( v 1 ( u ) , . . . , v n ( u )) , . . . , f n ( v 1 ( u ) , . . . , v n ( u ))) V i ( u ) = v i ( u ) mo d Q ( u ) , i = 1 , . . . , n . Let A j b e the su pp ort of f j , d an u pp er b oun d for deg( f j ), j = 1 , . . . , n , and D = deg q . Firs t, the sub routine computes slp’s of length O ( nD log D ) for the p olynomials v i , i = 1 , . . . , n . T h e gcd Q ( u ) is compu ted su ccessiv ely as follo ws: F or j = 1 , . . . , n , 20 the s ubroutine computes an slp of length L j = O ( n log d # A j ) f or the p olynomial f j and, b y m ultip oin t ev aluation and in terp olation, the d ense representat ion of F j ( u ) = f j ( v 1 ( u ) , . . . , v n ( u )) within complexit y O ( M ( dD )( L j + n D log D )); then, it compu tes Q j ( u ) := gcd( Q j − 1 , F j ( u )) within O ( M ( dD )) add itional op erations. Finally , the p oly- nomials V i ( u ) for i = 1 , . . . , n are obtained within complexit y O ( nM ( D )). The ov er all complexit y of the pr o cedure is of order O ( M ( dD )( n log d P n j = 1 # A j + n 2 D log D )). Algorithm Points InEquidCo mps INPUT: A sparse represen tation of a s ystem f = ( f 1 , . . . , f n ) of polynomials in Q [ X 1 , . . . , X n ] supp orted on A = ( A 1 , . . . , A n ), a lifting function ω = ( ω 1 , . . . , ω n ) for A ∆ = ( A 1 ∪ ∆ , . . . , A n ∪ ∆) and the m ixed cells in the induced sub d ivision of A ∆ . 1. Cho ose r andomly co efficient s for a p olynomial system g = ( g 1 , . . . , g n ) supp orted on A ∆ . 2. Apply the algorithm in [19, Section 5] to g to obtain a geometric resolution R g of its zero es in C n . 3. Cho ose randomly n − 1 affin e lin ear form s L 1 , . . . , L n − 1 in the v ariables X = ( X 1 , . . . , X n ) and n ( n − 1) intege r n um b ers ( b ij ) 1 ≤ i ≤ n, 1 ≤ j ≤ n − 1 . 4. F or k = 0 , . . . , n − 1: (a) Obtain the sparse r epresen tation of the p olynomials h ( k ) i ( X ) = f i ( X )+ P k j = 1 b ij L j ( X ) for 1 ≤ i ≤ n . (b) Apply the algorithm in [19, Section 6] to h ( k ) = ( h ( k ) 1 , . . . , h ( k ) n ) to obtain from R g a geometric resolution of a finite set P k whic h con tains th e isolated common zero es of h ( k ) in C n . (c) Apply sub routine CleanGR to the previous geometric resolution and f to obtain a geometric r esolution R ( k ) of P k ∩ V ( f ). OUTPUT: The n geometric r esolutions R ( k ) for 0 ≤ k ≤ n − 1. In the sequel we w ill estimate the complexit y of this p ro cedure. Steps 1 and 3 are fulfi lled by taking a r andom c hoice of num b ers. W e will not consider the cost of th is random c h oice in the o verall complexit y (see Remark 23). The complexit y of Step 2 is O (( n 3 N ∆ log( d ) + n 1+Ω ) M ( D ∆ ) M ( M ∆ )( M ( D ∆ ) + M ( E ∆ ))) where • N ∆ := P n j = 1 #( A j ∪ ∆); • d := max 1 ≤ j ≤ n { deg f j } ; • D ∆ := M V n ( A 1 ∪ ∆ , . . . , A n ∪ ∆); 21 • M ∆ := max {k µ k} wh er e the maximum r anges o ver all primitiv e normal v ectors to the mixed cells in the fine mixed su b division of A ∆ induced by ω ; • E ∆ := M V n +1 (∆ × { 0 } , ( A 1 ∪ ∆)( ω 1 ) , . . . , ( A n ∪ ∆)( ω n )) wh er e ( A j ∪ ∆)( ω j ) for ev ery 1 ≤ j ≤ n is the set A j ∪ ∆ lifted by ω . In Step 4b, we compute a fi n ite set wh ich includes the affine isolated zero es of th e system h ( k ) . By applyin g the result in [19 , Pr op osition 6.1], w e ha v e that the complexit y of this step is b ounded b y O (( n 2 N ∆ log d + n 1+Ω ) M ( D ∆ ) M ( E ′ ∆ )) wh er e E ′ ∆ := M V n +1 ( { 0 } × ∆ , { 0 , 1 } × ( A 1 ∪ ∆) , . . . , { 0 , 1 } × ( A n ∪ ∆)). Finally , the complexit y of Step 4c is of ord er O ( M ( dD ∆ )( n log d P n j = 1 # A j + n 2 D ∆ log D ∆ )). Note that the parameters E ∆ and E ′ ∆ in the p revious complexities can b e b ounded as follo ws: E ′ ∆ = n X j = 1 M V n (∆ , A 1 ∪ ∆ , . . . , \ A j ∪ ∆ , . . . , A n ∪ ∆) ≤ nD ∆ and, if ω max := max j,a { ω j ( a ) | 1 ≤ j ≤ n, a ∈ A j ∪ ∆ } , E ∆ ≤ M V n +1 (∆ × { 0 } , ( A 1 ∪ ∆) × { 0 , ω max } , . . . , ( A n ∪ ∆) × { 0 , ω max } ) ≤ ω max nD ∆ . T aking in to account these b oun ds, we ha v e: Theorem 21 L e t f = ( f 1 , . . . , f n ) b e n p olynomials in Q [ X 1 , . . . , X n ] supp orte d on A = ( A 1 , . . . , A n ) . Poi ntsInEqui dComps is a pr ob abilistic algorithm which c omputes a family of n ge ometric r esolutions ( R (0) , R (1) , . . . , R ( n − 1) ) su c h that, for ev e ry 0 ≤ k ≤ n − 1 , R ( k ) r epr esents a finite set c ontaining deg V k ( f ) p oints in the e quidimensional c omp onent V k ( f ) of dimension k of V ( f ) . Using the pr evious notation, the c omplexity of the algorithm is of or der O ( ω max n 4 N ∆ log( d ) M ( dD ∆ ) M ( D ∆ ) M ( M ∆ )) . Example 22 Consider the p olynomial system introd uced in Example 15, giv en b y the p olynomials f =      f 1 = X 1 X 2 − X 1 − X 2 + 1 f 2 = X 1 X 3 − X 1 − X 3 + 1 f 3 = X 2 X 3 − X 2 − X 3 + 1 supp orted on A 1 = { (1 , 1 , 0) , (1 , 0 , 0) , (0 , 1 , 0) , ( 0 , 0 , 0) } , A 2 = { (1 , 0 , 1) , (1 , 0 , 0 ) , (0 , 0 , 1) , (0 , 0 , 0) } and A 3 = { (0 , 1 , 1) , (0 , 1 , 0) , (0 , 0 , 1 ) , ( 0 , 0 , 0) } resp ectiv ely . Algorithm Points InEquidCom ps fir s t c ho oses (at r andom) a system supp orted on ( A 1 ∪ ∆ , A 2 ∪ ∆ , A 3 ∪ ∆), for example: g =      2 X 1 X 2 − 2 X 1 + X 2 − X 3 + 1 X 1 X 3 − X 1 + 2 X 2 + 2 X 3 + 2 X 2 X 3 + X 1 − 2 X 2 + X 3 − 1 and computes a geometric resolution R g of its isolated common ro ots in C 3 : R g =              u 5 − 9 2 u 4 − 17 u 3 + 80 u 2 − 2 u − 155 2 = 0 X 1 = − 9 100 u 4 + 7 40 u 3 + 351 200 u 2 − 543 200 u − 137 40 X 2 = − 1 200 u 4 − 1 80 u 3 − 1 400 u 2 + 233 400 u + 7 80 X 3 = − 1 10 u 4 + 3 20 u 3 + 7 4 u 2 − 51 20 u − 13 4 22 Then, in Step 3, th e algorithm tak es 2 linear form s: L 1 = X 1 + X 2 + 2 X 3 L 2 = X 1 + 2 X 2 In S tep 4, for k = 0 , 1 , 2, the isolated ro ots of the system h ( k ) obtained by adding to f generic linear com binations of L i , i = 0 , . . . , k , are computed: h (0) = f , h (1) =    f 1 ( X ) + L 1 ( X ) = X 1 X 2 + 2 X 3 + 1 f 2 ( X ) − L 1 ( X ) = X 1 X 3 − 2 X 1 − 3 X 3 − X 2 + 1 f 3 ( X ) + 2 L 1 ( X ) = X 2 X 3 + X 2 + 3 X 3 + 2 X 1 + 1 h (2) =    f 1 ( X ) + L 1 ( X ) + L 2 ( X ) = X 1 X 2 + X 1 + 2 X 2 + 2 X 3 + 1 f 2 ( X ) − L 1 ( X ) + 2 L 2 ( X ) = X 1 X 3 + 3 X 2 − 3 X 3 + 1 f 3 ( X ) + 2 L 1 ( X ) + L 2 ( X ) = X 2 X 3 + 3 X 2 + 3 X 3 + 3 X 1 + 1 In ord er to do this, the algorithm deforms th e geometric r esolution R g to geometric r eso- lutions R h ( k ) of the sets of isolated ro ots of h ( k ) : R h (0) =          u 3 − 7 u 2 + 2 u + 40 = 0 X 1 = 1 X 2 = − 1 14 u 2 + 9 14 u − 3 7 X 3 = 1 7 u 2 − 2 7 u − 1 7 R h (1) =            u 5 − 9 2 u 4 − 13 u 3 + 68 u 2 − 64 u = 0 X 1 = 57 200 u 4 − 369 400 u 3 − 953 200 u 2 + 647 50 u − 3 X 2 = − 269 600 u 4 + 1573 1200 u 3 + 1567 200 u 2 − 2599 150 u + 1 X 3 = 367 600 u 4 − 2039 1200 u 3 − 2181 200 u 2 + 3407 150 u + 1 R h (2) =            u 5 + 49 2 u 4 − 1549 9 u 3 + 538 9 u 2 + 679 6 u − 769 18 = 0 X 1 = − 101214 1803049 u 4 − 2537721 1803049 u 3 + 15987545 1803049 u 2 + 3986650 1803049 u − 7719426 1803049 X 2 = 58338 9015245 u 4 + 277533 1803049 u 3 − 11347628 9015245 u 2 + 5343077 9015245 u + 8036523 9015245 X 3 = 389394 9015245 u 4 + 1982655 1803049 u 3 − 57242469 9015245 u 2 − 21604159 9015245 u + 22524084 9015245 Finally , subroutine CleanGR remov es spur ious factors from R h ( k ) to obtain a geometric res- olution R ( k ) of a finite set contai ning a set of representati v e p oin ts of the equidimensional comp onen t of dimension k R (0) =          u 3 − 7 u 2 + 2 u + 40 = 0 X 1 = 1 X 2 = − 1 14 u 2 + 9 14 u − 3 7 X 3 = 1 7 u 2 − 2 7 u − 1 7 R (1) =          u 3 + 2 u 2 − 8 u = 0 X 1 = 1 2 u 2 + u − 3 X 2 = − 1 6 u 2 + 1 3 u + 1 X 3 = − 1 6 u 2 − 2 3 u + 1 R (2) = ∅ Since u 3 − 7 u 2 + 2 u + 40 = ( u + 2)( u − 4)( u − 5) and u 3 + 2 u 2 − 8 u = ( u + 4) u ( u − 2), substituting their r o ots in R (0) and R (1) resp ectiv ely , we get the follo wing sets of p oints in V ( f ): W 0 = { (1 , − 2 , 1) , (1 , 1 , 1) , (1 , 1 , 2) } W 1 = { ( − 3 , 1 , 1) , (1 , 1 , − 1) , ( 1 , − 3 , 1) } . 23 Note that W 1 con tains exactly one repr esentati v e p oint for eac h of the 3 lin es in V ( f ). Moreo v er, the fact that R (2) = ∅ imp lies that V ( f ) do es not hav e irr educible comp onen ts of dimension 2. Ho w ever, although there are n o isolate d p oints in V ( f ), W 0 ⊂ V ( f ) is not empt y . Remark 23 All the random c h oices of p oints made by our algorithms lead to correct computations pro vided these p oints do not annihilate certain p olynomials w hose d egrees can b e exp licitly b ound ed. These b oun ds dep end p olynomially on the degrees of affine v arieties asso ciated to the input systems, w hic h in turn, can b e estimate d in terms of mixed vol umes due to Theorem 16. Th e Scwhartz-Zipp el lemma allo ws us to con trol th e bit size of the constants to b e c h osen at random in order that the err or probab ility of the algorithms is less than a fi xed n um b er within the stated complexity b ound s. 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