A complex analogue of Todas Theorem

A COMPLEX ANALOGUE OF TOD A’S THEOR EM SAUG A T A BASU Abstract. T oda [28] pr o v ed in 1989 that the (discrete) polynomial time hi- erarch y , PH , is con tained in the class P # P , namely the class of languages that can be decide d by a T uring mac hine in polynomial time giv en access to an oracle with the p o wer to compute a function in the coun ting complexity class # P . This result, which illustrates the p ow er of counting is considered to be a seminal result in computat ional complexity theory . An analogous result (with a compactness hypothesis) in the complexit y theory ov er the reals (in the sense of Blum- Sh ub-Smale real m ac hines [5]) wa s pr o ve d in [2]. Unlike T oda’s proof i n the discr ete case, which relied on sophisticate d com binatorial argumen ts, the proof in [2] is top ological in nature in whic h the proper ties of the topological join is used i n a fundamental wa y . How ever, the constructions used in [2] were semi-algebraic – they used real inequalities in an essential w ay and as suc h do not extend to the complex case. In this paper, w e exte nd the tec hni ques developed i n [2] to the complex pro jective case. A key role is play ed b y the complex join of quasi-pro jectiv e complex v ar ieties. A s a consequence we obt ain a complex analogue of T o da’s theorem. The results con tained in this paper, taken toget her with those con tained in [ 2], i llustrate the central role of the Poincar ´ e polynomial in algorithmic algebraic geometry , as we ll as, in com- putational complexity theory ov er the complex and real num bers – namely , the ability to compute it efficiently enables one to decide in polynomial time all languages in the (compact) p olynomial hierarch y o ver the appropriate field. 1. Introduction and Main Resul ts 1.1. History and Bac kground. The primar y motiv ation for this pa per co mes from class ical (i.e. discrete) computatio nal complexity theory . In classical com- plexity theory , ther e is a seminal res ult due to T oda [2 8] linking the complexity of counting with that of deciding s e ntences with a fix e d n umber of quantifier alter na- tions. More precis ely , T oda’s theorem gives the following inclusio n (see Sec tio n 1.3.1 below or r efer to [2 1] for precise definitions of the complexity classes app earing in the theorem). Theorem 1 . 1 (T oda [28]) . PH ⊂ P # P . In other words, any languag e in the (discrete) p olyno mial hie r arch y can b e de- cided by a T uring machin e in p o lynomial time, given a ccess to an ora cle with the power to co mpute a function in # P . Date : Nov ember 9, 2018 . 1991 Mathematics Subje ct Classific ation. P r imary 14F25, 14Q20; Secon dary 68Q15. Key wor ds and phr ases. Polynomial hierarc h y , Betti num b ers, constructible sets, T o da’s theorem. The author was supp orted in part b y N SF grants CCF-0634907 and CCF-091595 4. C ommu- nicated by Peter B ¨ urgisser. 1 2 SAUGA T A BASU R emark 1.2 . The pro of of Theorem 1.1 in [28] is quite non-trivial. While it is obvious that t he c la sses P , NP , coNP are co n tained in P # P , the pro o f fo r the higher lev els of the polynomia l hierar ch y is quite in tric a te and procee ds in tw o steps: first pr oving that the PH ⊂ BP · ⊕ · P (using previous r esults of Sch¨ oning [23], and V alian t and V azirani [29]), and then s howing tha t BP · ⊕ · P ⊂ P # P . Aside from the obvious question a bo ut wha t should b e a pro p er analo g ue of the complexity class # P over the r eals o r complex num ber s, b ecause o f the presence of c o mplexity classes such as BP in the pro of, there seems to b e no direct wa y of extending suc h a pro of to r eal or c o mplex complexity clas ses in the sens e of Blum- Sh ub-Smale mo del of computation [5 , 2 4]. This is no t e n tirely sur pr ising, s ince complexity res ults in the Blum- Sh ub-Smale ov er different fields, while sup erficially similar, often requir e completely different pro of tec hniques. F or example, the fa ct that the p olynomial hiera rch y , PH is contained in the cla s s EXPTIME is obvious ov er finite fields, but is non- trivial to pr ov e over r eal closed o r algebra ically closed fields (where it is a consequence of efficient quantifier elimination algor ithms). The pro o f of the main theor em (Theorem 2.1) of this pap er, whic h ca n b e se en as a c omplex analogue o f Theorem 1.1, proceeds along completely differen t lines from the classica l (that is ov er finite fields) case, and is mainly top ological in natur e. In the late eighties Blum, Shub, a nd Smale [5, 24] in tr o duce d the notion o f T ur- ing machines ov er mor e general fields, thereby generaliz ing the clas s ical problems of computational complexity theor y s uch a s P vs. NP to corr esp onding pro blems over arbitrar y fields (such as the real, complex, p - adic num ber s etc.) If one considers languages accepted by a Blum-Shub-Smale machine ov er a finite field, one recov- ers the classical notions of discre te complexity theory . Over the last tw o dec a des there has been a lot o f resea rch activity towards proving real as well as co mplex analogues of well kno wn theorems in discre te complexity theo r y . The first steps in this directio n were taken by the a uthors Blum, Shub, a nd Smale (henceforth B-S-S) themselves, when they pr oved the NP C -completeness of the problem of deciding whether a systems of p olynomial equations has a solution (in affine space ) (this is the complex analo gue of Co ok-Lev in’s theorem that t he satisfia bilit y problem is NP -complete in the discrete ca se), and subsequently throug h the work o f s ev- eral resea rchers (Koiran, B ¨ urgisser, Cuck er, Meer to name a few) a well-established complexity theory ov er the r eals a s well a s complex n umbers hav e b een built up, which mirr ors closely the dis crete case. Indeed, one of the main attractions of the Blum-Shub-Smale co mputational mo del is that it pr ovides a fr a mework to prove complexity results ov e r more general structures than just finite fields, with the hop e that such results w ill help to un- rav el the algebro- geometric underpinnings o f the bas ic separation questions amongst complexity c lasses. It is also often interesting to inv estigate complex (as well as real) a nalogues of results in discre te co mplex it y theor y , b eca us e do ing so r eveals underlying geo metric and top olo gical phenomena not vis ible in the discr ete c a se. F rom this viewp oint it is quite natural to seek complex (as well as r eal) analog ues of T o da’s theor em; a nd as w e will see in this paper (see als o [2]), T o da’s theo rem prop- erly in terpreted ov er the real and complex num b ers gives an unexpec ted connection betw een tw o impo rtant but dis tinct s trands of a lgorithmic alg ebraic g eometry – namely , de cision pr oblems inv olving quantifier elimina tion on one hand, and the problems of c omputing top olo gic al invariants of constructible sets o n the other. In- deed, the o riginal result of T o da, tog ether with its r eal and complex c o un ter-par ts 3 seem to sug gest a deep er connection o f a mo del-theor e tic nature, betw een the prob- lems of efficie nt quantifi er-eliminatio n and efficient c o mputation of cer tain discrete inv ariants of definable sets in a structure, which mig ht b e an int eresting problem on its own to explore further in the future. 1.2. Recen t W ork. There has b een a large b o dy of recent resear ch on obtaining appropria te real (as well a s complex) a na logues of results in discrete co mplexity theory , especia lly thos e related to co un ting complexit y cla sses (see [20, 6, 8, 7]). In [2] a real analogue of T oda ’s theorem w as proved (with a compactness hypothes is ). In t his pap er w e prove a similar result in the complex case. Even though the basic approach is simila r in bo th cases , the top ologica l to ols in the complex case are different eno ug h to merit a sepa r ate tre atmen t. This is elab orated further in the next sectio n (the main difficulty in extending the real arg umen ts in [2] to the complex case is that we ca n no longer use inequa lities in our cons tructions). 1.3. Definitions of compl exit y classes. In o rder to formulate o ur result it is first necessary to define precisely complex counter-parts of the discr ete polynomia l time hierarch y PH a nd the discrete complexity c la ss # P , and this is what we do next. 1.3.1. Complex c oun ter-p arts of PH and # P . F or the rest of the paper C will denote an algebr aically closed fie ld of characteristic zero (ther e is no essential loss in assuming that C = C ) (indeed by a transfer argumen t it suffices to pr ov e all o ur results in this case). By a c omplex machine we will mean a mac hine in the sense of Blum-Shub-Smale [5]) ov er the g round field C. Notational c onvent ion. Since in what follows w e will b e forc ed to deal with mu ltiple blo cks of v aria bles in our for m ulas, we follo w a notationa l co nv ention b y which w e denote blo cks of v ar iables by b old letters with supe rscripts (e.g. X i denotes the i -th blo c k), and we use non-b o ld letters with subscr ipts to denote single v ar iables (e.g. X i j denotes the j -th v ar iable in the i -th blo ck). W e use x i to denote a sp ecific v alue of the blo ck o f v a riables X i . Definition 1. 3. W e will c a ll a qua n tifier-free firs t-order formula (in the languag e of fields), φ ( X 1 ; · · · ; X ω ), having several blo cks of v ariables ( X 1 , . . . , X ω ) to be multi-homo gene ous if ea ch p olynomial app earing in it is multi-homogeneous in the blo cks of v aria bles ( X 1 , . . . , X ω ) and such that φ is satisfied whenever a n y one of the blocks X i = 0. Recall that a polynomial P ∈ C[ X 1 ; · · · ; X ω ] is multi- homogeneous of multi-degree ( d 1 , . . . , d ω ) if and only if it sa tisfies the identit y P ( λ 1 X 1 ; · · · ; λ ω X ω ) = λ d 1 1 · · · λ d ω ω P ( X 1 ; · · · ; X ω ) . Clearly such a for mu la defines a c onstructible subset of P k 1 C × · · · × P k ω C where the blo ck X i is assumed to hav e k i + 1 v a riables. If ω = 1 , tha t is there is only one blo ck o f v a riables, then we call φ a homo gene ous formula. Notation 1 .4 (Rea liz a tion) . More genera lly , let Φ( X 1 ; . . . ; X σ ) def = (Q 1 Y 1 ) · · · (Q ω Y ω ) φ ( X 1 ; · · · ; X σ ; Y 1 ; · · · ; Y ω ) be a (quantified) multi-homogeneous formula, with Q i ∈ {∃ , ∀} , 1 ≤ i ≤ ω , φ a quantifier-free m ulti-homogeneo us formula, and X i (resp. Y j ) is a blo ck of k i + 1 4 SAUGA T A BASU (resp. ℓ j + 1) v ar iables. W e denote by R (Φ) ⊂ P k 1 C × · · · × P k σ C the constructible set which is the r e alizati on o f the formula Φ; i.e., R (Φ( X )) = { ( x 1 , . . . , x σ ) ∈ P k 1 C × · · · × P k σ C | (Q 1 y 1 ∈ P ℓ 1 C ) · · · (Q ω y ω ∈ P ℓ ω C ) φ ( x 1 ; · · · ; x σ ; y 1 ; · · · ; y ω ) } . Sometimes, in or de r to emphasize the blo ck structure in a multi-homogeneous for- m ula, we will write the quantifications as ( ∃ Y ∈ P ℓ C ) (resp. ( ∀ Y ∈ P ℓ C )) instead of just ( ∃ Y ) (resp. ( ∀ Y )). This is pur ely no ta tional and does not affect the syn tax of the formula. Notation 1.5 (Negation of a m ulti-homo g eneous formula) . It is clear that th e prop erty of multi-homogeneity is pres erved b y the Bo o le a n op eratio ns of conjunc- tion and dis junction. In order for it to b e preser v ed also under neg ation, we will adopt the con ven tio n that the negation, ¬ Φ( X 1 ; · · · ; X ω ), of a m ulti- ho mogeneous formula Φ( X 1 ; · · · ; X ω ) is by definition equal to ˜ Φ ∨ _ 1 ≤ i ≤ ω ( X i = 0 ) where ˜ Φ is the usua l negatio n of φ as a Bo o lean for m ula. It is clear that defined this w ay , ¬ Φ is mult i-homog e neous, and R ( ¬ Φ) = P k 1 C × · · · × P k ω C \ R (Φ) . W e say that tw o multi-homogeneous for m ulas, Φ and Ψ, are e quivalent if R (Φ) = R (Ψ). Clearly , equiv ale nt multi-homogeneous formulas must hav e identi- cal num b er of blo cks of free v ariable s , and the corr esp o nding blo ck sizes must also be equal. Since the no tio n of m ulti-homo g eneous formulas mig h t lo o k a bit un usual at first glance from the p oint of view of log ic , we illustrate b elow ho w to homogenize non-homogene o us formulas by co nsidering the following simple example (whic h is a building block for the “ rep eated squaring” tec hnique used to prov e doubly exp o- nent ial low er bounds for (real) qua n tifier elimination [10]). Example 1.6. Let Φ( X ) b e the following (existentially) qua n tified non-homoge neo us formula expr essing the fact, that X 4 = 1 . Φ( X ) def = ∃ Y ( Y 2 − 1 = 0) ∧ ( Y − X 2 = 0) . A multi-homogeneous v ersion of the same formula is given b y: Φ h ( X 0 : X 1 ) def = ∃ (( Y 0 : Y 1 ) ∈ P 1 C )( Y 2 1 − Y 2 0 = 0 ) ∧ ( X 2 0 Y 1 − X 2 1 Y 0 = 0) . Notice that the quantifier-free bi-homogeneous formula Ψ h ( X 0 : X 1 ; Y 0 : Y 1 ) def = ( Y 2 1 − Y 2 0 = 0 ) ∧ ( X 2 0 Y 1 − X 2 1 Y 0 = 0) defines a co nstructible subset of P 1 C × P 1 C , and that the a ffine part of the constr uctible subset of P 1 C defined by Φ h coincides with the constructible subset of C 1 defined by Φ( X ). 5 1.4. Complex analogue of PH. The definition o f the polyno mial hier arch y ov er C mirrors that o f the discrete case (see [27]) very closely . Definition 1. 7 (The class P C ) . A sequence ( T n ⊂ C n ) n> 0 of constructible subsets is said to b elong to the c lass P C if there exists a B-S- S machine M over C (see [5, 4]), such that for all x ∈ C n , the machine M decide s mem b ership of x in T n in time b ounded by a p olynomial in n . More genera lly , suppos e that k ( n ) is s ome fixed p olyno mial w hich is non- negative and increas ing . Let ( T n ⊂ C k ( n ) ) n> 0 be a s equence o f c o nstructible sets. W e will say that ( T n ⊂ C k ( n ) ) n> 0 belo ngs to P C if the sequence ( S n ⊂ C n ) n> 0 belo ngs to P C , where S n is defined by S k ( n ) = T k ( n ) , for all n > 0 , S m = ∅ , otherwise . Definition 1.8 (The clas ses Σ C ,ω and Π C ,ω ) . Let ω ≥ 0 b e a fixed integer. A sequence ( S n ⊂ C n ) n> 0 of constructible subsets is said to b e in the complexity cla ss Σ C ,ω , if for each n > 0, the constructible set S n is descr ibed by a first or der form ula (1.1) (Q 1 Y 1 ) · · · (Q ω Y ω ) φ n ( X 1 , . . . , X n , Y 1 , . . . , Y ω ) , with φ n a quantifier free formula in the fir st order theory of C, and for each i, 1 ≤ i ≤ ω , Y i = ( Y i 1 , . . . , Y i n ) is a blo ck of n v ariables, Q i ∈ { ∃ , ∀} , with Q j 6 = Q j +1 , 1 ≤ j < ω , Q 1 = ∃ , and the sequence   T n ⊂ C n × C n × · · · × C n | {z } ω times   n> 0 of constr uctible subsets defined by the q ua n tifier-free for m ulas ( φ n ) n> 0 belo ngs to the class P C . Similarly , the co mplexit y class Π C ,ω is defined as in Definition 1.8, with the difference that the a lternating quantifiers in (1.1) start with Q 1 = ∀ . R emark 1.9 . Notice that in Definition 1.8 there is no loss of generality in assuming that the sizes of the blo cks o f v ariables X , Y 1 , . . . , Y ω are all e q ual. T o b e mor e precise, supp ose tha t the size of blo ck X is k ( n ), and that of Y i is k i ( n ) fo r 1 ≤ i ≤ ω , where k ( n ) , k 1 ( n ) , . . . , k ω ( n ) are fixed no n- negative p olyno mia ls. Let  S n ⊂ C k ( n )  n> 0 be a sequence of co ns tructible subsets descr ibed by a first or der formu la (1.2) (Q 1 Y 1 ) · · · (Q ω Y ω ) φ n ( X , Y 1 , . . . , Y ω ) , with φ n a quantifier fre e fo rmula in the fir st o rder theory of C, and Q i ∈ {∃ , ∀} , with Q j 6 = Q j +1 , 1 ≤ j < ω , such that the sequenc e  T n ⊂ C k ( n ) × C k 1 ( n ) × · · · × C k ω ( n )  n> 0 6 SAUGA T A BASU of constr uctible subsets defined by the q ua n tifier-free for m ulas ( φ n ) n> 0 belo ngs to P C . Let ˜ k ( n ) b e any non-negative p olynomial whic h ma jorizes k ( n ) , k 1 ( n ) , . . . , k ω ( n ), and let ˜ X = ( X , X ′ ) , ˜ Y i = ( Y i , Y i ′ ) , 1 ≤ i ≤ ω , b e blocks o f v aria bles obtaine d from the blo cks X , Y i , of size ˜ k ( n ) b y padding b y an appr opriate n um b er of extra v ariables , X ′ , Y i ′ , respectively . By identifying the subs pa ce o f C ˜ k ( n ) defined by setting the v ar ia bles in the block X ′ to 0 , with C k ( n ) (and thus iden tifying S n with its image under the corres po nding inclusio n in C ˜ k ( n ) ), we hav e a sequence  S n ⊂ C ˜ k ( n )  n> 0 of constructible subsets describ ed by the formula (1.3) (Q 1 ˜ Y 1 ) · · · (Q ω ˜ Y ω ) φ n ( X , Y 1 , . . . , Y ω ) ∧ ( X ′ = 0) . It is cle a r that the sequence  T n ⊂ C k ( n ) × C k 1 ( n ) × · · · × C k ω ( n )  n> 0 belo ngs to the clas s P C if and o nly if the sequence  ˜ T n ⊂ C ˜ k ( n ) × C ˜ k ( n ) × · · · × C ˜ k ( n )  n> 0 defined by ˜ φ n ( ˜ X , ˜ Y 1 , . . . , ˜ Y ω ) := φ n ( X , Y 1 , . . . , Y ω ) ∧ ( X ′ = 0) . belo ngs to the class P C . In other words (up to padding by some additional v a r iables X ′ as ab ov e) there is no loss of g enerality in assuming that all the blo ck sizes ar e equal. Since, adding an additional blo ck o f quantifiers on the o utside (with new v ariables that do not appea r in the qua n tifier-free formula φ n ) do es not change the set defined by a q uantifi ed for m ula w e hav e the following inclusions: Σ C ,ω ⊂ Π C ,ω +1 , and Π C ,ω ⊂ Σ C ,ω +1 . Note that by the ab ov e definition the class Σ C , 0 = Π C , 0 is the class P C , the class Σ C , 1 = N P C and the cla ss Π C , 1 = co - NP C . Definition 1. 10 (Complex po lynomial hier arch y) . The complex p olynomia l time hierarch y is defined to b e the union PH C def = [ ω ≥ 0 ( Σ C ,ω ∪ Π C ,ω ) = [ ω ≥ 0 Σ C ,ω = [ ω ≥ 0 Π C ,ω . As in the r eal ca se s tudied in [2] for technical reas ons we need to r estrict to compact constructible sets. How ever, unlike in [2 ] where the compa c t lang uages consisted of closed semi-algebra ic subsets o f spheres, in this pap e r w e consider closed subsets of pro jective spa ces instead. This is a muc h mor e natural ch oice for defining compact complex complexity classes. W e now define the compact analog ue of PH C that w e will denote PH c C . Unlike in the non-co mpact case, we will assume all v a riables v ary over ce r tain compact sets (namely co mplex pro jective spaces o f v a rying dimensions). W e fir st need to b e pr ecise ab out what we mean by a complexity c lass of se- quences of constructible subsets of complex pro jective spaces. 7 Notation 1.11 (Affine cones) . F or any co ns tructible subset S ⊂ P k C we denote by C ( S ) ⊂ C k +1 the affine cone ov er S . More generally , if S ⊂ P k 1 C × · · · × P k ω C is a constructible subset, then C ( S ) ⊂ C k 1 +1 × · · · × C k ω +1 will denote the union o f L 1 × · · · × L ω such that each L i ⊂ C k i +1 is a line throug h the orig in, such that the po in t in P k 1 C × · · · × P k ω C represented by ( L 1 , . . . , L ω ) is in S . Definition 1. 12. W e say that a sequence    S n ⊂ P n C × · · · × P n C | {z } n times    n> 0 of co ns tructible subsets is in the c omplexity class P C , if the s equence o f a ffine cones   C ( S n ) ⊂ C n +1 × · · · × C n +1 | {z } n times   n> 0 belo ngs to the complex it y class P C . R emark 1.13 . The subspa c es spa nned by the increasing sequence of standard basis elements o f C = h e 0 i ⊂ C 2 = h e 0 , e 1 i ⊂ · · · ⊂ C n +1 = h e 0 , . . . , e n i ⊂ · · · after pro jectivization gives a flag P 0 C ⊂ P 1 C ⊂ · · · ⊂ P n C ⊂ · · · F or 0 ≤ m ≤ n , let ι m,n : P m C ֒ → P n C denote the c o rresp onding inclusion. Now, if ( S n ⊂ P n C ) n> 0 is a se q uence of constr uctible sets, we ca n after ident ifying P n C with the subspa c e P n C × ι 0 ,n ( P 0 C ) × · · · × ι 0 ,n ( P 0 C ) | {z } n − 1 times of P n C × · · · × P n C | {z } n times ident ify the s equence ( S n ⊂ P n C ) n> 0 with the sequenc e ( ˜ S n ⊂ P n C × · · · × P n C | {z } n times ) n> 0 . where ˜ S n = S n × ι 0 ,n ( P 0 C ) × · · · × ι 0 ,n ( P 0 C ) | {z } n − 1 times . W e will (b y abuse of la nguage) s ay that the sequence ( S n ⊂ P n C ) n> 0 belo ngs to the class P C if the sequence ( ˜ S n ⊂ P n C × · · · × P n C | {z } n times ) n> 0 . belo ngs to class P C . More g enerally , supp ose that m ( n ) is a non-neg ative p olynomial in n a nd, ( k i ( n )) i> 0 a sequence of non-neg ative p oly no mials such that there exists a p o ly- nomial k ( n ) which ma jorize s m ( n ) , k 1 ( n ) , k 2 ( n ) , . . . , k m ( n ) ( n ) for all n > 0. F or example, we could hav e m ( n ) = n , a nd k i ( n ) = in . Clea rly , in this case the p oly- nomial k ( n ) = n 2 + 1 ma jorizes m ( n ) , k 1 ( n ) , k 2 ( n ) , . . . , k m ( n ) ( n ) are for all n > 0. 8 SAUGA T A BASU W e say that a sequence  S n ⊂ P k 1 ( n ) C × · · · × P k m ( n ) ( n ) C  n> 0 is in P C , if the sequence    T n ⊂ P n C × · · · × P n C | {z } n times    n> 0 is in P C , where T k ( n ) = ˜ S n for all n > 0 , T n = ∅ , otherwise and ˜ S n ⊂ P k ( n ) C × · · · × P k ( n ) C | {z } k ( n ) times is defined by ˜ S n = ι k 1 ( n ) ,k ( n ) × · · · × ι k m ( n ) ,k ( n ) ( S n ) × ι 0 ,k ( n ) ( P 0 C ) × · · · × ι 0 ,k ( n ) ( P 0 C ) | {z } k ( n ) − m ( n ) times . Definition 1. 14 (Compa ct pro jective version of Σ C ,ω ) . W e say that a sequence    S n ⊂ P n C × · · · × P n C | {z } n times    n> 0 of constructible s ubsets is in the co mplexit y c la ss Σ c C ,ω , if for each n > 0, S n is describ ed by a first or der form ula (Q 1 Y 1 ∈ P n C ) · · · (Q ω Y ω ∈ P n C ) φ n ( X 1 ; · · · ; X n ; Y 1 ; · · · ; Y ω ) , with φ n a qua n tifier-free fir s t o rder multi-homogeneous formula defining a close d (in the Zar iski topo lo gy) subset of P n C × · · · × P n C | {z } n times × P n C × · · · × P n C | {z } ω times , Q i ∈ { ∃ , ∀} , Q 1 = ∃ , and the sequence of constructible sets ( T n ) n> 0 defined b y the formulas ( φ n ) n> 0 belo ngs to the clas s P C . R emark 1.15 . As rema r ked b efore (cf. Remar k 1.9), it is not essential to hav e all the blo ck sizes to b e equal in the a bove definition a s lo ng as all the nu mber and the sizes of the blo cks are poly nomially b ounded, and we will by a slight abuse of lang uage allow p olynomia lly b ounded num b er o f blo cks w ith poly nomially b ounded, but not necessarily equal, blo ck sizes in what follows without further re ma rk. Example 1.16. W e give a very natural exa mple o f a langua ge in Σ c C , 1 (i.e. the compact version of NP C ). Let k ( n, d ) =  n + d d  and identify P k ( n,d ) − 1 C × · · · × P k ( n,d ) − 1 C | {z } n +1 times 9 with systems o f n + 1 homog eneous p olynomia ls in n + 1 v a riables of deg ree d . Let S n,d ⊂ P k ( n,d ) − 1 C × · · · × P k ( n,d ) − 1 C | {z } n +1 times be defined by S n,d = { ( P 1 ; · · · ; P n +1 ) | P i ∈ P k ( n,d ) − 1 C and ∃ x = ( x 0 : · · · : x n ) ∈ P n C with P 1 ( x ) = · · · = P n +1 ( x ) = 0 } . In other words, S n,d is the set o f systems of ( n + 1) homogeneo us polynomial equations of deg ree d , which have a z e r o in P n C . Then it is c le a r from the definition of the class Σ c C , 1 that for any fixed d > 0,    S n,d ⊂ P k ( n,d ) − 1 C × · · · × P k ( n,d ) − 1 C | {z } n +1 times    n> 0 ∈ Σ c C , 1 . Note that it is not known if for any fixed d    S n,d ⊂ P k ( n,d ) − 1 C × · · · × P k ( n,d ) − 1 C | {z } n +1 times    n> 0 is NP C -complete, while the no n- compact version of this language i.e. the lang uage consisting of systems of polynomia ls ha ving a zero in C n (instead of P n C ), has b e en shown to be NP C -complete for d ≥ 2 [4]. W e define a nalogously the class Π c C ,ω , and finally define: Definition 1 .17. The c omp act pr oje ctive p oly nomial hier ar chy o ver C is defined to b e the unio n PH c C def = [ ω ≥ 0 ( Σ c C ,ω ∪ Π c C ,ω ) = [ ω ≥ 0 Σ c C ,ω = [ ω ≥ 0 Π c C ,ω . Notice that the constructible subsets belo nging to a n y languag e in PH c C are all compact (in fact Zariski closed subsets o f complex pro jective spa ces). R emark 1.1 8 . The compa ct clas ses in tro duced ab ov e might b e of interest in their own r ight. As remarked earlier it is not known whether the co mpa ct language    S n,d ⊂ P k ( n,d ) − 1 C × · · · × P k ( n,d ) − 1 C | {z } n +1 times    n> 0 in Ex ample 1.16 is NP C -complete. It is impor tant to resolve this question in order to understand whether the hardnes s of so lv ing po ly nomial systems o ver C is due to the no n-compactness of the (affine) solution space, or due to some intrinsic algebra ic reasons . 10 SAUGA T A BASU 1.4.1. Complex pr oje ctive analo gue of # P . W e now define the complex ana logue of # P (cf. the class # P † R defined in [2 ] in the real ca se). W e first need a notation. Notation 1.19 (Poincar´ e po lynomial) . In ca se C = C , for any constr uctible subset S ⊂ P k C we deno te by b i ( S ) the i -th Betti num b er (that is the rank of the singular homology gro up H i ( S ) def = H i ( S, Q )) o f S . W e also let P S ∈ Z [ T ] denote the Poinc ar´ e p olynomial of S , namely (1.4) P S ( T ) def = X i ≥ 0 b i ( S ) T i . R emark 1.20 . Since we are only g oing to b e co ncerned with the Betti num b ers o f constructible sets, w e do not lose an y information b y co nsidering homology groups with co efficients in Q rather than in Z , noting that H i ( S, Q ) = H i ( S, Z ) ⊗ Z Q . Note also that in this case by the universal co efficient theor em for cohomo lo gy [2 6], we have that the cohomolog y g roups H i ( S ) def = H i ( S, Q ) ∼ = Hom(H i ( S, Q ) , Q ) . R emark 1.21 . Ov er an arbitrary a lgebraically closed field C of c harac ter istic 0 , or- dinary s ingular homolo gy is not well defined. W e use a mo dified homolog y theory (whic h ag r ees with singular homolog y in case C = C and which is homotop y inv ari- ant) as done in [1] in ca se of semi-alg e braic sets ov er arbitrar y real closed fields (see [1], page 2 79). No te that by taking r eal a nd imaginar y parts, every constructible set over C is a semi-algebra ic set o ver an appr opriate r eal c lo sed subfield – namely , a maximal rea l subfield of C. F or the rest of the pap er we will assume C = C , noting that all the results g e n- eralize to a rbitrary algebra ically closed fields of characteristic 0 using the transfer principle. Definition 1. 22 (The class # P † C ) . W e say a sequence of co ns tructible functions ( f n : P n C → Z [ T ]) n> 0 is in the cla ss # P † C , if there ex is ts a sequence    S n ⊂ P n C × P n C × · · · × P n C | {z } n times    n> 0 ∈ P C , such that f n ( x ) = P S n , x for each x ∈ P n C , where S n, x = S n ∩ π − 1 n ( x ) and π n : P n C × P n C × · · · × P n C | {z } n times → P n C is the pro jection alo ng the last co-or dinates. 11 R emark 1.2 3 . W e make a few remarks a b out the class # P † C defined ab ov e. First of all notice that the class # P † C is quite robust. F or instance, given tw o sequences ( f n ) n> 0 , ( g n ) n> 0 ∈ # P † C it follows (by taking disjoint union of the co rresp onding constructible sets) that ( f n + g n ) n> 0 ∈ # P † C , and also ( f n g n ) n> 0 ∈ # P † C (b y taking Cartesian product of the corres po nding constructible sets and us ing the m ultiplicative pr op erty of the Poincar´ e p olynomials, which itself is a co ns equence of the Kunneth for m ula in homo logy theory [26].) R emark 1.24 . The connection b e t ween coun ting p oints o f v a rieties and their Betti nu mbers is mo re dir ect over fields of p ositive characteristic via the zeta function. The zeta function of a v ariety defined over F p is the ex po nential g e ne r ating function of the sequence whose n -th ter m is the num b er of p o int s in the v ar iety o ver F p n . T he zeta function o f such a v a riety turns out to be a rationa l function in o ne v ar iable (a deep theorem of algebraic g eometry first conjectured by Andre W eil [3 0] and prov ed by Dwork [13] and Deligne [11, 1 2]), and its numerator a nd denomina tor are pr o ducts of po lynomials whose deg rees a r e the Betti num ber s o f the v ariety with resp ect to a certain ( ℓ -adic) co- homology theo ry . The p oint o f this remark is that the problems of “ counting” v arieties and computing their Betti num b ers , are co nnected at a dee per level, and th us our choice of definition for a complex analogue of # P is not altogether ad ho c. R emark 1.2 5 . A different definition of the cla ss # P † C (more in line with prev io us work of B ¨ urgisser et a l. [8]) would be obtained by replacing in Definition 1.22 the Poincar´ e p olynomial, P S ( T ), by the Euler -Poincar´ e c haracter istic i.e. the v alue o f P S at T = − 1. The E uler-Poincar´ e characteristic is additive (at lea st when r e- stricted to complex v arieties ), and th us has some attributes of b eing a dis c rete ana- logue of volume. But a t the same time it should b e noted that the Euler-Poincar´ e characteristic is a rather weak inv aria n t – for insta nce, it do es not determine the nu mber of connected comp onents o f a given v ariety . Also notice that in the ca se of finite fields referred to in Rema r k 1 .24, all the Betti num b ers, not just t heir alternating sum, ent er (as degrees of factors) in the ra tional expre ssion for the zeta function of a v a riety . While it w ould certainly b e a m uch stronger re ductio n r esult if one could obtain a T o da-type theorem using only the Euler- Poincar ´ e characteristic instead of the whole P oincar´ e p olynomia l, it is at presen t unclear if such a theore m can b e proven (see also Sec tio n 5(C)). 2. St a tements of the main theorems W e can now state the ma in r esult of this pa p er . Theorem 2 . 1 (Complex a nalogue of T o da’s theo rem) . PH c C ⊂ P # P † C C . R emark 2.2 . Note that following the usua l conven tion P # P † C C denotes the class of languages accepted by a B-S-S machine ov er C in p olynomia l time with acces s to an oracle which can compute functions in # P † C . R emark 2.3 . W e lea ve it as an op en problem to prov e Theore m 2.1 with P H C instead of PH c C on the left ha nd side. How ever, we also note that many theorems of complex algebraic geometry tak e t heir most satisfacto ry for m in the case of complete v arieties , which is the setting considered in this pape r. 12 SAUGA T A BASU As a co nsequence o f our metho d, we obtain a reductio n (Theorem 2.6) that might be of indep enden t in terest. W e first define the following t wo problems: Definition 2.4 (Compact gener al decision pr oblem with at most ω qua n tifier al- ternations ( GDP c C ,ω )) . The input and output for this problem ar e as follows. • Input. A s e n tence Φ (Q 1 X 1 ∈ P k 1 C ) · · · (Q ω X ω ∈ P k ω C ) φ ( X 1 ; . . . ; X ω ) , where for e ach i, 1 ≤ i ≤ ω , Q i ∈ {∃ , ∀} , with Q j 6 = Q j +1 , 1 ≤ j < ω , and φ is a quantifier-free m ulti-homogene o us formula defining a close d subset S of P k 1 × · · · × P k ω . • Output. T r ue or F alse dep ending on whether Φ is true or false. Definition 2.5 (Computing the Poincar ´ e polyno mial of constructible sets ( Poinc ar´ e )) . The input and output for this proble m a re as follows. • Input. A quantifier-free homog eneous for m ula defining a cons tructible s ub- set S ⊂ P k C . • Output. The Poincar´ e polynomial P S ( T ). Theorem 2.6. F or every ω > 0 , t her e is a deterministic p olynomial time r e duction in t he Blum - Shub-Smale mo del of GDP c C ,ω to Poinc ar´ e . R emark 2 .7 . W e remark that (in contrast to the r eal cas e) in the co mplex case, we a re a ble to prov e a slig h tly str onger r esult than stated ab ov e in Theorem 2.6. Our pro of of Theor em 2.6 gives a p olynomia l time reduction of GDP c C ,ω to the problem of computing the pseudo-Poinc ar´ e polynomial (defined b elow, see Eq n. 3.2) of constructible sets. The pseudo - Poincar´ e p olynomia l is easily co mputable from the Poincar´ e p olynomial. 2.1. Outline of the main ideas and con tributions. The basic idea b ehind the pr o o f of a real analogue of T o da’s theo rem in [2] is a top o logical co ns truction, which g iven a semi-alg ebraic set X ⊂ R m × R n , p ≥ 0 , and pr 1 : R m × R n ⊂ R n the pro jection on R m constructs efficiently a semi-algebr aic s e t, D p ( X ), such that (2.1) b i (pr 1 ( X )) = b i ( D p ( X )) , 0 ≤ i < p. Moreov er, mem b ership in D p ( X ) can b e tested efficiently if the same is true for X . Note that this la st prop erty will not hold in general for the s e t pr 1 ( X ) itse lf (unless of course P R = NP R ). The topo logical constructio n used in the definition of D p ( X ) in [2] is the iterated fiber ed join, J p pr 1 ( X ), of a semi-algebr aic set X with itself over a pro jection map pr 1 . There is a lso an induced s urjective map J p pr 1 ( X ) → pr 1 ( X ) which we deno te by pr ( p ) 1 . The fib ers of this induced map pr ( p ) 1 : J p pr 1 ( X ) → pr 1 ( X ), ov er a p oint x ∈ pr 1 ( X ), ar e then or dinary ( p + 1)-fo ld jo ins of the fib er (pr ( p ) 1 ) − 1 ( x ), and by connectivity pr op e r ties of the join are p -connected. It is now p oss ible to pr ov e using a v ersion of the Vietoris -Beagle theorem that the map pr ( p ) 1 is a p -equiv a lence (see [2] for the precise definition of p - equiv alence). The main cons truction in [2] w as to realize efficiently the fib ered jo in J p pr 1 ( X ) up to homo top y by a semi-algebr aic s et. This construction how ever is semi-a lgebraic in na tur e, i.e. it uses re a l ineq ualities in an essential wa y and thus do es not generaliz e in a straig h tforward wa y to the complex case. Thus, a differen t constr uction is needed in the complex case. 13 In the complex cas e, the role of the fib ered join is play ed by the c omplex join fib er e d over a pr oje ct ion pr 1 : C m × C n → C m defined b e low (see Definition 3.21). The fib ers of the ( p + 1)-fold complex join fiber ed over a pro jection pr 1 , J p C ( X ), of a compa c t c onstructible set X a re no t quite p -co nnected as in the r eal case, but are reasonably nice – namely they are homologically equiv a len t to a pro jective space o f dimension p (s e e P rop osition 3 .16). This allows us to relate the Poincar´ e po lynomial of X with that of its image pr 1 ( X ), even though the relation is not as straightforward as in the r eal case (see Theor em 3.23 be low). W e remar k that Theo rem 3.23 can b e used to express dir e c tly the Betti num b ers of the image under pro jection of a pro jective v a riety in terms of those another pro jective v ariety o btained direc tly without having to p erform effective quantifier elimination (which has e xpo nen tial complexity). The des cription of this s e cond v ariety is much simpler and algebr aic in nature co mpared to the one used in [2] in the real semi-alg e braic ca se, and th us might b e of indep endent interest. Theorem 3.23 can also b e viewed as an impr ov ement over the descent spectral sequence argument used in [14] to b ound the Betti num b er s of pro jections (of semi-algebr aic sets) in the co mplex pro jective case. A similar construction using the pr o jective join is also av ailable in the real case (using Z / 2 Z coe fficie nts) but we omit its description in the current paper . Finally , we b elieve that the compact pro jective versions of the complex complex- it y classes intro duced in this paper deserve further in vestigations on their own (se e Remark 1.1 8 b elow), since many num erical algorithms fo r co mputing solutions of complex po lynomial systems as s ume so me form of compa ctness (see, for instance, [25, 3, 9]). The rest of the pap er is o rganized as follows. In Section 3 we state and prove the necessary ingre die nts from alg ebraic top ology needed to pr ove the main theor ems. In Section 4 we prov e the main results of the pap er. Finally , in Section 5, w e p ose some op en problems and discuss p ossible extensio ns to the current w ork. 3. Topological Ingredients In this section w e state and prov e the main top olo gical ingr edient s necess ary for the pro of of the ma in theorems. 3.1. Alexander-Lefsc hetz duali t y . W e will need the cla ssical Alexander-Lefschetz duality theor e m in order to r elate the Be tti num b ers o f a closed constr uctible subset S ⊂ P k 1 C × · · · × P k ℓ C with those of its complement P k 1 C × · · · × P k ℓ C \ S . Theorem 3. 1 (Alexander-Lefschetz duality) . L et S ⊂ P k 1 C × · · · × P k ℓ C b e a close d c onstruct ible subset. Then for e ach o dd i , 1 ≤ i ≤ 2 k + 1 with k = k 1 + · · · + k ℓ , we have that (3.1) b i − 1 ( S ) − b i − 2 ( S ) = b 2 k − i ( P k 1 C ×· · ·× P k ℓ C − S ) − b 2 k − i +1 ( P k 1 C ×· · ·× P k ℓ C − S )+ b i − 1 ( P k 1 C ×· · ·× P k ℓ C ) . Pr o of. Lefshetz duality theorem [26] gives for each i , 0 ≤ i ≤ 2 k , b i ( P k 1 C × · · · × P k ℓ C − S ) = b 2 k − i ( P k 1 C × · · · × P k ℓ C , S ) . The theorem now follo ws from the lo ng exact sequence of homology , · · · → H i ( S ) → H i ( P k 1 C × · · · × P k ℓ C ) → H i ( P k 1 C × · · · × P k ℓ C , S ) → H i − 1 ( S ) → · · · after noting that H i ( P k 1 C × · · · × P k ℓ C ) = 0 , for all i 6 = 0 , 2 , 4 , . . . , 2 k .  14 SAUGA T A BASU F or technical reas ons (see Coro llary 3.4 b elow) w e need to consider the ev en and o dd parts of the Poincar ´ e polynomial o f constructible sets. Given P = P i ≥ 0 a i T i ∈ Z [ T ], we write P def = P even ( T 2 ) + T P od d ( T 2 ) , where P even ( T ) = X i ≥ 0 a 2 i T i , and P od d ( T ) = X i ≥ 0 a 2 i +1 T i . W e introduce for any S ⊂ P n C , a r elated p olyno mial, Q S ( T ), which we call the pseudo-Poinc ar´ e p oly nomial o f S defined as follows. (3.2) Q S ( T ) def = X j ≥ 0 ( b 2 j ( S ) − b 2 j − 1 ( S )) T j . In other words, (3.3) Q S = P even S − T P od d S . W e intro duce b elow notation for se veral op erato rs on p olynomials tha t we will use later. Notation 3.2 (Op erato rs on po ly nomials) . F or any polynomia l Q = P i ≥ 0 a i T i ∈ Z [ T ] with deg( Q ) ≤ n , we will denote by: (A) Rec n ( Q ) the p olynomial T n Q ( 1 T ); (B) for 0 ≤ m ≤ n , T runc m ( Q ) the p olynomial P 0 ≤ i ≤ m a i T i ∈ Z [ T ]; and, (C) M P ( Q ) the p olyno mia l P Q , for an y p olynomial P ∈ Z [ T ]. R emark 3.3 . Notice that all the o per ators intro duced above a re computable in po lynomial time. Using the no ta tion in tro duced ab ov e w e h av e the following easy corolla ry of Theorem 3.1. Corollary 3.4. L et S ⊂ P k 1 C × · · · × P k ℓ C b e any close d (r esp. op en) c onstructible subset, and k = k 1 + · · · + k ℓ . Then, Q S = Q P k 1 C ×···× P k ℓ C − Rec k ( Q P k 1 C ×···× P k ℓ C − S ) . 3.2. The complex join o f subsets of comp l ex pr o jectiv e spaces. W e fir st give a purely geo metric definition o f the c omplex join o f tw o sets follow e d by one using co-o rdinates. The geometr ic definition is useful in understanding the top o- logical pr op erties of the join proved la ter. The definition involving co-ordinates and formulas is nec e ssary for the co mplexit y theoretic arguments. Let V , W b e finite dimensional v ector s paces ov er C and let X ⊂ P ( V ) and Y ⊂ P ( W ) b e t wo arbitr a ry (not necessarily constructible) subs e ts. Note that P ( V ) ∼ = P ( V ⊕ 0 ) ⊂ P ( V ⊕ W ) and P ( W ) ∼ = P ( 0 ⊕ W ) ⊂ P ( V ⊕ W ) a re tw o disjoint subspaces o f P ( V ⊕ W ) a nd thus X a nd Y a re embedded as disjoint subsets of P ( V ⊕ W ). With the ab ov e notation 15 Definition 3.5 (Geometric definition of complex join) . The c omplex j oin , J C ( X, Y ) , is defined to be the union of pr o jective lines in P ( V ⊕ W ) which meets both X and Y if X and Y are both non- empt y . W e let J C ( X, Y ) = Y if X is emp ty , and J C ( X, Y ) = X if Y is empty . W e now give a definition of the complex join which inv olve co -ordinates which we a re going to use in this pap er. Let X ⊂ P k C and Y ⊂ P ℓ C be t wo cons tr uctible sets defined by homog eneous formulas Φ( X 0 , . . . , X k ) and Ψ( Y 0 , . . . , Y ℓ ) r esp ectively , where ( X 0 : · · · : X k ) (re- sp ectively ( Y 0 : · · · : Y ℓ )) are homo geneous co-ordina tes in P k C (resp ectively P ℓ C ). Definition 3.6 (Complex join in terms of co- ordinates) . The complex join, J C ( X, Y ), of X and Y is the c onstructible subset o f P k + ℓ +1 C defined by the formula J C (Φ , Ψ) def = φ ( Z 0 , · · · , Z k ) ∧ ψ ( Z k +1 , · · · , Z k + ℓ +1 ) , where ( Z 0 : · · · : Z k + ℓ +1 ) are homo geneous co ordinates in P k + ℓ +1 C . R emark 3.7 . Firstly , notice that the r ealization, R ( J C (Φ , Ψ)), do es not de p end on the for m ulas φ a nd ψ used to define X a nd Y resp ectively . Also, notice that if X and Y are b oth empty then so is J C ( X, Y ). Indeed, if X = ∅ (resp ectively , Y = ∅ ) then J C ( X, Y ) is iso morphic to Y (respe ctiv ely , X ). T o see this notice that by definition (cf. Definition 1.3) the homogeneous formula Φ( X ) (resp. Ψ( Y )) is tr ue whenever X (resp. Y ) is the 0 -vector. Now consider the following t wo co ns tructible subsets of P k + ℓ +1 C . ˜ X = { ( x : 0 : · · · : 0) | x ∈ X } , ˜ Y = { (0 : · · · : 0 : y ) | y ∈ Y } . W e hav e tha t ˜ X (resp. ˜ Y ) is isomo r phic to X (resp. Y ). Mo reov er ˜ X and ˜ Y are contained in J C ( X, Y ), s inc e as remar ked ear lier Ψ( Y ) (resp. Φ( X )) is true whenever Y (resp. X ) is the 0-vector. Mo r eov er, ˜ X (resp. ˜ Y ) is equal to J C ( X, Y ) in case Y (r esp. X ) is empty . Also, clea rly ˜ X and ˜ Y are disjoint, and if X and Y are b oth non-empty then, J C ( X, Y ) is o btained by taking the union of pro jective lines in P k + ℓ +1 C meeting b oth ˜ X and ˜ Y . Example 3.8. It is easy to chec k from the ab ov e definition that the join, J C ( P k C , P ℓ C ), of t wo pro jectiv e spaces is again a pr o jective space, namely P k + ℓ +1 C . R emark 3.9 . The pro jective join as defined a b ove is a classical o b ject in a lg ebraic geometry . Amo ng st many other applications, the complex susp ension of a pro jective v ariety X (i.e. the complex join J C ( X, P 0 C )) plays an imp orta n t role in defining Lawson homo logy of pro jective v arieties [17]. Within the a rea o f computational complexity theory , the pro jective join of a v ar iet y with a p oint w a s use d in [22] for proving har dnes s of the problem of computing Betti n umbers of complex v arieties . Definition 3.1 0. F or p ≥ 0, w e denote b y J p C ( X ) the ( p + 1)-fold itera ted complex join of X with itself. More precise ly , J 0 C ( X ) := X , J p +1 C ( X ) := J C ( J p C ( X ) , X ) , for p ≥ 1 . 16 SAUGA T A BASU If X ⊂ P k C is defined by a first-order homogene o us for m ula Φ( X 0 , . . . , X k ), then J p C ( X ) ⊂ P ( p +1)( k +1) − 1 C is defined by the homogene o us form ula J p C (Φ)( X 0 0 , . . . , X 0 k , . . . , X p 0 , . . . , X p k ) def = p ^ i =0 φ ( X i 0 , . . . , X i k ) . where ( X 0 0 : · · · : X p k ) are homo geneous co-or dina tes in P ( p +1)( k +1) − 1 C . Note that by Remark 3.7, if X is empt y then J p C ( X ) is empt y for every p ≥ 0 . 3.3. Prop erties of the top ological joi n. W e a lso nee d to intro duce the top o- lo gic al join of tw o spa c e s. The following is classical. Definition 3. 11. The join , X ∗ Y , of tw o top ologica l spaces X and Y is defined by (3.4) X ∗ Y def = X × Y × ∆ 1 / ∼ , where ∆ 1 = { ( t 0 , t 1 ) | t 0 , t 1 ≥ 0 , t 0 + t 1 = 1 } denotes the standard geometric realization of the 1- dimensional simplex, and ( x, y , t 0 , t 1 ) ∼ ( x ′ , y ′ , t 0 , t 1 ) if and only if t 0 = 1 , x = x ′ or t 1 = 1 , y = y ′ . Int uitively , X ∗ Y is obta ine d by jo ining each p oint o f X with each po in t o f Y by a n in terv al. W e will nee d the well-kno wn fact that the itera ted join o f a top ologica l space is highly connected. In or der to make this statement prec is e w e fir s t define Definition 3 .12 ( p -equiv alence) . A map f : A → B b etw een tw o top ologica l spaces is called a p -e qu ivalenc e if the induce d homomorphism f ∗ : H i ( A ) → H i ( B ) is a n iso mo rphism for all 0 ≤ i < p , and an epimorphism for i = p , and w e say that A is p -e qu ivalent to B . The following is well k nown. (se e, for instance, [1 8, Prop ositio n 4.4.3 ]). Theorem 3.1 3. L et X b e a non-empty c omp act semi-algebr aic set. Then, the ( p + 1) -fold join X ∗ · · · ∗ X | {z } ( p +1) times is p -e quivalent to a p oint. W e will need a particular pro per t y of pro jection maps that w e ar e going to consider later in the pap er. Notation 3 .14. F or any constructible s et A , we denote by K( A ) the collection of all compact (in the Euclidean top ology) subse ts of A . Definition 3.15. Let f : A → B be a map b etw ee n tw o constructible sets A and B . W e say that f c omp act c overing if for any L ∈ K( f ( A )), there exists K ∈ K( A ) such that f ( K ) = L. 17 3.4. T op ological prop erties of the complex join. Prop ositio n 3.1 6 . L et X ⊂ P k C b e a non- empty semi-algebr aic subset and p > 0 . L et ι : J p C ( X ) ֒ → P ( p +1)( k +1) − 1 C denote the inclusion map. Then t he induc e d homomorphisms ι ∗ : H j ( J p C ( X )) → H j ( P ( p +1)( k +1) − 1 C ) ι ∗ : H j ( P ( p +1)( k +1) − 1 C ) → H j ( J p C ( X )) ar e isomorphisms for 0 ≤ j < p . Before proving Prop osition 3.16 we first fix some notation. Notation 3.1 7 (Hopf fibra tion) . F or any k ≥ 0 , w e will deno te by π : C k +1 \ { 0 } → P k C the tautolog ic al line bundle ov er P k C , and by ˜ π : S 2 k +1 → P k C , the Hopf fibr ation , namely the r estriction of π to the unit sphere in C k +1 defined by the equation | z 1 | 2 + · · · + | z k +1 | 2 = 1. Finally for a ny subset S ⊂ P k C , we will denote by e S the subset ˜ π − 1 ( S ) ⊂ S 2 k +1 . Restr icting the map ˜ π to e S we obtain the restriction of the Hopf fibration to the base S i.e. we hav e the following comm utative diagram. e S   ι / / ˜ π   S 2 k +1 ˜ π   S   ι / / P k C W e need the following lemma . Lemma 3.18. L et X ⊂ P k C , Y ⊂ P ℓ C b e semi-algebr aic s ubsets. Then ^ J C ( X, Y ) ⊂ S 2( k + ℓ )+3 is home omorphic to t he (top olo gic al) join e X ∗ e Y . Pr o of. Consider x ∈ X and y ∈ Y and the pro jective line L ⊂ J C ( X, Y ) joining x and y . It is easy to s e e that the pr eimage ˜ L = ˜ π − 1 ( L ) ∼ = S 3 is a topolog ic al join o f ˜ π − 1 ( x ) and ˜ π − 1 ( y ) (each homeomorphic to S 1 ). N ow since ˜ X (resp. ˜ Y ) is fiber ed by the v a rious ˜ π − 1 ( x ) (resp. ˜ π − 1 ( y )), it follo ws that ^ J C ( X, Y ) is homeo mo rphic to e X ∗ e Y .  Pr o of of Pr op osition 3.16. W e first tr e at the cases p = 1 , 2. p = 1 : It is an easy exe rcise to show that the join, J 1 C ( X ) = J C ( X, X ) is no n- empt y and connected whenever X is non-empty . This proves the prop ositio n in this case. p = 2 : It is easy to see that J 2 C ( X ) = J C ( J 1 C ( X ) , X ) is non-empty and connected, whenever X is non-empty . It is o nly a slig h tly more difficult exercise to prov e that H 1 ( J 2 C ( X )) (and in fact, H 1 ( J p C ( X )) for all p > 1 ) v a nishes. This follows fr o m the statement that H 1 ( J C ( Y , Z )) = 0 whenever Y is connected, since J p C ( X ) = J C ( J p − 1 C ( X ) , X ) a nd we hav e that J p − 1 C ( X ) is co nnected for p > 1. Proving that H 1 ( J C ( Y , Z )) = 0 whenev er Y is connected, after an applicatio n o f May er -Vietoris exa ct se q uence, r educes to proving that 18 SAUGA T A BASU H 1 ( J C ( Y , Z )) = 0 whenever b oth Y and Z ar e connected. This can b e chec ked by a direct calculation using the fact that the top ologica l join Y ∗ Z is s imply connected whenever Y , Z are b oth connected. Note tha t, this also prov es that J p C ( X ) is simply connected for a ll p > 1. Now let p ≥ 2. It follows from rep eated applications of Lemma 3.18 that ^ J p C ( X ) is homeomorphic to e X ∗ · · · ∗ e X | {z } ( p +1) times . W e also hav e the commutativ e square ^ J p C ( X )   i / / ˜ π   S 2( p +1)( k +1) − 1 ˜ π   J p C ( X )   i / / P ( p +1)( k +1) − 1 C and a corr espo nding square H ∗ ( ^ J p C ( X )) ι ∗ / / ˜ π ∗   H ∗ ( S 2( p +1)( k +1) − 1 ) ˜ π ∗   H ∗ ( J p C ( X )) ι ∗ / / H ∗ ( P ( p +1)( k +1) − 1 C ) of induced homomor phisms in the homolog y groups. It follows from Theorem 3.13 that if X 6 = ∅ , then H 0 ( ^ J p C ( X )) ∼ = Q , H i ( ^ J p C ( X )) ∼ = 0 , 0 < i < p . Since, for p > 1 , J p C ( X ) is simply connected (see ab ov e) ^ J p C ( X ) is a simple S 1 -bundle (i.e. a S 1 -bundle with a simply connected ba se) ov er J p C ( X ). It now follows by a standard arg umen t (whic h we expand b elow) inv olving the sp ectral sequence of the bundle ˜ π : ^ J p C ( X ) → J p C ( X ), that for 0 ≤ i < p , H i ( J p C ( X )) ∼ = Q , for i even , (3.5) H i ( J p C ( X )) ∼ = 0 for i o dd . (The abov e claim also follows from the Gysin se q uence of the S 1 -bundle ˜ π : ^ J p C ( X ) → J p C ( X ) but we g ive an indep endent pro of b elow). Consider the E 2 -term of the (homological) sp ectral sequence o f the bundle ˜ π : ^ J p C ( X ) → J p C ( X ) . F or i, j ≥ 0, we ha ve that E 2 i,j = H i ( J p C ( X )) ⊗ H j ( S 1 ) . 19 F rom this we deduce that E 2 i, 0 = E 2 i, 1 = H i ( J p C ( X )) . Also, from the fact that H 0 ( ^ J p C ( X )) = Q , we g et that E 2 0 , 0 = Q , and hence, E 2 0 , 1 = Q as well. Moreover, we ha ve that E 3 i,j = E 4 i,j = · · · = E ∞ i,j for a ll i ≥ 0 and j = 0 , 1. Now fr o m the fact that the sp ectral sequence E r conv e r ges to the homolog y of ^ J p C ( X ) w e deduce that E 3 i,j = 0 for 0 ≤ i ≤ p − 1 and all j, E 3 0 , 0 = Q . This implies that the differential d 2 : E 2 i, 0 → E 2 i − 2 , 1 is an isomorphis m for 1 ≤ i ≤ p − 1. T ogether with the fact that E 2 i, 0 = E 2 i, 1 = H i ( J p C ( X )) , this immediately implies (3.5). The claim that ι ∗ is a n iso morphism follows directly from the ab ov e. The dual statement abo ut ι ∗ follows immedia tely from the universal co efficient theor em for cohomolo gy (see e.g. [2 6]).  In our application we will need the follo wing (rather tec hnica l) gener alization of Prop osition 3.16. Let p, α 0 , . . . , α ω ≥ 0, N = Q 0 ≤ j ≤ ω ( α j + 1). Let I deno te the set of tuples ( i 0 , . . . , i ω ) with 0 ≤ i j ≤ α j , 0 ≤ j ≤ ω , and for ea c h tuple ( i 0 , . . . , i ω ) ∈ I , let π ( i 0 ,...,i ω ) denote the pro jection × ( j 0 ,...,j ω ) ∈ I P k C − → P k C defined by ( x ( j 0 ,...,j ω ) ) ( j 0 ,...,j ω ) ∈ I 7→ x ( i 0 ,...,i ω ) , and for any subset X ⊂ P k C we denote X ( i 0 ,...,i ω ) = π − 1 ( i 0 ,...,i ω ) ( X ) . Prop ositio n 3.19. L et X ⊂ P k C b e a semi-algebr aic subset. Also, let fo r e ach i, 0 ≤ i ≤ ω , Λ i ∈ { T , S } , and let S ⊂ × ( j 0 ,...,j ω ) ∈ I P k C denote the semi-algebr aic subset S def = Λ 0 0 ≤ i 0 ≤ α 0 · · · Λ ω 0 ≤ i ω ≤ α ω ( J p C ( X )) ( i 0 ,...,i ω ) , with ι : S ֒ → × ( j 0 ,...,j ω ) ∈ I P ( p +1)( k +1) − 1 C 20 SAUGA T A BASU denoting the inclusion map. Then, the induc e d homomorphisms ι ∗ : H j ( S ) → H j ( × ( j 0 ,...,j ω ) ∈ I P ( p +1)( k +1) − 1 C ) ι ∗ : H j ( × ( j 0 ,...,j ω ) ∈ I P ( p +1)( k +1) − 1 C ) → H j ( S ) ar e isomorphisms for 0 ≤ j < p . Pr o of. Notice that, if ω = 0 and Λ 0 = \ , then \ 0 ≤ i 0 ≤ α 0 J p C ( X ) ( i 0 ) = × ( j 0 ,...,j ω ) ∈ I J p C ( X ) , and the claim fo llows in this case from Prop o sition 3 .16 and the Kunneth for m ula. If ω = 0 and Λ 0 = [ , the claim follows from the previous case and a s tandard argument using the May er-Vietoris double complex. The gener al case is easily proved using induction on ω .  3.5. Complex joi n fib ered ov er a pro jection and its prop erties. In our application we need the complex join fib ered ov er ce rtain pro jections. W e first give a geometric definition followed by one inv o lving co-ordinates. Let V , W be finite dimensiona l C-v ector spaces and A ⊂ P ( V ) × P ( W ) a subset. Let pr 1 : P ( V ) × P ( W ) → P ( V ) denote the pro jection on the first comp onent. Then, for p ≥ 0 , the p -fold c omplex joi n of A fib er e d over the pr oje ction pr 1 is defined by Definition 3.2 0 (Geometric definition of complex join fib ere d ov er a pro jection) . J p C , pr 1 ( A ) = { ( x , y ) | x ∈ P k C , y ∈ J p C ( A x ) } , were A x = pr − 1 1 ( x ) ∩ A . W e now give a definition in terms o f co-or dinates. Definition 3.21 (Complex join fibered o ver a pro jection in terms of co- ordinates) . Let A ⊂ P k C × P ℓ C be a constructible s et defined b y a first-o r der mult i-homog e neous formula, Φ( X 0 , . . . , X k ; Y 0 , . . . , Y ℓ ) and let pr 1 : P k C × P ℓ C → P k C be the pro jection map to the first comp onent. F or p ≥ 0, the p -fold complex join of A fib ered over the map pr 1 , J p C , pr 1 ( A ) ⊂ P k C × P ( ℓ +1)( p +1) − 1 C , is defined by the formula (3.6) J p C , pr 1 (Φ)( X 0 , . . . , X k ; Y 0 0 , . . . , Y 0 ℓ , . . . , Y p 0 , . . . , Y p ℓ ) def = p ^ i =0 φ ( X 0 , . . . , X k ; Y i 0 , . . . , Y i ℓ ) . R emark 3.22 . The pro jection map pr 1 : P k C × P ( ℓ +1)( p +1) − 1 C → P k C clearly restricts to a surjection pr ( p ) 1 : J p C , pr 1 ( A ) → pr 1 ( A ) sending ( x 0 : · · · : x k ; y 0 0 : · · · : y p ℓ ) ∈ J p C , pr 1 ( A ) to ( x 0 : · · · : x k ) ∈ pr 1 ( A ). 21 Now, let A ⊂ P k C × P ℓ C be a semi-algebr aic subset pr 1 : P k C × P ℓ C → P k C be the pro jection on the first co mpo nen t. Suppo se tha t pr 1 restricted to A is a compact covering. The following theor em relates the Poincar´ e p olyno mial of J p C , pr 1 ( A ) to that of the imag e pr 1 ( A ). Theorem 3 . 23. F or every p ≥ 0 , we have that P pr 1 ( A ) = (1 − T 2 ) P J p C , pr 1 ( A ) mo d T p . (3.7) R emark 3.24 . Note that the compact cov er ing prop erty is crucial for Theor em 3.2 3. to ho ld. In our applica tions, pr 1 is g oing to be either an op en or a closed map, and will th us auto ma tically hav e the compact cov ering proper t y . Pr o of. W e fir st as s ume that A is semi-alg e br aic a nd compac t, and let B deno te pr 1 ( A ) × P ( p +1)( ℓ +1) − 1 C . W e have the following comm utative square. J p C , pr 1 ( A )   i / / pr ( p ) 1   B pr 1   pr 1 ( A ) Id / / pr 1 ( A ) The dia gram a bove induces a morphism, φ i,j r : E i,j r → ′ E i,j r betw een the Ler ay- Serre spectr al sequences of the tw o vertical maps in the ab ov e diag ram. Here, E r (resp. ′ E r ) denotes the Ler ay-Serr e sp ectral seq uence of the map pr 1 : B → pr 1 ( A ) (resp. pr ( p ) 1 : J p C , pr 1 ( A ) → pr 1 ( A ) ). The sp ectral sequence, E r , deg enerates at the E 2 -term where E i,j 2 = H i (pr 1 ( A ) , R j pr 1 ∗ Q B ) , and Q B denotes the cons tan t sheaf with stalk Q o n B , and R ∗ pr 1 ∗ denotes the higher direct ima g e functor. (The a bove for m ulation of Ler ay-Serre sp ectral sequence of a map is standa rd; w e refer the r eader to [15, Th´ eor` eme 4.17 .1] for a purely sheaf theoretic statement without reference to hig her derived ima g es.) Similarly we hav e ′ E i,j 2 = H i (pr 1 ( A ) , R j pr ( p ) 1 ∗ Q J p C , pr 1 ( A ) ) . W e also hav e that for ea ch x ∈ pr 1 ( A ) ( R j pr 1 ∗ Q B ) x ∼ = H j ( P ( p +1)( ℓ +1) − 1 C ) ∼ = H j ((pr ( p ) 1 ) − 1 ( x )) ∼ = ( R j pr ( p ) 1 ∗ Q J p C , pr 1 ( A ) ) x , where the first and the last isomorphisms are consequences of the proper base change theorem (see [15, Remar que 4.17 .1]) noting that pr 1 , pr ( p ) 1 are b oth prop er maps, and the middle o ne is a cons equence of Prop ositio n 3.16. It follows that the sheav es R j pr 1 Q B and R j pr ( p ) 1 Q J p C , pr 1 ( A ) are isomorphic by the sheaf map induced by the inclusions (pr ( p ) 1 ) − 1 ( x ) ֒ → { x } × P ( p +1)( ℓ +1) − 1 C , x ∈ pr 1 ( A ) and hence, φ i,j 2 : E i,j 2 → ′ E i,j 2 are isomorphisms for i + j < p . 22 SAUGA T A BASU It n ow follows fr om a general result about spectral sequences (see [19, page. 66]) that E i,j ∞ ∼ = ′ E i,j ∞ for 0 ≤ i + j < p . This implies that H q ( J p C , pr 1 ( A )) ∼ = H q (pr 1 ( A ) × P ( p +1)( ℓ +1) − 1 C ) for 0 ≤ q < p , and thus (3.8) P J p C , pr 1 ( A ) = P pr 1 ( A ) × P ( p +1)( ℓ +1) − 1 C mo d T p . W e also hav e that P pr 1 ( A ) × P ( p +1)( ℓ +1) − 1 C = P pr 1 ( A ) × P P ( p +1)( ℓ +1) − 1 C (3.9) = P pr 1 ( A ) × (1 + T 2 + · · · + T 2(( p +1)( ℓ +1) − 1) ) = P pr 1 ( A ) × (1 − T 2 ) − 1 mo d T p . Equation (3.7) now follows from Equations (3.8) and (3.9). The genera l case follows by taking direc t limit over all compa ct subs e ts o f A . More precisely , for K 1 ⊂ K 2 compact subsets of A , we hav e for 0 ≤ q < p the following c omm utative s quare after switching to homolog y (cf. Remark 1.2 0). (Note that following Definition 3.2 0 the complex join fib ered ov er a pro jection is defined for arbitra ry not necessarily constructible subsets o f A .) H q ( J p C , pr 1 ( K 1 )) ι ∗ / / ∼ =   H q ( J p C , pr 1 ( K 2 )) ∼ =   H q (pr 1 ( K 1 ) × P ( p +1)( ℓ +1) − 1 C ) ι ∗ / / H q (pr 1 ( K 2 ) × P ( p +1)( ℓ +1) − 1 C ) where the vertical maps are iso morphisms by the previo us case. If we take the direct limit as K range s in K ( A ), we obtain the following: lim − → H q ( J p C , pr 1 ( K )) ∼ = / / ∼ =   H q ( J p C , pr 1 ( A ))   lim − → H q (pr 1 ( K ) × P ( p +1)( ℓ +1) − 1 C ) ∼ = / / H q (pr 1 ( A ) × P ( p +1)( ℓ +1) − 1 C ) The isomorphism o n the top level comes from the fact that homo logy and dire ct limit comm ute [26]. F or the bo tto m isomor phism, w e need the additional fact that since we assume that pr 1 is a compact covering we hav e lim − → { H q (pr 1 ( K ) × P ( p +1)( ℓ +1) − 1 C ) | K ∈ K( A ) } = lim − → { H q ( L × P ( p +1)( ℓ +1) − 1 C ) | L ∈ K(pr 1 ( A )) } . This prov es that the right v ertical arrow is also an is omorphism.  Using the same notation a s in Theor em 3 .23 a nd Eq n. (3.2) we have the following easy corolla r y o f Theorem 3.2 3. Corollary 3. 25. L et p = 2 m + 1 with m ≥ 0 . Then Q pr 1 ( A ) = (1 − T ) Q J p C , pr 1 ( A ) mo d T m +1 . (3.10) Pr o of. The cor ollary follows directly from Theo rem 3.23 and the fact tha t for any po lynomial P ∈ Z [ T ] w e hav e ((1 − T 2 ) P ) even = (1 − T )( P ) even , ((1 − T 2 ) P ) od d = (1 − T )( P ) od d . 23  As b efore we need a slightly more g eneral version of Theore m 3 .2 3 as well as Corollar y 3 .25. Let α 0 , . . . , α σ ≥ 0 , and N = Q 0 ≤ j ≤ ω ( α j + 1). Let φ be a homo geneous formula defining a co ns tructible subset o f P k 0 × · · · × P k σ C × P ℓ C . Also, let for ea ch i, 0 ≤ i ≤ σ , Λ i ∈ { W , V } , and let Φ denote the multi-homogeneous formula defined by Φ def = Λ 0 0 ≤ i 0 ≤ α 0 · · · Λ σ 0 ≤ i σ ≤ α σ φ ( X 0 ; · · · ; X σ ; Y i 0 ,...,i σ ) . Let A = R (Φ) ⊂ P k 0 × · · · × P k σ C × P ℓ C × · · · × P ℓ C | {z } N times , and let pr [0 ,σ ] : P k 0 C × · · · × P k σ C × P ℓ C × · · · × P ℓ C | {z } N times → P k 0 C × · · · × P k σ C be the pro jection onto the first σ + 1 comp onents, a nd supp ose that pr [0 ,σ ] restricted to A is a compact cov e ring. F or p ≥ 0, let J p C , pr [0 ,σ ] ( A ) ⊂ P k 0 × · · · × P k σ C × P ( ℓ +1)( p +1) − 1 C × · · · × P ( ℓ +1)( p +1) − 1 C | {z } N times be defined by the formula (3.11) J p C , pr [0 ,σ ] (Φ) def = Λ 0 0 ≤ i 0 ≤ α 0 · · · Λ ω 0 ≤ i σ ≤ α σ J p C , pr [0 ,σ ] φ ( X 0 ; · · · ; X σ ; Y i 0 ,...,i σ ) . Theorem 3 . 26. F or every p ≥ 0 , we have that P pr [0 ,σ ] ( A ) = (1 − T 2 ) N P J p C , pr [0 ,σ ] ( A ) mo d T p . (3.12) Pr o of. The pr o of is iden tical to that of Theo rem 3.23 ab ov e using Prop osition 3.1 9 instead of Pr op osition 3.16 and noticing that by the Kunneth formula for homo logy , the Poincar´ e polyno mia l of P ( ℓ +1)( p +1) − 1 C × · · · × P ( ℓ +1)( p +1) − 1 C | {z } N times equals (1 − T 2 ) − N mo d T p .  As b efore we ha ve the followin g corolla r y . Corollary 3. 27. L et p = 2 m + 1 with m ≥ 0 . Then Q pr [0 ,σ ] ( A ) = (1 − T ) N Q J p C , pr [0 ,σ ] ( A ) mo d T m +1 . (3.13) It is clear fro m the definition that the c o mplex joins o f langua ges in P C also belo ng to th e c omplexity cla ss P C . W e record this obs erv ation for mally in t he following pr o po sition. Prop ositio n 3.2 8 (Polynomial time membership testing) . Supp ose that the se- quenc e of c onstructible sets ( S n ⊂ P k ( n ) C × P ℓ ( n ) C ) n> 0 ∈ P C , and X n = ( X 0 : · · · : 24 SAUGA T A BASU X k ( n ) ) Y n = ( Y 0 : · · · : Y ℓ ( n ) ) ar e homo gene ous c o-or dinates of P k ( n ) C and P ℓ ( n ) C r esp e ctively. L et p ( n ) b e a n on- ne gative p olynomial, and for e ach n > 0 let pr 1 : P k ( n ) C × P ℓ ( n ) C → P k ( n ) C denote the pr oje ction on the first c omp onent. Then,  J p ( n ) C , pr 1 ( S n ) ⊂ P k ( n ) C × P ( p ( n )+1)( ℓ ( n )+1) − 1 C  n> 0 ∈ P C . Pr o of. Obvious fr om the definition of ( J p ( n ) C , pr 1 ( S n )) n> 0 .  4. P r oof of the main th eorem W e are no w in a p osition to pr ov e Theorem 2.1. The pr o of relies on the following key propos itio n. Prop ositio n 4.1. L et m ( n ) , k 1 ( n ) , . . . , k ω ( n ) b e p olynomials, and let (Φ n ( X , Y )) n> 0 b e a se qu enc e of multi-homo gene ous formulas Φ n ( X , Y ) def = (Q 1 Z 1 ∈ P k 1 C ) · · · (Q ω Z ω ∈ P k ω C ) φ n ( X ; Y ; Z 1 ; · · · ; Z ω ) , having fr e e variables ( X ; Y ) = ( X 0 , . . . , X k ( n ) ; Y 0 , . . . , Y m ( n ) ) , with Q 1 , . . . , Q ω ∈ {∃ , ∀} , and φ n a multi-homo gene ous quantifier-fr e e formula defining a close d (re sp. op en) c onstruct ible subset S n ⊂ P k C × P m C × P k 1 C × · · · × P k ω C . Supp ose also that  R ( φ n ( X ; Y ; Z 1 ; · · · ; Z ω ))  n> 0 ∈ P C . Then ther e ex ist : (A) a se quenc e of quantifier-fr e e multi-homo gene ous formulas  Θ n ( X ; V 1 ; · · · ; V N )  n> 0 , with V i = ( V 0 , . . . , V p i ) , and N , p 1 , . . . , p N p olynomials in n , such that for al l x ∈ P k C Θ n ( x ; V 1 ; · · · ; V N ) defines a c onstructible su bset T n ⊂ P p 1 C × · · · × P p N C , with ( T n ) n> 0 ∈ P C ; (B) p olynomial time c omputable maps F n : Z [ T ] → Z [ T ] , such that for al l x ∈ P k C Q R (Φ n ( x ; Y )) = F n ( Q R (Θ n ( x ; V 1 ; ··· ; V N )) ) . The idea behind the pro of o f Prop osition 4 .1 is to use induction on the n umber, ω , of quantifier blo cks. When ω = 0, the prop osition is obvious. When ω > 0, then using Co rollary 3.25, we can construct a new for m ula (sa y Φ ′ n ) suc h that Φ ′ has one less blo ck of qua n tifiers, but such that Q R (Φ n ) is easily c o mputable from Q R (Φ ′ n ) . One can then use inductio n to finish the pr o o f. How ever, a tec hnical complica tion arises due to the fact that in the pro jective s ituation (unlik e in the affine case) we 25 cannot immediately repla ce tw o adjacent blo cks of the same quantifier b y a single blo ck. This is the logica l ma nifestation of the elementary fact that the pro duct of t w o pro jective spaces is not itself a pro jective space. In o rder to ov ercome this difficulty a nd car r y through the inductive step pr op e rly , we need to prove a slightly stronger , but technically mor e inv olved prop osition, which we state next. Prop osition 4.1 will b e a n immediate coro llary of this more g eneral prop osition. Prop ositio n 4.2. L et σ, ω ≥ 0 b e c onstant s, and a 0 ( n ) , α 0 ( n ) , a 1 ( n ) , α 1 ( n ) , . . . , a σ ( n ) , α σ ( n ) , k ( n ) , k 1 ( n ) , . . . , k ω ( n ) fixe d p olynomials in n taking n on-ne gative values for n ∈ N . L et X = ( X 0 : · · · : X k ( n ) ) denote a blo ck of k ( n ) + 1 variables. F or 0 ≤ j ≤ σ , let W j denote the tuple of variable s ( . . . , W j i 0 ,...,i j , . . . ) , 0 ≤ i 0 ≤ α 0 , . . . , 0 ≤ i j ≤ α j , wher e e ach W j i 0 ,...,i j is a blo ck of a j ( n ) + 1 variables. L et  Φ n ( X ; W 0 ; W 1 ; . . . ; W σ )  n> 0 , b e a se qu enc e of multi-homo gene ous formulas define d by Φ n ( X ; W 0 ; W 1 ; · · · ; W σ ) def = Λ 0 0 ≤ i 0 ≤ α 0 · · · Λ σ 0 ≤ i n ≤ α σ (Q 1 Z 1 ∈ P k 1 C ) · · · (Q ω Z ω ∈ P k ω C ) φ n ( X ; W 0 i 0 ; W 1 i 0 ,i 1 ; · · · ; W σ i 0 ,...,i σ ; Z 1 ; · · · ; Z ω ) , with Λ 0 , . . . , Λ σ ∈ { _ , ^ } , Q 1 , . . . , Q ω ∈ {∃ , ∀} , and e ach φ n a mu lti-homo gene ous quantifier-fr e e formula, multi-homo gene ous in the blo cks of variables, X , Z 1 , . . . , Z ω and ( W j i 0 ,...,i j , 0 , . . . , W j i 0 ,...,i j ,α j ) for 0 ≤ j ≤ σ . Supp ose also that e ach φ n defines a close d (r esp. op en ) c onstru ctible set , and t hat ( R ( φ n )) n> 0 ∈ P C . Then ther e ex ist s: (A) a se quenc e of quantifier-fr e e multi-homo gene ous formulas  Θ n ( X ; V 1 ; · · · ; V N )  n> 0 , with V i = ( V 0 , . . . , V p i ) , and N , p 1 , . . . , p N p olynomials in n , such that Θ n ( x ; V 1 ; · · · ; V N ) d efines a c onstructible subset T n ⊂ P p 1 C × · · · × P p N C , with ( T n ) n> 0 ∈ P C ; (B) p olynomial time c omputable maps F n : Z [ T ] → Z [ T ] , such that Q R (Φ n ( x ; · )) = F n ( Q R (Θ n ( x ; V 1 ; ··· ; V N )) ) . 26 SAUGA T A BASU Pr o of of Pr op osition 4.2. The pro of is by induction on ω . If ω = 0, we let Θ n = Φ n , a nd F n to b e the identit y ma p. Since there ar e no quantifiers, for each n ≥ 0 the constr uctible set defined by Θ n and Φ n are the s ame, and th us the Betti num b ers o f the sets defined b y Θ n and Φ n are equal. If ω > 0, we have the fo llowing tw o cases. (A) Case 1, Q 1 = ∃ : First note that Φ n defines a constructible subset of P k ( n ) C × U n , where U n = ( P ( a 0 +1)( α 0 +1) − 1 C ) m 0 × · · · × ( P ( a j +1)( α j +1) − 1 C ) m j × · · · × ( P ( a σ +1)( α σ +1) − 1 C ) m σ , where for 0 ≤ j ≤ σ , m j ( n ) = Y 0 ≤ i ≤ j − 1 ( α i ( n ) + 1) . The formula Φ n is equiv a lent to the following form ula:  · · · ( ∃ Z 1 ,i 0 ,...,i σ ∈ P k 1 C ) · · ·  ¯ Φ n where where the blo cks of existential quantifiers in the b eginning a re in- dexed by the tuples ( i 0 , . . . , i σ ) , 0 ≤ i 0 ≤ α 0 ( n ) , . . . , 0 ≤ i σ ≤ α σ ( n ) . and the num b er of such blocks is α ( n ) = σ Y i =0 ( α i ( n ) + 1) , and ¯ Φ n def = Λ 0 0 ≤ i 0 ≤ α 0 · · · Λ σ 0 ≤ i σ ≤ α σ (Q 2 Z 2 ∈ P k 2 C ) · · · (Q ω Z ω ∈ P k ω C ) φ n ( X ; W 0 i 0 ; · · · , W σ i 0 ,...,i σ ; Z 1 i 0 ,...,i σ ; Z 2 · · · ; Z ω ) , Let m ( n ) = σ X j =0 m j ( n )(( a j ( n ) + 1)( α j ( n ) + 1) − 1 ) . (Note that m ( n ) is the (complex ) dimension of U n defined previo us ly .) Let pr 1 , 2 : P k ( n ) C × U n × ( P k 1 C ) α ( n ) → P k ( n ) C × U n denote the pr o jection on the first tw o comp onents. Consider the s equence  J 2 m ( n )+1 C , pr 1 , 2 ( ¯ Φ n )  n> 0 . Note that by 3.11 we ha ve J 2 m ( n )+1 C , pr 1 , 2 ( ¯ Φ n ) = Λ 0 0 ≤ i 0 ≤ α 0 · · · Λ σ 0 ≤ i σ ≤ α σ Λ σ +1 0 ≤ i σ +1 ≤ α σ +1 (Q 2 Z 2 ∈ P k 2 C ) · · · (Q ω Z ω ∈ P k ω C ) φ n ( X ; W 0 i 0 ; · · · ; W σ i 0 ,...,i σ ; W σ +1 i 0 ,...,i σ ,i σ +1 ; Z 2 ; · · · ; Z ω ) . 27 with Λ σ +1 = V , α σ +1 = 2 m + 1 , and W σ +1 i 0 ,...,i σ ,i σ +1 = Z 1 i 0 ,...,i σ ,i σ +1 . W e will denote by W σ +1 the tuple ( . . . , W σ +1 i 0 ,...,i σ ,i σ +1 , . . . ) , 0 ≤ i 0 ≤ α 0 , . . . , 0 ≤ i σ +1 ≤ α σ +1 . Observe that the num ber of quant ifiers in the formulas J 2 m ( n )+1 C , pr 1 , 2 ( ¯ Φ n ) , is ω − 1. Moreov er, J 2 m ( n )+1 C , pr 1 , 2 ( ¯ Φ n ) s a tisfy by P r op osition 3.28 the requir ed po ly- nomial time hypo thesis, and hav e the sa me shap e a s the formulas Φ n . W e can thus apply the induction hypothesis to this sequence to obtain a se- quence (Θ n ) n> 0 , as well as a seque nce of poly nomial time computable maps ( G n ) n> 0 . By inductive h yp othesis we can suppose tha t for each x ∈ P k ( n ) C Q R ( J 2 m ( n )+1 C , pr 1 , 2 ( ¯ Φ n )( x ; · )) = G n ( Q R (Θ n ( x ; · )) ) . Using Coro lla ry 3.27 and noticing that the map pr 1 , 2 is either op en or closed and henc e a compact covering, Q R (Φ n ( x ; · )) = (1 − T ) α ( n ) Q R ( J 2 m ( n )+1 C , pr 1 , 2 ( ¯ Φ n )( x ; · )) mo d T m ( n )+1 = (1 − T ) α ( n ) G n ( Q R (Θ n ( x ; · )) ) mod T m ( n )+1 . W e set F n = T runc m ( n ) ◦ M (1 − T ) α ( n ) ◦ G n (see Notation 3 .2). This completes the induction in this cas e. (B) Case 2, Q 1 = ∀ : The formula Φ n is equiv a lent to the following form ula:  · · · ( ∀ Z 1 ,i 0 ,...,i σ ∈ P k 1 C ) · · ·  ¯ Φ n where the blo c ks of universal quantifiers in the beg inning ar e indexed by the tuples ( i 0 , . . . , i σ ) , 0 ≤ i 0 ≤ α 0 ( n ) , . . . , 0 ≤ i σ ≤ α σ ( n ) , the num b er of such blocks is α ( n ) = σ Y i =0 ( α i ( n ) + 1) , and ¯ Φ n def = Λ 0 0 ≤ i 0 ≤ α 0 · · · Λ σ 0 ≤ i σ ≤ α σ (Q 2 Z 2 ∈ P k 2 C ) · · · (Q ω Z ω ∈ P k ω C ) φ n ( X ; W 0 i 0 ; · · · ; W σ i 0 ,...,i σ ; Z 1 i 0 ,...,i σ ; Z 2 · · · ; Z ω ) . Consider the s equence  J 2 m +1 C , pr 1 , 2 ( ¬ ¯ Φ n )  n> 0 . 28 SAUGA T A BASU Note that the formula J 2 m +1 C , pr 1 , 2 ( ¬ ¯ Φ n ) = ¯ Λ 0 0 ≤ i 0 ≤ α 0 · · · ¯ Λ σ 0 ≤ i σ ≤ α σ Λ σ +1 0 ≤ i σ +1 ≤ α σ +1 ( ¯ Q 2 Z 2 ∈ P k 2 C ) · · · ( ¯ Q ω Z ω ∈ P k ω C ) ¬ φ n ( X ; W 0 i 0 ; · · · ; W σ i 0 ,...,i σ ; W σ +1 i 0 ,...,i σ +1 ; Z 2 ; · · · ; Z ω ) with Λ σ +1 = V , α σ +1 = 2 m + 1, W σ +1 i 0 ,...,i σ +1 = Z 1 i 0 ,...,i σ ,i σ +1 , and ¯ Λ i = _ if Λ i = ^ , ¯ Λ i = ^ if Λ i = _ , ¯ Q i = ∃ if Q i = ∀ , ¯ Q i = ∀ if Q i = ∃ . Observe that the n um b er of quantifiers in the for m ulas J 2 m +1 C , pr 1 , 2 ( ¬ ¯ Φ n ) , is ω − 1. Moreov er, J 2 m +1 C , pr 1 , 2 ( ¬ ¯ Φ n ) satisfy by Pro po sition 3.2 8 the r equired p oly- nomial time hypo thesis, and hav e the sa me shap e a s the formulas Φ n . W e can thus apply the induction hypothesis to this sequence to obtain a se- quence (Θ n ) n> 0 , as well as a seque nce of poly nomial time computable maps ( G n ) n> 0 . By inductive h yp othesis we can suppose tha t for each x ∈ P k ( n ) C Q R ( J 2 m +1 C , pr 1 , 2 ( ¬ ¯ Φ n )( x ; · )) = G n ( Q R (Θ n ( x ; · )) ) . Using Coro lla ry 3.27 and noticing that the map pr 1 , 2 is either op en or closed and henc e a compact covering, we ha ve Q R ( ¬ Φ n ( x ; · )) = (1 − T ) α ( n ) Q R ( J 2 m +1 C , pr 1 , 2 ( ¯ Φ n )( x ; · )) mo d T m ( n )+1 = (1 − T ) α ( n ) G n ( Q R (Θ n ( x ; · )) ) mo d T m ( n )+1 . The sets K n = R (Φ n ( x ; · )) are constr uc tible a nd open (re sp. clo s ed); so by Corollary 3.4 (corolla ry to Theorem 3.1), we ha ve Q K n ( T ) = Q U n − Rec m (T runc m ( Q U n − K n )) . W e set F n to be the op era tor defined by F n ( Q ) = Q U n − Rec m (T runc m ( M (1 − T ) α ( n ) ( G n ( Q )))) . This completes the induction in this case as well.  Pr o of of Pr op osition 4.1. Prop osition 4.1 is a sp ecial ca se o f Pro po sition 4.2 with σ = 0, α 0 = 0, and Y = W 0 .  Pr o of of The or em 2.1. F o llows immediately from Pr op osition 4 .1 in the sp ecial cas e m = 0. In this ca se the sequence of formulas (Φ n ) n> 0 corres p onds to a language in the p olynomial hierarch y and for each n , x = ( x 0 : · · · : x k ( n ) ) ∈ S n ⊂ P k ( n ) C if and only if F n ( Q R (Θ n ( x ; · )) )(0) > 0 and this last condition can b e chec ked in p olynomial time using an oracle from the class # P † C .  29 R emark 4.3 . It is in teresting to obser ve that in c o mplete analogy with the pro o f of the classical T o da’s theorem the pr o of o f Theo rem 2.1 also requires just o ne ca ll to the ora cle at the end. Pr o of of The or em 2.6. F o llows fro m the pro of of Prop ositio n 4.1 since the formula Θ n is clearly computable in p olynomial time from the given form ula Φ n as long as the num b er of qua n tifier alternatio ns ω is b ounded by a constant.  5. Future Directions In this section, we s ket ch a few direc tions in which the work pr esented in this pap er could b e developed further . (A) Remov e the compactness hypothesis from the main theor e m. (B) The compact frag men t of the p oly no mial hierar chy introduced in this pa- per , and esp ecially the class Σ c C , 1 (whic h is the compact fragment of NP C ), is p ossibly interesting on their own, and it w ould b e nice to develop a the- ory of compact reductions and compact ha rdness, and hav e N P c C -complete problems. The compact feasibility problem discussed in E xample 1.1 6 is a go o d candidate for being a NP c C -complete problem. (C) As rema r ked earlier , one would obtain a strong er reduction r e s ult if one could prove a T o da-type theorem using only the E uler-Poincar´ e character- istic instea d of the whole Poincar´ e po lynomial. This s e e ms to be rather difficult. An in termedia te goal could be to use the virtual Poinc ar´ e p olyno- mial . The vir tual Poincar´ e p olynomial o f a co mplex v ariety X is defined by P X ( T ) = H X ( − T , − T ) , where H X ( u, v ) ∈ Z [ u, v ] is the Ho dge-Delig ne p olyno mia l uniquely deter- mined by the following properties . (1) The map X 7→ H X gives an additive and multiplicativ e in v aria n t fro m the Gro thendiec k ring of equiv alence classes of co mplex v arie ties to Z [ u, v ]. (2) H X ( u, v ) coincides with P ( − 1) p + q h p,q ( X ) u p v q when X is smo oth and pro jective, where h p,q ( X ) are the Ho dge num ber s. Clearly , the v irtual Poincar´ e po lynomial is additive, and coincides with the ordinary Poincar´ e po lynomial, P X , in the case when X is smo oth and pro- jective. Thus, one co uld try to prov e the results in this pap er using the virtual Poincar´ e p olynomial instea d of the Poincar´ e p olyno mial. Unfortu- nately , the virtual Poincar´ e p olynomial is an algebro-g eometric, rather than top ological inv ariant, and the topo logical metho ds used in this pap er are not sufficient to obtain such a result. In particular, Theore m 3.23 do es not hold for the virtual P o incar´ e p olynomial except in the tr ivial case when A = P k C × P ℓ C . (D) Theorem 3 .23 can b e used to b ound the Betti num b ers of the images o f complex v ar ieties under regular maps (in conjunction with tigh t b ounds on the Betti num b ers o f co mplex pro jective v arieties due to Ka tz [1 6]), instead of first using e limina tion metho ds, and then applying the b ounds due to Katz. A simila r method was used in [1 4] to obtain bo unds on the Betti nu mbers of pr o jections of semi-algebr a ic sets in the real c a se. 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