An Almost Optimal Rank Bound for Depth-3 Identities

An Almost Optimal Rank Bound for Depth-3 Identities Nitin Saxena ∗ C. Seshadhri † Abstract W e show that the r ank of a depth-3 circuit (o ver an y ﬁeld) that is simple, min- imal and zero is at most O ( k 3 log d ). The previous bes t rank bo und known was 2 O ( k 2 ) (log d ) k − 2 by Dvir and Shpilk a (STOC 2005 ). This almost res olves the ra nk question ﬁrst p osed b y Dvir and Shpilk a (as w e also provide a simple a nd minimal ident ity of rank Ω( k log d )). Our rank b ound signiﬁcantly impro ves (dependence on k exp onentially re duced) the b est known deterministic black-box iden tit y tests for depth-3 circ uits by Karnin and Shpilk a (CCC 20 08). Our techniques also shed light on the factorization pattern of nonzero depth-3 circuits, most strikingly: the rank of linear factors of a simple, minimal and no nz e r o depth-3 circ uit (ov er any ﬁeld) is at most O ( k 3 log d ). The nov el feature of this w ork is a new notion of maps between s e ts o f linear forms, called ide a l matchings , used to s tudy depth-3 circuits. W e prove interesting str uctural results ab out depth-3 iden tities using these techniques. W e b elieve that these can lead to the goal of a deter ministic po lynomial time identit y tes t for these circuits. 1 In tro duction P olynomial identit y testing (PIT) ranks as one of the most imp ortan t op en prob lems in th e in tersection of algebra and computer s cience. W e are provided an arithmetic circuit that computes a polynomial p ( x 1 , x 2 , · · · , x n ) o v er a ﬁeld F , and w e wish to test if p is iden tically zero. In the blac k-b o x setting, the circuit is provided as a blac k-b o x and we are only allo w ed to ev aluate th e p olynomial p at v ario us d omain p oints. T h e main goal is to devise a deterministic p olynomial time algorithm for PIT. Kabanets and Impagliazzo [KI 04] and Agra w al [Agr05] ha v e shown connections b etw een deterministic algorithms for iden tit y testing and circuit lo w er b ound s , emphasizing th e imp ortance of this problem. The ﬁrst rand omized p olynomial time PIT algorithm, which w as a blac k-box algorithm, w as giv en (indep endent ly) b y S c h w artz [Sc h80] and Zip p el [Zip79]. Randomized algorithms that u se less randomness w ere giv en b y Chen & Kao [CK 00], Lewin & V adhan [L V98], and Agra w al & Biswa s [AB03]. Kliv a ns and S pielman [KS01] observ ed that eve n for depth-3 circuits for b oun ded top fanin , d eterministic identit y testing was op en. Progress to w ards this w as ﬁ rst made b y Dvir and Sh p ilk a [DS06], w ho ga v e a quasi-p olynomial time algorithm, although with a d oubly-exp onen tial dep endence on the top fanin. The problem wa s resolv ed b y a p olynomial time algo rithm giv en b y Kay al and Saxena [KS07], ∗ Hausdorﬀ Center for Mathematics, Bonn 53115, Germany . E- mail: ns@hcm.uni-bonn. de † IBM Almaden Research Center, San Jose - 95123, USA . E- mail: csesha@us.ibm.co m 1 with a running time exp onentia l in the top f an in . F or a sp ecial case of depth -4 circuits, Saxena [Sax08] has designed a determin istic p olynomial time alg orithm for P I T. Why is progress restricted to small depth circuits? Agra w al and Vina y [A V08] recent ly sho w ed that an eﬃcien t b lac k-b o x identit y test for d ep th-4 circuits will actually giv e a quasi- p olynomial blac k-b ox test for circuits of al l depths . F or d eterministic blac k-b o x testing, the ﬁrst results w ere giv en by Karnin and Sh- pilk a [KS08]. Base d on results in [DS06 ], they gav e an algorithm for depth-3 circuits ha ving a quasi-p olynomial running time (with a d oubly-exp onent ial dep endence on the top fanin) 1 . One of the consequen ces of our r esult will b e a signiﬁcan t imp ro v emen t in the runn ing time of their d etermin istic blac k-b o x tester. This w ork f o cuses on depth-3 circuits. A structural study of depth-3 iden tities w as initiated in [DS06] by deﬁning a notion of r ank of simple and minimal ident ities. A dep th-3 circuit C o v er a ﬁeld F is: C ( x 1 , . . . , x n ) = k X i =1 T i where, T i ( a multiplic at ion term ) is a p ro duct of d i linear functions ℓ i,j o v er F . Note that for the pu rp oses of studying id en tities w e can assume w log (by homo genization ) that ℓ i,j ’s are linear forms (i.e. lin ear p olynomials with a zero constan t co eﬃcien t) and that d 1 = · · · = d k =: d . S uc h a circuit is referred to as a ΣΠΣ( k , d ) circuit, where k is the top fanin of C and d is th e de gr e e of C . W e give a few deﬁn itions from [DS06]. Deﬁnition 1. [Simple Circuits] C is a simp le cir cuit if ther e i s no nonzer o line ar form dividing al l the T i ’s. [Minimal Circuits] C is a minimal cir cuit if for every pr op er subset S ⊂ [ k ] , P i ∈ S T i is nonzer o. [Rank of a circuit] The rank of the cir c u it, r an k ( C ) , i s deﬁne d as the r a nk of the line a r forms ℓ i,j ’s viewe d as n -dimensional ve ctors over F . Can all th e forms ℓ i,j b e indep enden t, or must there b e r elations b etw een them? T he rank can b e inte rpr eted as the minimum num b er of v ariables that are required to exp r ess C . Th er e exists a linear trans f ormation con v erting the n v ariables of the circuit in to r ank ( C ) indep endent v ariables. A trivial b oun d on the rank (for an y ΣΠΣ-circuit) is k d , since th at is th e total n um b er of linear f orms inv olv ed in C . The r ank is a fun damen tal prop erty of a ΣΠΣ( k, d ) circuit and it is crucial to understand ho w large th is can b e for iden tities. A substantial ly sm aller rank b ound th an k d shows that id en tities d o n ot hav e as many “degrees of freedom” as general circuits, and lead to deterministic iden tit y tests 2 . F ur thermore, th e tec hniques u sed to pr o v e rank b ounds show u s structural p rop erties of iden tities that ma y su ggest directions to resolv e PIT for ΣΠΣ( k , d ) circuits. Dvir and Shp lik a [DS06] prov ed that the rank is b ounded by 2 O ( k 2 ) (log d ) k − 2 , an d this b ound is translated to a p oly( n )exp(2 O ( k 2 ) (log d ) k − 1 ) time blac k-b o x id en tit y tester by Karnin and Shpilk a [KS08]. Note that when k is larger than √ log d , these b ounds are trivial. Our p r esen t und erstanding of ΣΠΣ ( k , d ) iden tities is v ery p o or when k is larger than a constan t. W e present the ﬁrst result in th is direction. 1 [KS08] had a b etter running time for read- k depth-3 circuits, where each v ariable app ears at most k times. But even there the dep endence on k is doubly- exp onentia l. 2 W e usually do n ot get a p olynomial time algorithm. 2 Theorem 2 (Main Theorem) . The r ank of a simple and minimal ΣΠΣ( k, d ) identity is O ( k 3 log d ) . This giv es an exp onen tial improv emen t on the previously known d ep endence on k , and is strictly b etter than the previous r ank b ound f or ev ery k > 3. W e also giv e a simple construction of identit ies with rank Ω ( k log d ) in Section 2, s ho wing that the ab o v e theo- rem is almost optimal . As m entioned ab ov e , we can in terpret this b ound as sa ying that an y simple and minimal ΣΠΣ( k , d ) iden tit y can b e expressed u sing O ( k 3 log d ) indep en- den t v ariables. One of the most inte resting features of this result is a nov el tec hnique dev elop ed to study depth-3 circuits. W e introd uce the concepts of ide al matchings and or der e d matchings , that allo w u s to analyze the structure of depth-3 iden tities. These matc hings are studied in detail to get the rank b ound. Along th e w a y we initiate a th eory of matc hings, viewin g a matc hing as a fund amen tal map b etw ee n sets of linear forms. Wh y are the simplicit y and minimalit y restrictions required? T ak e the non-simple ΣΠΣ(2 , d ) iden tit y ( x 1 x 2 · · · x d ) − ( x 1 x 2 · · · x d ). This has rank d . Similarly , w e can tak e the n on-minimal Σ ΠΣ(4 , d + 1) id en tit y ( y 1 y 2 · · · y d )( x 1 − x 1 ) + ( z 1 z 2 · · · z d )( x 2 − x 2 ) th at has rank (2 d + 2). In some sense, these restrictions only ignore ident ities th at are comp osed of smaller identit ies. 1.1 Consequences Apart from b eing an in teresting s tructural result ab out ΣΠΣ iden tities, w e can use th e rank b ound to get n ice algorithmic results. Our ran k b ound immediately giv es faster deterministic blac k-b o x identit y testers for ΣΠΣ( k , d ) circuits. A dir ect application of Lemma 4.10 in [KS08] to our rank boun d give s an exp onen tial improv emen t in the dep endence of k compared to previous blac k-b ox testers (that had a runn in g time of p oly( n )exp(2 O ( k 2 ) (log d ) k − 1 )). Theorem 3. Ther e is a deterministic b lack- b o x identity tester for ΣΠΣ( k , d ) cir cuits that runs in p oly ( n , d k 3 log d ) time. The ab o ve blac k-b o x tester is n o w muc h closer in complexit y to the b est non blac k-b o x tester kno wn ( pol y ( n, d k ) time by [KS07]). Our result also app lies to blac k-b o x identit y testing of r e ad- k ΣΠΣ( k , d ) circuits, wh ere eac h v ariable o ccurs at most k times. W e get a similar immediate impro vemen t in the dep endence of k (the previous run ning time w as n 2 O ( k 2 ) .) Theorem 4. Ther e is a deterministic black-b ox identity tester for r e ad- k ΣΠΣ( k , d ) cir- cuits that runs in O ( n k 4 log k ) time. Although it is not immediate from Theorem 2, our tec h nique also p r o vides an in- teresting algebraic result ab out p olynomials compu ted by simp le, minimal, and nonzero ΣΠΣ( k , d ) circuits 3 . Consider such a circuit C that compu tes a p olynomial p ( x 1 , · · · , x n ). Let u s facto rize p into Q i q i , w here eac h q i is a nonconstan t and irr educible p olynomial. W e d enote b y L ( p ) the set of line ar factors of p (that is, q i ∈ L ( p ) iﬀ q i | p is linear). Theorem 5. If p is c ompute d by a simple, minimal, nonzer o ΣΠΣ( k , d ) cir cuit then the r ank of L ( p ) is at most k 3 log d . 3 Here we can also consi der circuits where the diﬀeren t terms in C have d iﬀeren t degrees. The parameter d is then an up p er b oun d on the degree of C . 3 1.2 Organization W e ﬁrs t giv e a simple construction of identit ies with rank Ω ( k log d ) in S ection 2. Section 3 con tains th e p ro of of our main theorem. W e give some preliminary n otatio n in Section 3.1 b efore explaining an in tuitiv e picture of our ideas (Section 3.2). W e then explain our main to ol of ide al matchings (Section 3.3) and pr o ve some usefu l lemmas ab out them. W e mov e to Section 3.4 wh er e the concepts of or der e d matchings and simple p arts of cir cuits are in tro duced. W e motiv ate these deﬁnitions and then pro v e s ome easy facts ab out them. W e are no w ready to tackl e the p roblem of b ound in g the rank. W e describ e our p ro of in terms of an iterativ e pro cedure in S ection 3.5. Eve rything is put together in Section 3.6 to b ound the rank. Finally (it should h op efully b e ob vious by then), we sh o w how to apply our tec h niques to pro ve T h eorem 5. 2 High Rank Iden tities The follo w ing id entit y was constructed in [KS07]: ov er F 2 (with r > 2), C ( x 1 , . . . , x r ) := Y b 1 ,...,b r − 1 ∈ F 2 b 1 + ··· + b r − 1 ≡ 1 ( b 1 x 1 + · · · + b r − 1 x r − 1 ) + Y b 1 ,...,b r − 1 ∈ F 2 b 1 + ··· + b r − 1 ≡ 0 ( x r + b 1 x 1 + · · · + b r − 1 x r − 1 ) + Y b 1 ,...,b r − 1 ∈ F 2 b 1 + ··· + b r − 1 ≡ 1 ( x r + b 1 x 1 + · · · + b r − 1 x r − 1 ) It w as sho wn that, o v er F 2 , C is a simple and minimal ΣΠΣ zero circuit of degree d = 2 r − 2 with k = 3 multiplic ation terms and r ank ( C ) = r = log 2 d + 2. F or this section let S 1 ( x ), S 2 ( x ), S 3 ( x ) denote the thr ee m ultiplication terms of C . W e n o w b u ild a high rank iden tit y based on S 1 , S 2 , S 3 . Our basic step is giv en by the follo win g lemma that w as u sed in [DS06] to construct identities of rank (3 k − 2). Lemma 6. [DS06] L et D i ( y i, 1 , . . . , y i,r i ) := P k i j =1 T j b e a simple, minimal and zer o Σ ΠΣ cir cuit, over F 2 , with de gr e e d i , fanin k i and r ank r i . Deﬁne a new cir cuit over F 2 using D i and C : D i +1 ( y i, 1 , . . . , y i,r i + r ) :=   k i − 1 X j =1 T j   · S 1 ( y i,r i +1 , . . . , y i,r i + r ) − T k i · S 2 ( y i,r i +1 , . . . , y i,r i + r ) − T k i · S 3 ( y i,r i +1 , . . . , y i,r i + r ) Then D i +1 is a simple, minimal and zer o ΣΠΣ cir cuit with de gr e e d i +1 = ( d i + d ) , fanin k i +1 = ( k i + 1) and r ank r i +1 = ( r i + r ) . Pr o of. Since C is an id en tity , w e get that S 2 ( y i,r i +1 , . . . , y i,r i + r ) + S 3 ( y i,r i +1 , . . . , y i,r i + r ) = 4 − S 1 ( y i,r i +1 , . . . , y i,r i + r ). Th erefore, D i +1 ( y i, 1 , . . . , y i,r i + r ) =  k i − 1 X j =1 T j  S 1 ( y i,r i +1 , . . . , y i,r i + r ) − T k i ( S 2 ( y i,r i +1 , . . . , y i,r i + r ) + S 3 ( y i,r i +1 , . . . , y i,r i + r )) =  k i − 1 X j =1 T j  · S 1 ( y i,r i +1 , . . . , y i,r i + r ) + T k i S 1 ( y i,r i +1 , . . . , y i,r i + r ) =  k i X j =1 T j  · S 1 ( y i,r i +1 , . . . , y i,r i + r ) = 0 The terms T j do not share an y v ariables with S ℓ ( ℓ ∈ { 1 , 2 , 3 } ). Sin ce D i and C are simp le, D i +1 is also simple. Supp ose D i +1 is not minimal. W e h a ve some su bset P ⊂ [1 , k i − 1] suc h that C ′ := ( P j ∈ P T j ) S 1 − α 2 T k i S 2 − α 3 T k i S 3 = 0, where α 2 , α 3 ∈ { 0 , 1 } . If b oth α 2 and α 3 are 1, then w e get ( P j ∈ P T j ) S 1 + T k i S 1 = 0, now P must b e the whole set [1 , k i − 1], b ecause D i is minimal. On the other hand, if b oth α 2 , α 3 are 0, then ( P j ∈ P T j ) S 1 = 0 which is imp ossible as D i is min im al. The only remaining p ossibilit y is (wlog) ( P j ∈ P T j ) S 1 − T k i S 2 = 0. As S 1 is coprime to S 2 and T k i , this is imp ossible. Therefore, D i +1 is minimal. It is easy to see the parameters of D i +1 : k i +1 = ( k i + 1) and d i +1 = ( d i + 1). Because the T j ’s do not sh are an y v ariables w ith S ℓ ’s, the rank r i +1 = ( r i + r ). F amily of High Rank Iden tit ies: No w w e will start with D 0 := C ( y 0 , 1 , . . . , y 0 ,r ) and apply the ab ov e lemma ite rativ ely . Th e i -th circuit we get is D i with d egree d i = ( i + 1) d , fanin k i = i + 3 an d rank r i = ( i + 1) r = ( i + 1)(log 2 d + 2). So r i relates to k i , d i as: r i = ( k i − 2)  log 2 d i k i − 2 + 2  . Also it can b e seen that if d > i then d i k i − 2 ≥ √ d i . Thus after simpliﬁcation, w e h a ve for an y 3 ≤ i < d , r i > k i 3 · log 2 d i . This giv es us an inﬁnite family of ΣΠΣ( k , d ) iden tities o ver F 2 with rank Ω( k log d ). A similar f amily can b e obtained o v er F 3 as wel l. 3 Rank Bound Our tec h n ique to b ound the rank of Σ ΠΣ identit ies relies m ainly on tw o notions - form- ide als and matchings by them - that o ccur n aturally in study in g a ΣΠΣ circuit C . Using these to ols we can do a su r gery on the circuit C and extract out smaller circuits an d smaller identi ties. Before explaining our b asic idea we need to develo p a small theory of matc hin gs and deﬁ n e gc d and simple p arts of a sub c ir cu it in that framework. W e set do wn some preliminary deﬁn itions b efore giving an imprecise, yet in tuitiv e explanation of our idea and an o verall picture of h o w we b ound the rank. 3.1 Preliminaries W e w ill denote the set { 1 , . . . , n } by [ n ]. 5 In this p ap er we will stud y iden tities o ve r a ﬁ eld F . So the circu its compute m ultiv ariate p olynomials in the p olynomial ring R := F [ x 1 , . . . , x n ]. W e will b e studying ΣΠΣ( k , d ) cir cuits : su c h a circuit C is an expression in R giv en b y a depth-3 circuit, with th e top gate b eing an addition gate, the second lev el ha ving multiplicat ion gates, the last lev el having addition gates, and the lea ves b eing v ariables. The edges of the circuit ha v e elemen ts of F (constan ts) asso ciated with th em (signifying m ultiplication b y a constant). The top fanin is k and d is the degree of the p olynomial compu ted by C . W e will call C a ΣΠΣ- identity , if C is an identica lly zero ΣΠΣ-circuit. A line ar form is a linear p olynomial in R . W e will denote the set of all linear forms b y L ( R ) : L ( R ) := ( n X i =1 a i x i | a 1 , . . . , a n ∈ F ) Muc h of what w e do sh all deal with sets of linear forms, and v arious maps b et ween th em. A list L of linear forms is a m ulti-set of f orms with an arb itrary order asso ciated with them. The actual orderin g is unimp ortant : w e merely ha v e it to distinguish b et ween rep eated forms in th e list. O ne of the fundamen tal constructs w e u se are maps b et ween lists, which could h a ve man y copies of the same form. The ordering allo ws us to d eﬁ ne these maps u nam biguously . All lists w e consider will b e ﬁ nite. Deﬁnition 7. [Multiplication term] A multiplicat ion term f is an expr ession i n R given as (the pr o duct may have r ep e ate d ℓ ’s): f := c · Y ℓ ∈ S ℓ, wher e c ∈ F ∗ and S is a list of line ar forms. The list of linear forms in f , L ( f ) , is just the list S of forms o c curring in the pr o duct ab ove. # L ( f ) is natur al ly c al le d the degree of the multiplic ation term. F or a list S of line ar forms we deﬁne the m u ltiplication term of S , M ( S ) , as Q ℓ ∈ S ℓ or 1 if S = φ . Deﬁnition 8. [F orms in a Circuit] We wil l r epr esent a ΣΠΣ( k , d ) c i r cui t C as a sum of k multiplic ation terms of de gr e e d , C = P k i =1 T i . The list of linear f orm s o ccurring in C is L ( C ) := S i ∈ [ k ] L ( T i ) . Note that L ( C ) is a list of size exactly k d . The rank of C , r ank ( C ) , is just the numb er of line arly indep endent line ar forms in L ( C ) . 3.2 In tuition W e s et the scene, f or pro ving the r ank b ound of a ΣΠΣ ( k , d ) iden tit y , b y giving a com bi- natorial/graphical picture to ke ep in mind. Our circuits consist of k multiplica tion terms, and eac h term is a p r o duct of d linear forms. Think of there b eing k groups of d no d es, so eac h n o de corresp ond s to a form and eac h group represen ts a term 4 . W e will incremen tally construct a sm all basis for all these forms. This pro cess will b e d escrib ed as some kind of a c oloring pr o c e dur e . A t an y intermediate stage, we ha ve a partial basis of forms . These are all linearly indep end en t, an d the corresp onding no des (w e will use no de and form in terchange ably) are colored r e d . F orms not in the basis that are linear combinations of the basis forms (and are th erefore in the sp an of the basis) are colored gr e en . Once all the forms are 4 A form that app ears many times corresp onds t o that many no des. 6 colored, either green or red, all the red forms form a basis of al l forms . The n u m b er of red forms is the rank of the circuit. When we ha v e a partial basis, w e carefully c ho ose some uncolored forms and color them r ed (add th em to the basis). As a result, some other forms get “automatically” colored green (they get added to the span). W e “pa y” only for the red forms, and w e w ould lik e to get many green f orm s f or “free”. Note that w e are trying to prov e that the rank is k O (1) log d , when th e total n umber of forms is k d . Roughly sp eaking, for ev ery k O (1) forms we color red, w e need to show that the n umb er of gree n forms will double . So far nothing ingenious has b een done. Nonetheless, this imag e of coloring forms is v ery useful to get an in tuitiv e and clear idea of ho w the pro of w orks. T he m ain c h allenge comes in choosing the right forms to color r ed. Once that is done, ho w d o we k eep an accurate count on the forms that get colored green? One of the m ain conceptual con trib utions of th is w ork is th e idea of matchings , whic h aid us in these tasks. Let u s start fr om a trivial example. Sup p ose w e h a ve t wo terms th at sum to zero, i.e. T 1 + T 2 = 0. By uniqu e factorizatio n of p olynomials, for ev ery form ℓ ∈ T 1 , there is a un ique form m ∈ T 2 suc h that ℓ = cm , where c ∈ F ∗ (w e will denote th is by ℓ ∼ m ). By asso ciating the forms in T 1 to those in T 2 , we create a matching b et ween the forms in these t wo groups (or terms). This rather simple observ ation is the starting p oin t for the construction of matc hin gs. Let us no w mov e to k = 3, so we hav e a simple circuit C ≡ T 1 + T 2 + T 3 = 0. Therefore, there are no common factors in the terms. T o get matc hings, we w ill lo ok at C mo dulo some forms in T 3 . By lo oking at C mod u lo v arious f orms in T 3 , w e redu ce the fanin of C and get many matc hings. Then we can d ed uce structural resu lts ab out C . Similar id eas w ere used b y Dvir and Shpilk a [DS06] for th eir rank b ound . T aking a f orm q ∈ T 3 , we lo ok at C (mo d q ) whic h gives T 1 + T 2 = 0(mo d q ). By unique factorization of p olynomials mo dulo q , w e ge t a q -matching . Supp ose ( ℓ, m ) is an edge in this matc h ing. In terms of the coloring pro cedure, this means that if q is colored and ℓ gets colored, then m must also b e colored. A t some intermediate s tage of th e coloring, let us choose an u ncolored form q ∈ T 3 . A k ey structural lemma th at w e will prov e is that in the q -matc hin g (b et we en T 1 and T 2 ) any neigh b or of a colored form must b e unc olor e d . This crucially r equires the simplicit y of C . W e will color q red, and th us all neigh b ors of the colored form s in T 1 ∪ T 2 will b e colo red green. By coloring q red, w e ca n double the n u m b er of colored forms. It is the v arious matc hings (com bin ed with the abov e prop ert y) th at allo w us to show an exp onent ial gro w th in the colored forms as forms in T 3 are colored red. By con tin uing this pro cess, we can color all forms by coloring at most O (log d ) forms. Quite su rprisingly , the ab o v e v erb al argument can b e form alized easily to prov e th at rank of a minimal, simple circuit w ith top f anin 3 is at most (log 2 d + 2). F or this case of k = 3, the logarithmic rank b ound w as there in a lemma of Dvir and Sh p ilk a [DS06], though they did not present the pro of idea in this form, in p articular, their rank b ound grew to (log d ) 2 for k = 4. The ma jor d iﬃcult y arises when we try to p ush these argum en ts for h igher v alues of k . In essence, the ideas are the same, but there are man y tec hnical and conceptual issues that arise. Let us go to k = 4. The ﬁrst attempt is to take a form q ∈ T 4 and lo ok at C (mo d q ) as a fanin 3 circuit. Can w e no w simp ly app ly the ab o ve argumen t recursive ly , and co ve r all th e forms in T 1 ∪ T 2 ∪ T 3 ? No, the p ossible lac k of simplicit y in C (mo d q ) blo c ks this simple idea. It ma y b e the case that T 1 , T 2 and T 3 ha v e no common factors, but on ce w e go m o dulo q , there could b e many common factors! (F or example, let q = x 1 . 7 Mo dulo q , the forms x 1 + x 2 and x 2 w ould b e common f actors.) Instead of doing things r ecursiv ely (b oth [DS06] and [KS07] u sed r ecursiv e argum ents), w e lo ok at generating matc h ings iterativ ely . By p erforming a careful iterativ e analysis that k eeps trac k of man y relations b etw een the linear forms w e ac h iev e a stronger b ound f or k > 3. W e start with a form ℓ 1 ∈ T 1 , an d lo ok at C (mo d ℓ 1 ). F rom C (mo d ℓ 1 ), we remo v e all common factors. This common factor part we shall refer to as the gc d of C (mo d ℓ 1 ), the remov al of w hic h lea v es the simple part of C (mo d ℓ 1 ). No w, we c ho ose an app ropriate form ℓ 2 from the simple part, and lo ok at C (mo d ℓ 1 , ℓ 2 ). W e no w c ho ose an ℓ 3 and so on and so forth . F or eac h ℓ that we c ho ose, w e d ecrease the top fan in by at least 1, so we will end up with a matc hing mo dulo the ide al ( ℓ 1 , ℓ 2 , ..., ℓ r ), where r ≤ ( k − 2). W e call these sp ecial ideals form ide als (as they are generated by forms), and the main stru ctures that w e ﬁnd are matc hings mo dulo form ideals. Th e co loring pro cedure will color the f orms in the form ideal r ed. Of cour s e, it’s not as simple as the case of k = 3, since, for one thing, w e h av e to deal with the simp le and gcd parts. Many other problems arise, but w e will explain them as and when we see them. F or no w, it su ﬃces to understand the o verall picture and the concept of matc hings among the linear forms in C . W e now start b y setting some notation and giving some key deﬁnitions. 3.3 Ideal Matchings W e will use the concept of ide al matchings to develo p tools to p ro ve Theorem 2. In th is subsection, we pro vid e the necessary deﬁnitions and prov e some basic facts ab out these matc hin gs. First, we d iscuss similarity b et w een form s and form ide als . Deﬁnition 9. We give sever al deﬁnitio ns : • [Similar forms] F or any two p olynomials f , g ∈ R we c al l f similar to g if ther e is a c ∈ F ∗ such that f = cg . We say f is similar to g mo d I , for some ide al I of R , if ther e is a c ∈ F ∗ such that f = cg ( mo d I ) . We also denote this by f ∼ g ( mo d I ) or f is I -similar to g . • [Similar lists] L et S 1 = ( a 1 , . . . , a d ) and S 2 = ( b 1 , . . . , b d ) b e two lists of line ar forms with a bije ction π b etwe e n them. S 1 and S 2 ar e c al le d similar und er π if for al l i ∈ [ d ] , a i is similar to π ( a i ) . Any two lists of line ar forms ar e c al le d similar if ther e exists such a π . Empty lists of line ar forms ar e similar vacuously. F or any ℓ ∈ L ( R ) we deﬁne the list of forms in S 1 similar to ℓ as the fol lowing list (unique upto or dering): simi ( ℓ, S 1 ) := ( a ∈ S 1 | a is similar to ℓ ) We c al l S 1 , S 2 coprime lists i f ∀ ℓ ∈ S 1 , # simi ( ℓ, S 2 ) = 0 . • [F orm-ideal] A form-ideal I i s the ide al ( I ) of R gener ate d by some nonempty I ⊆ L ( R ) . Note that i f I = { 0 } then a ≡ b ( mo d I ) simply me ans that a = b absolutely. • [Spa n sp ( S ) ] F or any S ⊆ L ( R ) we let sp ( S ) ⊆ L ( R ) b e the linear span of the line ar forms in S over the ﬁeld F . 8 • [O rthogonal sets of forms] L et S 1 , . . . , S m b e sets of line ar forms for m ≥ 2 . We c al l S 1 , . . . , S m orthogonal if for al l m ′ ∈ [ m − 1] : sp  [ j ∈ [ m ′ ] S j  ∩ sp ( S m ′ +1 ) = { 0 } Similarly, we c an deﬁne orthogonalit y of form-ideals I 1 , . . . , I m . W e giv e a f ew simple facts b ased on these deﬁn itions. It will b e h elpful to h a ve these explicitly stated. F act 10. L et U, V b e lists of line ar form s and I b e a form-ide al. If U, V ar e similar then their sublists U ′ := ( ℓ ∈ U | ℓ ∈ sp ( I )) and V ′ := ( ℓ ∈ V | ℓ ∈ sp ( I )) ar e also similar. Pr o of. If U, V are similar th en for some c ∈ F ∗ , M ( V ) = cM ( U ). This implies: M ( V ′ ) · M ( V \ V ′ ) = cM ( U ′ ) · M ( U \ U ′ ) Since elemen ts of U \ U ′ are not in sp ( I ), for an y ℓ ∈ V ′ , ℓ do es n ot divide M ( U \ U ′ ). In other wo rds M ( V ′ ) divides M ( U ′ ), and vice ve rsa. Thus, M ( U ′ ) , M ( V ′ ) are similar and hence b y unique factorizatio n in R , lists U ′ , V ′ are similar. F act 11. L et I 1 , I 2 b e two ortho gonal form-ide als of R and let D b e a ΣΠΣ( k , d ) cir cu it such that L ( D ) has al l i ts line ar forms in sp ( I 1 ) . If D ≡ 0 ( mo d I 2 ) then D = 0 . Pr o of. As I 1 , I 2 are orthog onal w e can assume I 1 to b e { ℓ 1 , . . . , ℓ m } and I 2 to b e { ℓ ′ 1 , . . . , ℓ ′ m ′ } where the ordered set V := { ℓ 1 , . . . , ℓ m , ℓ ′ 1 , . . . , ℓ ′ m ′ } has ( m + m ′ ) linearly indep end en t linear form s. Clearly , th ere exists an in vertible linear trans f ormation τ on s p ( { x 1 , . . . , x n } ) that maps the elemen ts of V bijectiv ely , in that ord er , to x 1 , . . . , x m + m ′ . On applying τ to the equation D ≡ 0 (mo d I 2 ) w e get: τ ( D ) ≡ 0 (mo d x m +1 , . . . , x m + m ′ ) , w h ere τ ( D ) ∈ F [ x 1 , . . . , x m ] . Ob viously , this m eans that τ ( D ) = 0 wh ic h b y the inv ertibility of τ implies D = 0. W e no w come to the most im p ortan t deﬁnition of this section. W e m otiv ated the notion of ide al matchings in the intuition section. Thin king of tw o lists of linear forms as t wo sets of vertices, a matc hin g b et ween them signiﬁes some linear r elatio nship b etw een the forms mo du lo a form-ideal. Deﬁnition 12. [Ideal matching s] L et U, V b e lists of line ar forms and I b e a f orm-ide al. An ideal m atc hing π b et ween U, V b y I is a bije ction π b etwe en lists U, V suc h that: for al l ℓ ∈ U , π ( ℓ ) = cℓ + v for some c ∈ F ∗ and v ∈ sp ( I ) . The matching π is c al le d trivial if U, V ar e similar. Note that π b eing a bijection and c b eing nonzero together imply that π − 1 can also b e viewed as a matc hing b et ween V , U by I . W e will also u se the terminology I - matching b etwe en U and V for the ab o ve . S imilarly , an I -matching π b etwe en multiplic ation terms f , g is the one that matc hes L ( f ) , L ( g ). (F or con v enience, we will just sa y “matc hing” instead of “ideal matc h in g”.) The follo w ing is an easy fact ab out matc h ings. 9 F act 13. L et π b e a matching b etwe en lists of line ar f orms U, V by I a nd let U ′ ⊆ U , V ′ ⊆ V b e similar sublists. Then ther e exists a matching π ′ b etwe en U, V by I such that: U ′ , V ′ ar e similar under π ′ . Pr o of. Let ℓ ′ ∈ U ′ b e such that π ( ℓ ′ ) = d ′ ℓ ′ + v ′ (for some d ′ ∈ F ∗ and v ′ ∈ sp ( I )) is n ot in V ′ or is not similar to ℓ ′ . As V ′ is similar to U ′ there exists a form equal to αℓ ′ in V ′ , for some α ∈ F ∗ , and π b eing a matc h ing m ust b e mappin g some ℓ ∈ U to αℓ ′ in V ′ . Also from the matc hing condition ther e m u st b e some d ∈ F ∗ and v ∈ sp ( I ) suc h that π ( ℓ ) = dℓ + v = αℓ ′ . No w we deﬁn e a n ew matc hing e π by ﬂ ipping the images of ℓ and ℓ ′ under π , i.e., deﬁne e π to b e the same as π on U \ { ℓ, ℓ ′ } and: e π ( ℓ ) V := π ( ℓ ′ ) and e π ( ℓ ′ ) V := π ( ℓ ). Note that e π inherits the bijection pr op ert y from π and it is an I -matc h ing b ecause: e π ( ℓ ′ ) = αℓ ′ for α ∈ F ∗ and more imp ortantly , e π ( ℓ ) = π ( ℓ ′ ) = d ′ ℓ ′ + v ′ = d ′  dℓ + v α  + v ′ =  dd ′ α  ℓ +  d ′ v α + v ′  The form ( d ′ v α + v ′ ) is clearly in sp ( I ). Thus, we hav e obtained no w a m atc hing e π b et w een U, V by I suc h that the ℓ ′ ∈ U ′ is similar to e π ( ℓ ′ ) ∈ V ′ . Note that we incr eased the n umber of form s in U ′ that are matc hed to similar forms in V ′ . If we ﬁnd another form in U ′ that is not matc hed to a similar f orm in V ′ , we can just rep eat the ab ov e pro cess. W e will end up with the desired matc hin g π ′ in at most # U ′ man y iterations. W e are ready to present the most imp ortan t lemma of this section. The follo wing lemma sh o ws that there cann ot b e to o many matc hin gs b et w een t w o giv en nonsimilar lists of linear forms. It is at th e heart of our rank b ound pro of and the reason for the logarithmic dep end en ce of the rank on the degree. It can b e considered as an alg ebraic generalizat ion of the com b in atorial result us ed by Dvir & Shp ilk a (Corollary 2.9 of [DS06]). Lemma 14. L et U, V b e lists of line ar forms e ach of size d > 0 and I 1 , . . . , I r b e ortho gonal form-ide als such that for al l i ∈ [ r ] , ther e is a matching π i b etwe en U, V by I i . If r > (log 2 d + 2) then U, V ar e similar lists. Before giving the p r o of, let us ﬁr st p ut it in the con text of our ov erall app roac h. In the s k etc h that w e ga ve for k = 3, at eac h step, we we re generating orthogonal matc h in gs b et w een t wo terms . F or eac h orthogonal matc h ings w e got, we colo red one linear form red (added one f orm to our b asis) and double d the n umber of green forms (doubled the n umber of forms in the circuit that are in the span of the basis). This sh o wed that th ere is a logarithmic-sized basis for all L ( C ). If we tak e th e contrapositive of th is, we get that there c annot b e to o m an y orthogonal matc hings b et ween tw o (nonsimilar) lists of forms. F or dealing with larger k , it will b e con v enient to state things in this wa y . Pr o of. Let U 1 ⊆ U b e a s u blist su c h that: th ere exists a sub list V 1 ⊆ V similar to U 1 for whic h U ′ := U \ U 1 and V ′ := V \ V 1 are coprime lists. Let U ′ , V ′ b e of size d ′ . If d ′ = 0 then U, V are indeed similar and we are done already . So assume that d ′ > 0. By the h yp othesis and F act 13, f or all i ∈ [ r ], there exists a matc hing π ′ i b et w een U, V b y I i suc h that: U 1 , V 1 are similar und er π ′ i and π ′ i is a matc hing b et ween U ′ , V ′ b y I i . Ou r subsequent argument will only consider th e latter prop ert y of π ′ i for all i ∈ [ r ]. 10 In tuitiv ely , it is b est to thin k of the v arious π ′ i s as bip artite matc hings. Th e graph G = ( U ′ , V ′ , E ) h as v ertices lab elled with the resp ectiv e form. F or eac h π ′ i and eac h ℓ ∈ U ′ , w e add an (un directed) edge tagg ed with I i b et w een ℓ and π ′ i ( ℓ ). T here m a y b e man y tagged edges b et ween a pair of ve rtices 5 . W e call π ′ i ( ℓ ) the I i -neigh b or of ℓ (and vice versa, since the edges are u ndirected). Abusing n otatio n, w e u se vertex to refer to a form in U ′ ∪ V ′ . W e w ill denote S j ≤ i I i b y J i . W e will now show that there cannot b e more than (log 2 d + 2) such p erfect matc hings in G . The p ro of is done by follo w in g an iterativ e pro cess that has r phases, one for eac h I i . This is essen tially the coloring p ro cess that w e describ ed earlier. W e maintain a partial basis for the f orms in U ′ ∪ V ′ whic h will b e u p dated iterativ ely . This b asis is kept in the set B . Note that although w e only wa nt to span U ′ ∪ V ′ , w e will use forms in the v arious I i ’s for sp anning. W e s tart with emp ty B and in itialize by add ing some ℓ ∈ U ′ to B . In the i th roun d , w e will add all forms in I i to B . All forms of U ′ ∪ V ′ in sp ( { ℓ } ∪ J i ) are n o w spanned. W e then pro ceed to the n ext round. T o introd uce some colorful terminology , a gr e e n vertex is one th at is in the set sp ( B ) (a f orm in ( U ′ ∪ V ′ ) ∩ s p ( B )). Here is a nice fact : at the end of a round, the num b er of green v ertices in U ′ and V ′ are the same. Why? All forms of I 1 are in B , at the end of an y r ound. Let ve rtex v b e green, so v ∈ sp ( B ). Th e I 1 -neigh b or of v is a lin ear combinatio n of v and I 1 . Therefore, the neighbor is in sp ( B ) and is colored green. Th is shows that the num b er of green v ertices in U is equal to the n umber of those in V . Let i 0 ∈ [ r ] b e the lea st index such that { ℓ } , I 1 , . . . , I i 0 are not orthogonal, if it do es not exist then set i 0 := r + 1. No w we h a ve the follo wing easy claim. Claim 15. The sets { ℓ } , I 1 , . . . , I i 0 − 1 ar e ortho gonal and the sets: { ℓ } ∪ J i 0 , I i 0 +1 , . . . , I r ar e orth o gonal. Pr o of of Claim 15. Th e ideals { ℓ } , I 1 , . . . , I i 0 − 1 are orthogonal by the minimalit y of i 0 . As I 1 , . . . , I i 0 are orthogonal but { ℓ } , I 1 , . . . , I i 0 are not orthogonal w e deduce that { ℓ } ∈ sp ( J i 0 ). Th u s, { ℓ } ∪ sp ( J i 0 ) = sp ( J i 0 ) w h ic h is orthogonal to the sets I i 0 +1 , . . . , I r b y the orth ogonalit y of I 1 , . . . , I r .  W e now sh o w th at the green vertic es double in at least ( r − 2) many rounds. Claim 16. F or i 6∈ { 1 , i 0 } , the numb er of g r e e n vertic e s doubles in the i th r ound. Pr o of of Claim 16. Let ℓ ′ b e a green vertex, sa y in U ′ , at the end of the ( i − 1)th round ( B = { ℓ } ∪ J i − 1 ). C onsider the I i -neigh b or of ℓ ′ . This is in V ′ and is equal to ( cℓ ′ + v ) where c ∈ F ∗ and v is a nonzer o elemen t in sp ( I i ) (this is b ecause U ′ , V ′ are coprime). I f this neighbor is green, then v would b e a linear combinatio n of t wo green forms, im p lying v ∈ sp ( B ). But by Claim 15, I i is orthogonal to B , implying v ∈ sp ( B ) ∩ sp ( I i ) = { 0 } whic h is a con tradiction. Therefore, th e I i -neigh b or of an y green ve rtex is not green. On adding I i to B , all th ese neigh b ors will b ecome green. T his completes the p ro of.  5 It can b e shown, using the orthogonality of the I i ’s, that an edge can have at most two distinct tags. 11 W e s tarted oﬀ with one green ve rtex ℓ , and U ′ , V ′ eac h of size d ′ . Th is doub ling can happ en at most log 2 d ′ times, implying that ( r − 2) ≤ log 2 d ′ . Remark 17. The b ound of r = log 2 d + 2 is achievable by lists of line ar forms inspir e d by Se ction 2. Fix an o dd s and deﬁne: U := { ( b 1 x 1 + · · · + b s − 1 x s − 1 + x s ) | b 1 , . . . , b s − 1 ∈ { 0 , 1 } s.t. b 1 + · · · + b s − 1 is even } V := { ( b 1 x 1 + · · · + b s − 1 x s − 1 + x s ) | b 1 , . . . , b s − 1 ∈ { 0 , 1 } s.t. b 1 + · · · + b s − 1 is o dd } It is e asy to se e that over r ationals, # U = # V = 2 s − 2 and for al l i ∈ [ s − 1] , ther e is a matching b etwe en U, V by ( x i ) , furthermor e, ther e is a matching by ( x 1 + · · · + x s − 1 + 2 x s ) . Thus ther e ar e ( log 2 | U | + 2) many ortho gonal matchings b etwe en these nonsimilar U, V ; showing that our L emma is tight. 3.4 Ordered Matc hings and Simple P arts of Circuits Before w e delve into the d eﬁnitions and pro ofs, let us motiv ate them b y an intuitiv e explanation. 3.4.1 In tuition Our main goal is to d eal w ith th e case k > 3. T he ov erall p ictur e is still the same. W e k eep up dating a partial basis S for L ( C ). Th is pro cess go es through v arious r ounds , eac h round consisting of iter ations . A t the end of eac h roun d, w e obtain a form-ideal I that is orthogonal to S . In th e ﬁrst iteration of a round, we s tart by c ho osing a form ℓ 1 in L ( T 1 ) that is not in sp ( S ), and addin g it to I . W e lo ok at C (mo d ℓ 1 ) in the next iteration, whic h is ob viously an identit y , and try to rep eat this step. T he top fan-in has gone do wn b y at least one, or in other words, some multiplicatio n terms ha v e b ecome iden tically zero (mo d ℓ 1 ). W e will sa y that the other terms ha v e survive d . The ma jor obstacle to pro ceeding is th at our circuit is n ot simple an y more, b ecause there c an b e common factors among multiplica tion terms m o dulo ℓ 1 . Note ho w this seems to b e a d iﬃcult y , sin ce it app ears that our matc hings will not giv e us a pr op er handle on these common factors. Supp ose that form v is now a common factor. Th at means, in eve ry surviving term, there is a form that is v mo dulo ℓ 1 . So these forms can b e ℓ 1 -matc hed to eac h other! W e ha v e con verted the obstacle in to s ome kind of a partial matc hin g, wh ic h we can hop efully exploit. Let us go b ac k to C (mo d ℓ 1 ). Let u s r emo ve all common factors from th is circuit. T his stripp ed do wn ident it y circuit is the simple part, d enoted b y s im ( C mo d ℓ 1 ). The r emo ved p ortion, called the g cd part, is r eferr ed to as g cd ( C m o d ℓ 1 ). By the ab o v e obs er v ation, the g cd part has ℓ 1 -matc hings. A k ey observ ation is that all the form s in the g cd part are not similar to ℓ 1 . This is b ecause we were only lo oking at nonzero terms in C (mo d ℓ 1 ). Ha ving (somewhat) dealt with g cd ( C mo d ℓ 1 ) by ﬁnd ing I -matc hings, let us fo cus on the smaller circuit s im ( C mo d ℓ 1 ) W e try to ﬁnd an ℓ 2 ∈ L ( sim ( C mo d ℓ 1 )) that is not in sp ( S ∪ { ℓ 1 } ). Su p p ose w e can ﬁnd suc h an ℓ 2 . Then, we add ℓ 2 to I and pro ceed to the next iteration. In a giv en iteration, we start with a form-ideal I , and a circuit sim ( C m o d I ). W e ﬁnd a form 12 ℓ ∈ L ( sim ( C mo d I )) \ sp ( S ∪ I ). W e add ℓ to I (for con v enience, let us set I ′ = I ∪ { ℓ } ) and lo ok at the C (mo d I ′ ). W e no w ha v e new terms in the gcd p art, wh ic h w e can matc h through I ′ -matc hings. As w e obs er ved earlier, all the terms th at hav e f orms in I ′ are remo ved, so the terms we matc h h ere are all nonzero mo dulo I ′ . W e remo v e the g cd part to get s im ( C mo d I ′ ), and go to the next iteration with I ′ as the new I . When do es this stop? I f th ere is no ℓ in L ( sim ( C mo d I )) \ sp ( S ∪ I ), then this means that all of L ( sim ( C m o d I )) is in ou r curren t span. S o we h appily stop here w ith all the m atc hings obtained from the gcd parts. Also , if the fan-in reac hes 2, then w e can imagine that the whole circuit is itself in the g cd p ortion. A t eac h iteration, the fan-in go es down b y at least one, so we can hav e at most ( k − 2) iterations in a round , h ence the I in an y round is generated by at most ( k − 2) forms . When we ﬁn ish a round obtaining an id eal I , there are some m ultiplication terms in C that are nonzero mo d ulo I after the gc d parts in the v arious iterat ions are remo ved f rom th ese terms. Th ese we shall refer to as constituting the blo c k ing subset of [ k ], for that round. The w a y we pro ve rank b ounds is by in voki ng Lemma 14. Eac h round constructs a new orthogonal form ideal. At the end of a round, w e ha v e a set S , wh ic h is a partial basis. If S d o es not co ve r all of L ( C ), then we use the ab o ve pro cess (o f iteratio ns) to generate a form -id eal I orthogonal to S . Consider t wo term s T a and T b that su rviv e this pro cess (mo d I ). At eac h stage, when we add a form to I , w e remov e forms from T a and T b , I -matc hing them. When we stop with our form-ideal I , we can thin k of T a and T b as split in to tw o parts : one ha v in g forms from sp ( S ∪ I ), and the other wh ic h is I -matc hed. F or eac h orthogonal form-ideal w e generate, we matc h subsets of term s . W e use Lemma 14 to tell us that we cannot h a ve to o man y suc h f orm -ideals, w hic h leads to th e rank b ound . 3.4.2 Deﬁnitions W e s tart with lo oking at the particular kind of matc hings that we get. T ak e t wo terms T a and T b that survive a rou n d, wh ere we ﬁnd the form-ideal I generated by { ℓ 1 , ℓ 2 , · · · , ℓ r } . A t the end of the ﬁrst iteration, we add ℓ 1 to I . No form in L ( T a ) ∪ L ( T b ) can b e 0(mo d ℓ 1 ). W e matc h some forms in T a to T b via ℓ 1 -matc hings. Th ey are remo ved, and then w e pro ceed to the next iteration. W e no w matc h some forms via sp ( { ℓ 1 , ℓ 2 } ) matc hings and none of these f orms are in this sp an . So in eac h ite ration, the f orms that are matc hed (and then remo ved) are n on-zero mo d the partial I obtained b y that iteratio n. W e formalize this as an or der e d matching . Deﬁnition 18. [Ordered mat c hing] L et U, V b e lists of line ar forms and an or der e d set I = { v 1 , . . . , v i } b e a form-ide al having i ≥ 1 line arly indep endent line ar forms. A matching π b etwe en U, V by I is c al le d an ordered I -matc hing if : L et v 0 b e zer o. F or al l ℓ ∈ U , π ( ℓ ) = ( cℓ + w ) wher e c ∈ F ∗ , and w ∈ sp ( v 0 , . . . , v j ) for some j satisfying ℓ / ∈ sp ( v 0 , . . . , v j ) . W e add the zero elemen t v 0 , j ust to d eal with similar forms in U and V . Note that the in v erse bijection π − 1 is also an ordered matc hing b et wee n V , U by I . It is also easy to see that if π 1 and π 2 are ordered matc hings b et ween lists U 1 , V 1 and lists U 2 , V 2 resp ectiv ely b y the same ordered f orm-ideal I then their disjoint u nion , π 1 ⊔ π 2 , is an ord er ed matc h ing b et w een lists U 1 ∪ U 2 , V 1 ∪ V 2 b y I . W e will stic k to the n otatio n in Deﬁnition 18. F or con venience, let s p j := sp ( v 0 , · · · , v j ). Let π ( ℓ ) = dℓ + w , where w ∈ sp j but ℓ 6∈ sp j then the constan t d is un ique. If there we re 13 t wo such diﬀerent constants, sa y d and d ′ , th en b oth ( π ( ℓ ) − dℓ ) and ( π ( ℓ ) − d ′ ℓ ) w ould b e in sp j implying that ( d − d ′ ) ℓ ∈ sp j . That contradict s ℓ 6∈ sp j . Thus f or a ﬁxed ℓ and an ordered matc hing π , d is u niquely determin ed . Keeping the notation ab o ve, we can w ell deﬁne : Deﬁnition 19. [Scaling factor] The sc aling factor of an or der e d matching π b etwe en U and V is denote d by sc ( π ) . F or e ach ℓ ∈ U , let d ℓ b e the unique c onstant such that π ( ℓ ) = d ℓ ℓ + w , wher e w ∈ sp j but ℓ 6∈ s p j . Then sc ( π ) := Q ℓ ∈ U d ℓ . F or empty U , sc ( π ) is set to b e 1 . Deﬁnition 20. [Sub circuits and regular circuits] F or non-empty Q ⊆ [ k ] , the sub - circuit C Q of a Σ ΠΣ( k , d ) cir cu it C is the sum P j ∈ Q T j . F or a form-ide al I we c al l C Q regular mo d I if ∀ q ∈ Q , T q 6≡ 0 ( mo d I ) . W e wil l denote the c onstant factor in the multiplic ation term T q by α q ∈ F ∗ , thus T q = α q M ( L ( T q )) . W e are n o w ready to deﬁn e the g cd and sim p arts of a sub circuit. Although the ideas are qu ite s im p le and intuitiv e, we ha ve to b e careful in dealing with constant factors. Muc h of this notation has b een in tro duced for rigorous deﬁ nitions. T ak e a su b circuit C Q that is regular mo d I as we ll as an ident it y mo d I . A maximal list of forms, sa y U , that divides T q , for all q ∈ Q , is called the g cd of C Q (mo d I ). I n every T q , there is a list U q of forms that are I -similar to U . Therefore, we ha ve I -matc hings b et wee n U and U q . This is the gc d data of C Q mo dulo I , and represents that v arious matc hings that w e will later exploit. If we remo ve U q from eac h T q , th en (by accoun ting for constant s carefully) we get a simple (mo d I ) ident it y , th e sim part of C Q (mo d I ). W e formalize this b elo w. Let C Q b e regular mo du lo I . Fix a q 1 in Q . Let U b e a m aximal s u blist of L ( T q 1 ) s u c h that M ( U ) divides T q mo dulo I for all q ∈ Q . Since R/I is isomorp hic to a p olynomial ring, the nonconstan t p olynomials in R /I satisfy uniqu e factorizatio n prop er ty , i.e. any p olynomial in R that is nonconstant mod ulo I uniquely factors mo du lo the ideal ( I ) in to p olynomials irreducible mo du lo I . Since C Q is regular mo dulo I and U ⊆ L ( T q 1 ) is a maximal list such that ∀ q ∈ Q , M ( U ) | T q (mo d I ): • M ( U ) is a gcd of the p olynomials { T q | q ∈ Q } mo du lo the ideal ( I ). • F or all q ∈ Q , there exists a sublist U q ⊆ L ( T q ) and a c q ∈ F ∗ suc h th at M ( U q ) ≡ c q · M ( U ) (mo d I ). By unique f actoriza tion in R/I and r egularit y of C Q mo d I this giv es an ordered matc hing π q b et w een U, U q b y I . Also, by th e deﬁnition of scaling factor of a matc hin g, π q satisﬁes: ∀ q ∈ Q , M ( U q ) ≡ sc ( π q ) · M ( U ) (mo d I ). Note that giv en C Q and I there are man y p ossibilities to c ho ose the lists U and { U q | q ∈ Q } but th ey are all uniqu ely determined upto similarit y mo d ulo the ideal ( I ) and that will b e go o d enough for our purp oses. So w e c h o ose them in some wa y , say the lexicographically smallest one u nless sp eciﬁed otherwise, and deﬁne the gcd data. Using the gcd data of C Q mo d I w e can extract out a smaller circuit fr om C Q whic h we call the simple p art. Deﬁnition 21. [gcd and sim pa rts] The gcd data of C Q mo dulo I is the fol lowing set of # Q matchings: g cd ( C Q mo d I ) := { ( π q , U, U q ) | q ∈ Q } (1) 14 The gcd of C Q (mo d I ) is just g cd ( C Q mo d I ) := M ( U ) . The simp le part of C Q mo d I is the cir cuit: sim ( C Q mo d I ) := X q ∈ Q sc ( π q ) α q · M ( L ( T q ) \ U q ) Before a r ound, we ha v e a partial basis S . A t the en d of a round, w e p ro duce a form-ideal I that is orthogonal to S . W e call this a useful ide al . Let Q ⊂ [ k ] b e s u c h that all T q , q ∈ Q surviv e (mod I ). T h is is call ed the b lo cking subset . F or eac h suc h q , there are a list of forms V q ⊂ L ( T q ) that are mutually matc hed via ordered I -matc hings (these are really a collection of g cd datas). This is called the matching data . Even after w e remo ve V q from eac h term T q (carefully accoun ting for constan ts, as explained ab o ve), w e still ha ve an id en tit y mo d I . All forms of this identi t y are in sp ( S ∪ I ) \ sp ( I ), since we assume that th e round has end ed. F urth ermore by rearranging linear forms, all V q ’s can b e made d isjoin t to s p ( S ∪ I ) \ sp ( I ). Therefore th is round p artitions the L ( T q ) into V q and L ( T q ) ∩ ( sp ( S ∪ I ) \ sp ( I )) (for all q ∈ Q ). T hese end-of-a-round prop erties are formalized b y the f ollo wing d eﬁnition. Deﬁnition 22. [U seful ideals, blo cking subsets, and matc hing data] L et C = P j ≤ k T j , T j = α j M ( L ( T j )) . The set S ⊆ L ( R ) and I is an or der e d form-ide al ortho gonal to S . We c al l I useful in C w r t S if ∃ Q ⊂ [ k ] , 1 < # Q < k with the fol lowing pr op erties : F or al l q ∈ Q , let V q b e L ( T q ) \ ( sp ( S ∪ I ) \ sp ( I )) . (Ther efor e, L ( T q ) \ V q ⊂ sp ( S ∪ I ) \ sp ( I ) .) • Ther e exists a list of line ar forms V such that for al l q ∈ Q , ther e is an or der e d I - matching τ q b etwe en V , V q . • The cir cui t P q ∈ Q sc ( τ q ) α q · M ( L ( T q ) \ V q ) is a r e gu lar identity mo dulo I . Such a Q we c al l a blo c king subs et of C, S, I . By matc hing data of C, S, I , Q we wil l me an the set: mdata ( C, S, I , Q ) := { ( τ q , V , V q ) | q ∈ Q } We wil l c al l mdata ( C , S, I , Q ) trivial if the lists V q , q ∈ Q , ar e al l mutual ly si milar. F rom the matc hing data, w e will exploit th e fact that for eac h pair q 1 , q 2 ∈ Q , there is an ordered I -matc hing b et we en V q 1 and V q 2 . Nonetheless, w e w ill repr esen t these # Q matc hin gs via V b ecause it will b e more con v enien t to d eal with the in termediate g cd parts while we are building I . 3.4.3 Basic facts In this subsection, we pro v e some basic facts ab out ordered matc hings, s caling factors and g cd and sim parts of a circuit. These facts are not d iﬃcult to pro ve , but it will b e helpfu l later to hav e th em. The follo w ing tw o p rop erties are immediate fr om the deﬁn ition of scaling factor. F act 23. L et π 1 and π 2 b e or der e d I -matchings b etwe en lists U 1 , V 1 and lists U 2 , V 2 r e - sp e c tively. Then sc ( π − 1 1 ) = sc ( π 1 ) − 1 and sc ( π 1 ⊔ π 2 ) = sc ( π 1 ) · sc ( π 2 ) . Th us, ordered matc h ings ha v e inv erses, h a ve a union and the follo win g fact shows that they can also b e comp osed. 15 F act 24. L et π 1 and π 2 b e or der e d match ings b etwe en U 1 , V and V , U 2 r e sp e c tively by the same or der e d form-ide al I = { v 1 , . . . , v i } . Then the natur al ly deﬁne d c omp osite matching π 2 π 1 is also an or der e d match ing b etwe en U 1 , U 2 by I . F urthermor e, s c ( π 2 π 1 ) = sc ( π 1 ) · sc ( π 2 ) . Pr o of. Consider a linear form ℓ ∈ U 1 . There exists c 1 ∈ F ∗ and α 1 ∈ s p j 1 , ℓ / ∈ sp j 1 suc h that π 1 ( ℓ ) = c 1 ℓ + α 1 . Also, there exists c 2 ∈ F ∗ and α 2 ∈ sp j 2 , π 1 ( ℓ ) / ∈ sp j 2 suc h that π 2 ( π 1 ( ℓ )) = c 2 ( c 1 ℓ + α 1 ) + α 2 . Let j = max { j 1 , j 2 } . Ob viously , ( c 2 α 1 + α 2 ) ∈ sp j . If ℓ ∈ sp j then as ℓ 6∈ sp j 1 w e deduce that j = j 2 > j 1 , thus ℓ ∈ sp j 2 , imp lying π 1 ( ℓ ) = c 1 ℓ + α 1 ∈ sp j 2 , whic h is a con tr adiction. Therefore, ℓ / ∈ sp j . T his p ro ves that the composite bijection π 2 π 1 is an ordered matc hing. The contribution from th e image of ℓ ∈ U 1 to sc ( π 2 π 1 ) is c 1 c 2 while the corresp ondin g con trib utions of ℓ ∈ U 1 to sc ( π 1 ) is c 1 and of π 1 ( ℓ ) ∈ V to sc ( π 2 ) w as c 2 . Th us, sc ( π 2 π 1 ) = sc ( π 1 ) · sc ( π 2 ). The s caling factor nicely c haracterizes the ratio of M ( U ) and M ( V ) when U, V are similar. F act 25. L et π b e an or der e d matching b etwe en lists U, V of line ar forms, by an or der e d form-ide al I = { v 1 , . . . , v i } . If π is trivial then M ( V ) = sc ( π ) · M ( U ) . Thus al l the or der e d matchings, b etwe en a g i ven p air of similar lists, have the same sc aling factor. Pr o of. The pro of idea is identic al to the one seen in F act 13. Let ℓ ∈ U b e suc h that π ( ℓ ) = dℓ + v is not similar to ℓ , where d ∈ F ∗ , v ∈ sp j and ℓ / ∈ sp j . Since V is similar to U there exists a form equal to cℓ in V , for some c ∈ F ∗ . As π is an ord ered matc hing, it must b e map p ing s ome ℓ ′ ∈ U to cℓ in V , satisfying: π ( ℓ ′ ) = d ′ ℓ ′ + v ′ = cℓ , where d ′ ∈ F ∗ , v ′ ∈ sp j ′ , and ℓ ′ / ∈ sp j ′ . No w we deﬁn e a n ew matc hing e π by ﬂ ipping the images of ℓ and ℓ ′ under π , i.e., deﬁne e π to b e the same as π on U \ { ℓ, ℓ ′ } and: e π ( ℓ ) V := π ( ℓ ′ ) and e π ( ℓ ′ ) V := π ( ℓ ). The matc h in g e π is an ordered matc hin g b ecause: e π ( ℓ ) = cℓ for c ∈ F ∗ and m ore imp ortan tly e π ( ℓ ′ ) = dℓ + v = d ( d ′ ℓ ′ + v ′ c ) + v = ( dd ′ c ) ℓ ′ + ( dv ′ c + v ). Let j ∗ := max { j, j ′ } . Ob viously , ( dv ′ c + v ) ∈ sp j ∗ . If j ∗ = j ′ , we are done, b ecause we already know that ℓ ′ / ∈ sp j ′ . If j ∗ = j and ℓ ′ ∈ sp j , then cℓ = d ′ ℓ ′ + v ′ is in s p j (con tradiction). W e h a ve obtained no w an ord ered matc hin g e π b et w een U, V b y I where the num b er of forms mapp ed to a similar form h as strictly increased. Ob serv e that s c ( π ) had a un ique con trib ution of d , d ′ from the images of ℓ , ℓ ′ resp ectiv ely while sc ( e π ) has a corresp ond in g con trib ution of c , ( dd ′ c ). On all th e other elemen ts of U , e π is the same as π . Thus, w e h av e that sc ( e π ) = sc ( π ). The ab o ve pr o cess will yield an ord ered matc hing π ′ in at m ost # U man y iterations, suc h that U, V are similar under π ′ and s c ( π ′ ) = sc ( π ). But this means that, for all ℓ ∈ U , π ′ ( ℓ ) = λℓ , for some λ ∈ F ∗ . By d eﬁnition the con tribution by ℓ to sc ( π ′ ) wo uld b e then λ . This clearly implies that M ( V ) = sc ( π ′ ) · M ( U ) and ﬁnally M ( V ) = sc ( π ) · M ( U ). W e mo v e on to facts ab out the g cd and sim parts of a circuit. F act 26. If C Q is a r e g ular mo d I sub cir cuit of C then: C Q ≡ g cd ( C Q mo d I ) · sim ( C Q mo d I ) ( mo d I ) 16 A dditional ly, if C Q is an identity mo dulo I then sim ( C Q mo d I ) is a simple identity mo dulo I . Pr o of. Recall that C Q = P q ∈ Q T q and the g cd data g cd ( C Q mo d I ) is { ( π q , U, U q ) | q ∈ Q } . No w T q = α q M ( U q ) · M ( L ( T q ) \ U q ) and M ( U q ) ≡ sc ( π q ) · M ( U ) (mo d I ), w here M ( U ) is g cd ( C Q mo d I ). Th us, C Q ≡ X q ∈ Q α q sc ( π q ) M ( U ) · M ( L ( T q ) \ U q ) (mo d I ) ≡ g cd ( C Q mo d I ) · sim ( C Q mo d I ) (mo d I ) This p r o ves the ﬁrst p art. Assume n o w that C Q ≡ 0(mo d I ) whic h means s im ( C Q mo d I ) ≡ 0(mo d I ). If it is not a simple id entit y mo d I , then there is an ℓ ′ ∈ L ( sim ( C Q mo d I )) suc h that, ∀ q ∈ Q , ℓ ′ | M ( L ( T q ) \ U q ) mo d I . Then, M ( U ) cann ot b e the gcd of the p olynomials { T q | q ∈ Q } mo dulo the ideal ( I ) (con tradiction). When I = { 0 } we write gcd ( C Q ), g cd ( C Q ) and s im ( C Q ) instead of gcd ( C Q mo d I ), g cd ( C Q mo d I ) an d sim ( C Q mo d I ) resp ectiv ely . W e colle ct here some pr op erties of sim ( C Q ) that wo uld b e directly usefu l in our rank b ound pro of. F act 27. L e t ℓ ∈ L ( R ) ∗ and C Q b e a sub cir cuit of C . Then # simi ( ℓ, L ( sim ( C Q ))) > 0 iﬀ ∃ q 1 , q 2 ∈ Q such that # simi ( ℓ, L ( T q 1 )) 6 = # simi ( ℓ, L ( T q 2 )) . Pr o of. Note that # simi ( ℓ, L ( T q )) is the highest p o wer of ℓ that divid es T q . Th us, if # simi ( ℓ, L ( T q )) is the same, s ay r , f or all q ∈ Q then the highest p o wer of ℓ dividing g cd ( C Q ) is also r implying that for all q ∈ Q , the p olynomial T q g cd ( C Q ) is coprime to ℓ . By deﬁnition of the simple part of C Q this means th at # simi ( ℓ, L ( sim ( C Q ))) = 0. Con v ersely , if for an ℓ ∈ L ( R ) ∗ , ∃ q 1 , q 2 ∈ Q such that # simi ( ℓ, L ( T q 1 )) > # simi ( ℓ , L ( T q 2 )) then it is easy to see that T q 1 g cd ( C Q ) cannot b e coprime to ℓ . Th is implies that # simi ( ℓ, L ( sim ( C Q ))) > 0. F act 28. L et S ⊆ L ( R ) and Q 2 ⊆ Q 1 ⊆ [ k ] . If L ( sim ( C Q 1 )) has al l its line ar forms in sp ( S ) , then al l the line ar forms in L ( sim ( C Q 2 )) ar e also in s p ( S ) . Pr o of. F or an arbitrary ℓ ∈ L ( sim ( C Q 2 )), by F act 27, there are q 1 , q 2 ∈ Q 2 suc h that # simi ( ℓ, L ( T q 1 )) 6 = # simi ( ℓ, L ( T q 2 )). As q 1 , q 2 ∈ Q 1 , w e can again app ly F act 27 to deduce that # s imi ( ℓ , L ( sim ( C Q 1 ))) > 0. Therefore ℓ ∈ sp ( S ). F act 29. L et S ⊆ L ( R ) and Q 1 , Q 2 ⊆ [ k ] such that Q 1 ∩ Q 2 6 = φ . If L ( sim ( C Q 1 )) and L ( sim ( C Q 2 )) have al l their line ar forms in sp ( S ) then al l the line ar forms in L ( sim ( C Q 1 ∪ Q 2 )) ar e also in sp ( S ) . Pr o of. T ak e q 0 ∈ Q 1 ∩ Q 2 and an arbitrary ℓ ∈ L ( sim ( C Q 1 ∪ Q 2 )). By F act 27, there are q 1 , q 2 ∈ Q 1 ∪ Q 2 suc h that # simi ( ℓ, L ( T q 1 )) 6 = # simi ( ℓ, L ( T q 2 )). If q 1 , q 2 are in the same set (wlog , in Q 1 ), then F act 27 tells us that # simi ( ℓ , L ( sim ( C Q 1 ))) > 0, trivially implying that ℓ ∈ sp ( S ). Now assum e wlog th at q 1 ∈ Q 1 , q 2 ∈ Q 2 . F or some i ∈ { 1 , 2 } , # simi ( ℓ, L ( T q 0 )) 6 = # simi ( ℓ, L ( T q i )). Therefore, by F act 27, ℓ ∈ sp ( S ). 17 3.5 Getting Useful F orm-ideals Giv en a set S that do es not sp an all of L ( C ), we can ﬁn d a form-ideal that is useful wr t S . As w e ment ioned earlier, in a r ound we s tart with S , and end up with a useful I thr ough v arious iterations. W e will f ormally describ e this pro cess b elo w. An iteration starts with a partial I , and a simp le regular identit y E in the ring R /I , whic h h as multiplic ation terms with indices in [ k ]. At least one of the forms in E is not in sp ( S ∪ I ). A t th e b eginnin g of the ﬁrst iteration, E is set to C and I is { 0 } . A single itera tion 1. Let ℓ b e a form in E that is not in sp ( S ∪ I ). 2. Ad d ℓ to I . 3. C on s ider E mo dulo I and let Q b e th e subset of indices of nonzero m u ltiplicatio n terms. 4. Let U b e the g cd of E (mo d I ), and let the gcd d ata b e g cd = { ( π q , U, U q ) | q ∈ Q } . 5. If the fanin , | Q | , of E (mo d I ) is 2, stop the round . 6. If all forms in sim ( E (mo d I )) are con tained in sp ( S ∪ I ), stop the roun d. Oth er w ise, set E to b e sim ( E (mo d I )) and go to the next iteration. Lemma 30. L et C b e a simple ΣΠΣ( k , d ) identity in R . Supp ose S ⊆ L ( R ) and L ( C ) \ sp ( S ) is non-empty. Then ther e is a form-ide al I useful in C wrt S . Pr o of. As discussed b efore in the intuition, we generate I in one round and the pr o of w ill b e done by indu ction on the n umb er of iterations in this r ound. F or con v enience, w e set the end of th e zero iteration to b e the b eginning of the round . W e will prov e the follo win g claim: Claim 31. Consider the end of some i ter ation. Ther e exists a list V of f orms such that : for al l q in the curr ent Q , ther e is a list V q ⊆ L ( T q ) that has an or der e d I - matching to V . F urthermor e, M ( L ( T q ) \ V q ) i s similar to the term indexe d by q in sim ( E ( mo d I )) . Pr o of of Claim 31. This is prov en by induction on the iterations. At the end of the zero iteration, E is ju st C and I = { 0 } . By the simplicit y of C , s im ( E (mo d I )) is ju st C , and Q = [ k ]. So all th e V q ’s can b e tak en jus t empty . No w, su pp ose that at the end of the i th iteration, w e h a ve an ordered I -matc hing from V q to V for all q in the curr en t Q . In the ( i + 1)th iteration we will d enote by I ′ the set I ∪ { ℓ } , E ′ = sim ( E (mo d I )), and Q ′ ⊂ Q the su bset of in dices of non-zero terms in E ′ mo dulo I ′ . F or a q ∈ Q ′ , we hav e a list V q ⊆ L ( T q ) and an ordered I -matc h in g τ q b et w een V , V q . All forms of T q not in V q are in E ′ . No w consider the I ′ -matc hing π q b et w een U, U q obtained in this iteration. No forms in these can b e in sp ( I ′ ), since U is g cd ( E ′ (mo d I ′ )) and q ∈ Q ′ . Therefore, π q is an order ed matc hing. W e can tak e th e disj oin t u nion of these matc hin gs to get an ordered I ′ -matc hing τ q ⊔ π q b et w een V ∪ U and V q ∪ U q . All forms in L ( T q ) \ ( V q ∪ U q ) are in the q th term of sim ( E ′ (mo d I ′ )). This completes the pr o of of the claim.  18 The num b er of iterations in a roun d is at most ( k − 2). This is b ecause afte r eac h iteration, the fanin of the circuit E goes do wn b y at least 1. T herefore, there must b e a last iteration (signifying the end of the round ). C onsider the end of the last iteration. If the fanin | Q | of E (mo d I ) is 2, then by unique factorizati on, sim ( E (mo d I )) is empty . So, all the forms in sim ( E (mo d I )) are in s p ( S ∪ I ), at the end of a r ou n d. By the previous claim, there is a list V such that for eve ry su rviving q ∈ Q , there is a sublist V q ⊆ L ( T q ) and an ordered I -matc h ing τ q b et w een V and V q . By F act 26, we ha v e that E (mo d I ) is P q ∈ Q sc ( τ q ) α q · M ( L ( T q ) \ V q ) and is an iden tit y (in R/I ). Let V ′ q := V q \ ( sp ( S ∪ I ) \ sp ( I )) (similarly , deﬁ n e V ′ ). Note that τ q induces a m atc hing τ ′ q b et w een V ′ and V ′ q . F ur thermore, P q ∈ Q sc ( τ ′ q ) α q · M ( L ( T q ) \ V ′ q ) is a multiple of E (mo d I ) and is regular (eac h term in th e ab o ve sum is non-zero mo d I ). T h us, form- ideal I is useful in C wrt S . T o prov e a rank b ound for minimal and simple ΣΠΣ( k, d ) identit y C , our plan is to start with S = φ and exp an d it rou n d-b y-round by add ing the forms of a form-ideal, useful in C wr t S , to the cur r en t S . T r ivially , suc h a pr o cess has to stop in at most k d iterations (o ver all rounds) but we in tend to sho w that it actually ends up, co vering all the forms in L ( C ), in a m u c h faster w ay . T o formalize this pro cess w e n eed the notion of a chain of form-ide als . This is j ust a concise represent ation of the matc hings that w e get from the v arious round s. Deﬁnition 32 . [Chain of form-ideals] L e t C b e a ΣΠΣ( k , d ) cir cuit. We deﬁne a c h ain of f orm -ideals for C to b e the or der e d set T := { ( C, S 1 , I 1 , Q 1 ) , . . . , ( C , S m , I m , Q m ) } wher e, • F or al l i ∈ [ m ] , S i ⊆ L ( R ) , I i is a form-ide al ortho gonal to S i and Q i ⊆ [ k ] . • S 1 = φ and for al l 2 ≤ i ≤ m , S i = S i − 1 ∪ I i − 1 . • F or al l i ∈ [ m ] , I i is useful in C wrt S i . • F or al l i ∈ [ m ] , Q i is a blo cking subset of C , S i , I i . We wil l use sp ( T ) to me an s p ( S m ∪ I m ) and # T to denote m , the length of T . The chain T is maximal if L ( C ) ⊆ s p ( T ) . Note that by Lemma 30, if a c hain T of length m is not maximal, then w e can ﬁnd a form-ideal I m +1 that is u seful wrt S m ∪ I m . T his allo ws us to add a new ( C , S m +1 , I m +1 , Q m +1 ) to this c h ain. It is easy to construct a maximal c h ain f or C , and the length of this can b e used to b ound the r an k : F act 33. L et C b e a simple ΣΠΣ( k , d ) identity. Then ther e exists a maximal chain of form-ide als T for C . The r ank of C is at most ( k − 2)(# T ) . Pr o of. W e start with S 1 = φ and an ℓ ∈ L ( C ). By Lemma 30 there is a f orm-ideal I 1 (con taining ℓ ) u seful in C wrt S 1 with blo cki ng subset, sa y , Q 1 . So we ha v e a c hain of form-ideals { ( C, S 1 , I 1 , Q 1 ) } to start w ith . No w if L ( C ) h as all its elemen ts in sp ( S 1 ∪ I 1 ) then the c hain cannot b e extended an y further an d w e are done. Otherwise, we can again apply Lemma 30 to get a form-ideal I 2 useful in C wrt S 2 := S 1 ∪ I 1 with blo c king su bset, 19 sa y , Q 2 . Th us, we hav e a longer chain of form-ideals { ( C, S 1 , I 1 , J 1 ) , ( C , S 2 , I 2 , J 2 ) } no w. W e keep rep eating till w e ha ve a c hain of length m wh ere L ( C ) ⊆ s p ( S m ∪ I m ). Note th at S m ∪ I m = S i ≤ m I m . Eac h I i is generated by at most ( k − 2) forms, so there is a basis for L ( C ) ha ving at m ost ( k − 2) m forms. W e come to a stronger v ersion of the main theorem of this p ap er. Theorem 34. If C is a simple and minimal Σ ΠΣ( k , d ) identity then the length of any maximal chain of form-ide als for C is at most  k 2  (log 2 d + 3) + ( k − 1) . This theorem with F act 33 imply the main result, Theorem 2. W e prov e this theorem in the next s ection. 3.6 Coun t ing all Matc hings: Pro of of T heorem 34 Let a maximal c h ain of form-ideals T for C b e { ( C, S 1 , I 1 , J 1 ) , . . . , ( C, S m , I m , J m ) } . W e will partition the element s of the chain in to three types according to prop erties of the matc hin gs that they represent. Eac h of these t yp es will b e coun ted separately . W e ﬁ rst set some notation b efore explaining the d iﬀeren t typ es. Let the m matc hings data b e: mdata ( C, S i , I i , Q i ) =: { ( τ i,q , V i , V i,q ) | q ∈ Q i } W e will use mdata i as shorthand for the ab ov e. F or all q ∈ Q i , V i,q is a su blist of L ( T q ) and τ i,q is an ordered matc hing b et ween V i , V i,q b y I i . By the deﬁnition of u seful-ness of f orm -ideal I i w e ha ve that V i,q is d isjoin t to sp ( S i ∪ I i ) \ sp ( I i ). Th u s, V i,q can b e partitioned into tw o sublists: V i,q , 0 := ( ℓ ∈ V i,q | ℓ ∈ sp ( I i )) , and V i,q , 1 := ( ℓ ∈ V i,q | ℓ 6∈ sp ( S i ∪ I i )) . and analogously V i can b e partitioned int o t wo sublists V i, 0 and V i, 1 . It is easy to see th at these partitions induce a corresp onding partition of τ i,q as τ i,q , 0 ⊔ τ i,q , 1 , w here τ i,q , 0 (and τ i,q , 1 ) is an order ed matc h in g b et ween V i, 0 , V i,q , 0 (and V i, 1 , V i,q , 1 ) b y I i . Here are the thr ee t yp es of mdata i ’s: 1. [Type 1] Th ere exist q 1 , q 2 ∈ Q i suc h that V i,q 1 , 1 is not similar to V i,q 2 , 1 . 2. [Type 2] T here exist q 1 , q 2 ∈ Q i suc h that V i,q 1 is not similar to V i,q 2 , bu t for all r 1 , r 2 ∈ Q i , V i,r 1 , 1 and V i,r 2 , 1 are similar. 3. [Type 3] F or all q 1 , q 2 ∈ Q i , V i,q 1 is sim ilar to V i,q 2 . In other wo rds, mdata i is trivial. W e partition [ m ] in to sets N 1 , N 2 , N 3 , w hic h are the index sets for the mdata of t yp es 1 , 2 , 3 resp ectiv ely . 20 3.6.1 Bounding # N 1 and # N 2 The domin ant term in Theorem 34 comes from # N 1 . If # N 1 is large, then by an a veragi ng argumen t, for some pair ( a, b ), w e ﬁnd many matc h ings b et w een forms in T a and T b . These are all orthogonal matc hings, bu t are deﬁned on diﬀer ent sub lists of L ( T a ) and L ( T b ). Nonetheless, w e can ﬁn d t w o dissimilar lists that are matc hed to o m any times. Inv oking Lemma 14 giv es u s the required b ound. Lemma 35. # N 1 ≤  k 2  (log 2 d + 2) . Pr o of. F or the sak e of con tradiction, let us assume # N 1 >  k 2  (log 2 d + 2). F or eac h mdata i ( i ∈ N 1 ), c h o ose an unord ered pair of indices P i = { q 1 , q 2 } suc h that V i,q 1 , 1 and V i,q 2 , 1 are not similar. As there can b e only  k 2  distinct pairs, we get by an a veragi ng argument that, s > (log 2 d + 2) of the P i ’s are equal. Let P i 1 = · · · = P i s = { a, b } for i 1 < · · · < i s ∈ N 1 . No w we will fo cus our atten tion solely on th e ordered matc h ings µ i := τ i,b, 1 τ − 1 i,a, 1 b et w een V i,a, 1 , V i,b, 1 b y I i , for all i ∈ { i 1 , . . . , i s } . The source of con tradiction is the fact that all these matc hings are also wel l d eﬁned on the ‘last’ pair of s ublists V i s ,a, 1 , V i s ,b, 1 : Claim 36. F or al l i ∈ { i 1 , . . . , i s } , µ i induc es an or der e d matching b etwe en V i s ,a, 1 , V i s ,b, 1 by I i . Pr o of of Claim 36. The cla im is true for i = i s so let i < i s . T he matc hing µ i is an ordered I i -matc hing b et ween V i,a, 1 , V i,b, 1 . F or ℓ ∈ V i s ,a, 1 , ℓ 6∈ sp ( S i s ∪ I i s ). Since i < i s and L ( T a ) \ V i,a, 1 ⊂ sp ( S i ∪ I i ), ℓ cannot b e in L ( T a ) \ V i,a, 1 . Therefore, ℓ is in V i,a, 1 . So µ i maps ℓ to some elemen t in V i,b, 1 , sho wing µ i is deﬁn ed on the domain V i s ,a, 1 . So we know µ i maps ℓ ∈ V i s ,a, 1 to an elemen t µ i ( ℓ ) ∈ V i,b, 1 . As µ i is an I i -matc hing, µ i ( ℓ ) = ( cℓ + α ) for some c ∈ F ∗ and α ∈ sp ( I i ) ⊆ sp ( I i s ), thus µ i ( ℓ ) 6∈ sp ( S i s ∪ I i s ) (recall ℓ 6∈ sp ( S i s ∪ I i s )). Th us µ i ( ℓ ) cannot b e in L ( T b ) \ V i s ,b, 1 (whic h has all its elemen ts in sp ( S i s ∪ I i s )). As to b egin with µ i ( ℓ ) ∈ L ( T b ) we get that µ i ( ℓ ) ∈ V i s ,b, 1 . Th us, µ i maps an arb itrary ℓ ∈ V i s ,a, 1 to µ i ( ℓ ) ∈ V i s ,b, 1 . In other words, µ i induces an ordered matc hing b et ween V i s ,a, 1 , V i s ,b, 1 b y I i .  This claim means th at there are s > (log 2 d + 2) bip artite m atc hings b et w een V i s ,a, 1 , V i s ,b, 1 b y orth ogonal form-ideals I i 1 , . . . , I i s resp ectiv ely . Lemma 14 implies that the lists V i s ,a, 1 , V i s ,b, 1 are similar. This con tradicts the deﬁnition of P i s . Thus, # N 1 ≤  k 2  (log 2 d + 2). F or dealing with # N 2 , we use a sligh tly diﬀeren t argumen t to get a b etter b ound. W e sho w that a T yp e 2 matc hing can inv olve a pair of terms at most once. Lemma 37. # N 2 ≤  k 2  . Pr o of. F or the sak e of con tradiction, assume # N 2 >  k 2  . F or eac h mdata i ( i ∈ N 2 ), let P i b e an unord ered pair ( q 1 , q 2 ) s u c h that V i,q 1 is not similar to V i,q 2 . Note th at b ecause V i,q 1 , 1 is similar to V i,q 2 , 1 , it must b e that V i,q 1 , 0 is not s im ilar to V i,q 2 , 0 . By the p igeon-hole principle, at least t w o P i ’s are the same. Su pp ose P i 1 = P i 2 = { a, b } for i 1 < i 2 ∈ N 2 . Let ℓ ∈ V i 2 ,a, 0 then by the deﬁ n ition of V i 2 ,a, 0 w e hav e that ℓ ∈ sp ( I i 2 ). This coupled with i 1 < i 2 means th at ℓ cannot b e in L ( T a ) \ V i 1 ,a, 1 (whic h has all its element s in sp ( S i 1 ∪ I i 1 )). As to b egin with ℓ ∈ L ( T a ) we get that ℓ ∈ V i 1 ,a, 1 . Thus, V i 2 ,a, 0 ( V i 2 ,b, 0 ) is 21 a sublist of V i 1 ,a, 1 ( V i 1 ,b, 1 ). F rom the usefu l-ness of I i 2 , the su blist V i 2 ,a, 0 ( V i 2 ,b, 0 ) collects all the linear form s in L ( T a ) ( L ( T b )) that are in sp ( I i 2 ) while from the useful-ness of I i 1 the sub list L ( T a ) \ V i 1 ,a, 1 ( L ( T b ) \ V i 1 ,b, 1 ) is disjoin t fr om sp ( I i 2 ). Thus, the s ublist V i 2 ,a, 0 ( V i 2 ,b, 0 ) collects all th e linear forms in V i 1 ,a, 1 ( V i 1 ,b, 1 ) that are in sp ( I i 2 ). This tog ether with th e sim ilarity of V i 1 ,a, 1 and V i 1 ,b, 1 giv es us (b y F act 10) that V i 2 ,a, 0 and V i 2 ,b, 0 are similar, which contradicts the wa y P i 2 = { a, b } was deﬁned. Thus, # N 2 ≤  k 2  . 3.6.2 Bounding # N 3 This requires a d iﬀeren t argum en t th an the pigeon-hole ideas us ed for # N 1 and # N 2 . W e divide these t yp e 3 m atc hings further into internal and external ones. Ou r ﬁnal aim is to pro v e : Lemma 38. # N 3 ≤ ( k − 1) W e shall use a com b inatorial pictur e of ho w the c hain of form-id eals connects the v arious multiplicatio n terms through matc hings. W e will d escrib e an ev olving for est F and only deal w ith T yp e 3 m data i . Initially , the f orest F consists of k isolated v ertices, eac h repr esen ting the k terms T 1 , · · · , T k . W e pr o cess eac h mdata i in increasing order of the i ’s, and up date the forest F accordingly . W e will refer to this as adding mdata i to F . A t any intermediate state, the forest F w ill b e a collection of ro oted trees with a total of k lea v es. Deﬁnition 39. Consider F wh en mdata i is pr o c esse d. If al l of Q i b elongs to a single tr e e in F , then mdata i is c al le d in ternal . Otherwise, it is c al le d extern al . If mdata i is in ternal, F r emains unc hanged. While eac h time w e encoun ter an external mdata i , we up date the forest F as follo ws. W e create a new root no de labelled with mdata i (abusing notati on, w e refer to mdata i as a no de), and for an y tree of F that con tains a T q , q ∈ Q i , w e make th e ro ot of this tree a c hild of mdata i . F act 40. The total nu mb er of external matchings is at most ( k − 1) . Pr o of. Note that eac h external mdata i reduces th e num b er of trees in the forest F by at least on e. As initially F has k trees and at ev ery p oin t of the pr o cess it will hav e at least one tree, we get the claim. It remains to coun t the n umber of int ernal matchings. Whenever w e encounter an in ternal mdata i , we can alw ays asso ciate it w ith some ro ot mdata i ′ of F suc h that i ′ < i and all of Q i is in the tr ee ro oted at mdata i ′ . Lemma 41. If mdata i is internal, then the su b cir cuit C Q i is identic al ly zer o in R . Ther e - for e, by the minimality of C , no mdata i c an b e internal. This lemma with the previous fact immediately imp ly that # N 3 ≤ ( k − 1). W e no w set the stage to prov e this lemma. T ake any T yp e 3 mdata i . By the trivialit y of mdata i , the lists in { V i,q | q ∈ Q i } are m utually similar. By the useful-ness of I i the lists in { L ( T q ) \ V i,q | q ∈ Q i } h av e all their forms in sp ( S i ∪ I i ) \ sp ( I i ). F u rthermore, D i := P q ∈ Q i sc ( τ i,q ) α q M ( L ( T q ) \ V i,q ) is a r egular identit y mo dulo I i . Our aim is to r emo ve the forms in D i whic h are common factors ( not mo d I i , but mo d 0). This giv es us a new circuit 22 (quite naturally , th at w ill turn out to b e sim ( C Q i )) that is still an id entit y (mo d I i ). In other words, start with the sub circuit C Q i , and remo ve all common factors f r om th is sub circuit. This is exp ected to b e b oth sim ( C Q i ) and an iden tit y mod I i . Using this we will actually show that if mdata i is internal then sim ( C Q i ) is an ident it y (mo d 0). T hen w e can multi ply the common factors back, an d C Q i w ould b e an absolute iden tit y (violating m inimalit y of C ). W e pr o ceed to sho w this rigorously . W e hav e to carefully deal w ith ﬁeld constan ts to ensure that sim ( C Q i ) is indeed a factor of D i . Claim 42. F or T yp e 3 mdata i , the c ir cu it sim ( C Q i ) i s an identity mo d I i and has al l its forms in sp ( S i ∪ I i ) . Pr o of. Let the gcd data of D i b e: g cd ( D i ) := { ( π i,q , U i , U i,q ) | q ∈ Q i } where U i,q is a sublist of L ( T q ) \ V i,q and π i,q is an ordered matc hing b et ween U i , U i,q b y { 0 } . Note that this is not m o d I i , ev en though D i is an identit y only mo d I i . By F acts 23 and 26 w e can ‘stitc h’ U ’s and V ’s to get: • τ ′ i,q := τ i,q ⊔ π i,q is an ordered matc hing b et ween V ′ i := V i ∪ U i , V ′ i,q := V i,q ∪ U i,q b y I i . • D ′ i := P q ∈ Q i sc ( τ ′ i,q ) α q M ( L ( T q ) \ V ′ i,q ), is a r egular iden tity mo dulo I i . Let q m b e the min im u m elemen t in Q i . W e hav e that τ ′ i,q τ ′− 1 i,q m is an ordered I i -matc hing b et w een th e sim ilar lists V ′ i,q m , V ′ i,q . By F act 25, w e can constru ct an ordered m atching µ i,q b et w een V ′ i,q m , V ′ i,q b y { 0 } , with scaling factor equal to sc ( τ ′ i,q τ ′− 1 i,q m ) = sc ( τ ′ i,q ) /sc ( τ ′ i,q m ). The wa y D ′ i is constructed it is clear that D ′ i is a sim p le circuit. This combined with the similarit y of V ′ i,q m , V ′ i,q under µ i,q implies that the follo wing set of # Q i matc hin gs:  ( µ i,q , V ′ i,q m , V ′ i,q ) | q ∈ Q i  is a gcd d ata of C Q i mo dulo (0) and th e corresp onding simple p art is: sim ( C Q i ) = X q ∈ Q i sc ( µ i,q ) α q M ( L ( T q ) \ V ′ i,q ) = X q ∈ Q i sc ( τ ′ i,q ) sc ( τ ′ i,q m ) α q M ( L ( T q ) \ V ′ i,q ) = 1 sc ( τ ′ i,q m ) · D ′ i Th us, sim ( C Q i ) is a regular identit y mo d I i as w ell. Also, by the useful-ness of I i , sim ( C Q i ) has all its forms in sp ( S i ∪ I i ). Th is completes the pr o of. W e now u se the structure of F to sh o w relationships b et ween the v arious connected terms. Claim 43. At some stage, let mdata i b e a r o ot no de of F . L et X b e a subset of the le aves of mdata i . Then L ( sim ( C X )) is a subset of s p ( S i ∪ I i ) . 23 Pr o of. Let the indices of all the extern al Typ e 3 mdata b e (in order) i 1 , i 2 , · · · . W e p ro v e the claim by ind u ction on the ord er in wh ic h F is p r o cessed. F or the base case, let i := i 1 . Consider F j ust after mdata i is added. Th e lea ve s of mdata i are all in Q i . By Claim 42, L ( sim ( C Q i )) ⊂ sp ( S i ∪ I i ). An y X is a subset of Q i . By F act 28, L ( sim ( C X )) ⊂ sp ( S i ∪ I i ). F or the ind uction step, consider an external mdata i . When this is pro cessed, a series of trees r o oted at mdata j 1 , m data j 2 , · · · will b e made c hildren of mdata i . Ev ery j r is less than i . Let Y r denote the lea v es of the tree mdata j r . Note that Y r ∩ Q i 6 = φ . By the induction hypothesis, L ( sim ( C Y r )) is a sub set of sp ( S j r ∪ I j r ) ( ⊂ sp ( S i ∪ I i )). Let Z 1 b e Q i ∪ Y 1 . By F act 29 applied to sim ( C Y 1 ) and sim ( C Q i ), w e hav e that L ( sim ( C Z 1 )) is in sp ( S i ∪ I i )). Let Z 2 b e Z 1 ∪ Y 2 . W e can apply the same argument to sh o w that L ( sim ( C Z 2 )) is in sp ( S i ∪ I i )). With rep eated applications, w e ge t that for Z = S r Y r , L ( sim ( C Z )) ⊂ sp ( S i ∪ I i )). Note that Z is the set of all lea ves of the tree ro oted at mdata i . By F act 28, L ( C X ) ⊂ sp ( S i ∪ I i ), completing th e pro of. W e are ﬁnally armed with all th e tools to p ro ve Lemma 41. Pr o of. (of Lemma 41) Consider some int ernal mdata i . All the elemen ts of Q i are lea v es in the tree ro oted at some mdata j , for j < i . By Claim 43, L ( sim ( C Q i )) ⊂ sp ( S j ∪ I j ). But by Claim 42, s im ( C Q i ) ≡ 0 (mo d I i ). Since I i is orth ogonal to sp ( S j ∪ I j ), F act 11 tells us that s im ( C Q i ) is an identit y (mo d 0). Therefore, C Q i is an identit y . 3.7 F actors of a ΣΠΣ( k, d ) Circuit: Pro of of Theorem 5 The ideal matc hing tec hn ique is quite robust and can b e u sed to p ro ve Theorem 5. Let C b e a simp le, minimal, nonzero circuit with top fanin k and d egree d (so the diﬀerent terms ma y hav e diﬀeren t degrees) that computes a p olynomial p ( x 1 , · · · , x n ). W e remin d the reader of the deﬁnition of L ( p ). Let us factorize p in to Q i q i , where eac h q i is irreducible. Then L ( p ) den otes the set of line ar factors of p (that is, q i ∈ L ( p ) if q i is linear). F or an y q ∈ L ( p ), C ≡ 0 (mo d q ), th erefore w e can generate a form-ideal u seful in C in v olving q . Using these w e can create a c hain of form-ideals whose span conta ins L ( p ), and all our coun ting lemmas for the matc h ings of types 1 , 2 , 3 will follo w. As a result, w e get a b oun d of ( k 3 log d ) on the rank of L ( p ). 4 Concluding Remarks It w ould b e v ery in teresting to lev erage the matc hing tec hniqu e to design ident it y testing algorithms. By unique f actoriza tion, matc hin gs can b e easily detected in p olynomial time, and it is also not hard to searc h for I -matc hings inv olving a sp eciﬁc set of forms in I . W e prov e th at depth-3 identitie s exhibit structural prop erties describ ed b y the ideal matc hin gs. Can w e reve rse these theorems? In other w ords, can w e sh o w that certain collect ions of matc hings are present iﬀ C is an identit y? This wo uld lead to a p olynomial time identi t y tester for al l d epth-3 circuits. There is s till a gap b et ween our u pp er b oun d for the rank of O ( k 3 log d ) and the lo we r b ound of Ω( k log d ). W e feel that k log d is the right answer and a more careful analysis of the matchings could prov e this. More in terestingly , it is conjectured that when the c h aracteristic of the base ﬁeld is 0, the r an k is O ( k ), indep endent of d . W e b eliev e th at an adapation of our matc hing tec hniques to c haracteristic 0 ﬁelds could lead to such a b oun d. 24 References [AB03] M. Agra wal and S. Biswa s. Pr imalit y and iden tit y testing via c hinese remaind er- ing. JACM , 50(4):4 29–443 , 2003. [Agr05] M. Agra w al. Pro ving lo w er b oun ds via pseud o-random generators. In Pr o c e e dings of the 25th Annual F oundations of Softwar e T e chnolo gy and The or etic al Computer Scienc e (FSTTCS) , pages 92–105, 2005. [A V08] M. Agraw al and V. Vina y . Arithmetic circuits: A c h asm at depth f ou r . 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An Almost Optimal Rank Bound for Depth-3 Identities

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