cs.CC 2009-12-15 0

Deterministic Identity Testing of Read-Once Algebraic Branching Programs

In this paper we study polynomial identity testing of sums of $k$ read-once algebraic branching programs ($\Sigma_k$-RO-ABPs), generalizing the work in (Shpilka and Volkovich 2008,2009), who considered sums of $k$ read-once formulas ($\Sigma_k$-RO-fo…

Authors: Maurice Jansen, Youming Qiao, Jayalal Sarma

Deterministic Iden tit y T esting of Read-Once Algebraic Branc hing Programs Maurice Jansen ∗ Y ouming Qiao ∗ Ja yalal Sarma M.N. ∗ No ve m b er 4, 2018 Abstract In this pap er w e study p olynomial identit y testing o f sums of k re a d-once algebraic branc hing progra ms (Σ k -RO-ABPs), generalizing the work of Shpilk a and V olko vich [1 , 2], who co nsidered sums of k rea d-once for m ulas (Σ k -RO-form ulas). W e show that Σ k -RO-ABPs are strictly more powerful than Σ k -RO-form ulas, for any k ≤ ⌊ n/ 2 ⌋ , where n is the num ber of v a riables. Nev- ertheless, a s a sta rting obser v ation, we show that the genera tor given in [2] for testing a single R O-formula also works aga inst a single R O -ABP . F o r the main technical part of this pa per, we develop a prop ert y of p olynomials called alignment . Us ing this pr operty in conjunction with the har dness of r epr esentation appr o ach of [1, 2], we o bta in the following results for identit y testing Σ k -RO-ABPs, pr o vided the under lying field ha s eno ugh ele m ent s (more than k n 4 suffices): 1. Given free acce ss to the RO-ABPs in the sum, w e get a deterministic algor ithm that r uns in time O ( k 2 n 7 s ) + n O ( k ) , w he r e s bo unds the siz e of any larges t RO-ABP given o n the input. This implies we ha ve a deter ministic p olynomial time algorithm for testing whether the s um of a consta nt num b er of RO-ABPs computes the zero p olynomial. 2. Given bla ck-b o x a ccess to the RO-ABPs computing the individual po lynomials in the sum, we get a deterministic algor ithm that runs in time k 2 n O (log n ) + n O ( k ) . 3. Finally , g iv en only black-box access to the p olynomial co mpu ted by the sum of the k R O-ABPs, we o btain a n n O ( k +log n ) time deter ministic algor it hm. Items 1. and 3. ab o ve streng then tw o ma in results of [2] (Theor e ms 2 and 3, resp ectiv ely , for the case of non-prepro cessed Σ k -RO-form ulas). 1 In tro duction In this pap er w e m ake con tributions to the program of constructing in c reasingly more p o wer- ful pseud o -random generators useful against arithmetic circuits. As argued by Agra w al [3], this program is an approac h to wards resolving V alian t’s Hyp othesis, whic h states th a t the algebraic complexit y classes VP and VNP are d istin ct. Cen tr a l to this pr o gram is the PIT problem: giv en an arithm e tic circuit C with input v ari- ables x 1 , x 2 . . . x n o ve r a field F , test if C ( x 1 , x 2 , . . . , x n ) computes the zero p olynomial in the r in g ∗ Institute for Theoretical Computer Science, Tsingh u a Universit y , Beiji ng, P .R. China. Email: maurice.ju lien.jans en@gmail. com , jimmyqiao8 6@gmail.c om , jayalal @tsinghua .edu.cn . This w ork was supp orted in part by t h e National Natural Science F oundation of China Grant 6055300 1, and the National Basic Research Pro- gram of China Grant 2007CB 807900 ,2007CB807901. 1 F [ x 1 , x 2 , . . . x n ]. This is a well-studied algorithmic problem with a long history and a v ariety of connections and app li cations. See [4] f o r a recent survey . Efficien t rand omiz ed algorithms were prop osed indep enden tly by S c hw artz [5] and Zipp el [6]. Ob t aining a deterministic algorithm for the problem seemed sur prisingly elus i v e. It w as originally Kab an ets and Imp a gliazz o [7] wh o sho wed the strong connection b et ween deran - domizing PIT and pro ving circuit low er b ounds. They sh o wed that giving a deterministic p olyno- mial time (ev en sub exp onen tial time) identi t y testing algorithm means either th a t NEXP 6⊆ P /pol y , or that th e p ermanen t has no p olynomial size arithmetic circuits. This was fu rther strengthened in [3], wh er e it was sho wn that giving a blac k-b o x derandomization of PIT imp lie s that an explicit m u lt ilinear p olynomial has n o su bexp onenti al size arithmetic circuits. Since the seminal work of [7], there has b een a lot of atten tion and an imp ressiv e amoun t of progress in the area. Some of the sp ecial cases for whic h pr o gress has b een rep orted are: depth-2 arithmetic formulas [8, 9, 10], depth-3 and d e pth-4 arithm e tic circuits with b ounded top f a n - in [11 , 12, 13, 14, 15, 16], and non-comm utativ e arithmetic form ulas [17]. In a surp r ising result, Agra wa l and Vina y [18] show ed that the blac k-b o x derandomization of PIT for only depth-4 circuits is almost as hard as that for general arithmetic circuits. P artly aimed at making progress to wards an efficien t deterministic PIT algorithm f or multi- linear f ormulas, Shpilk a and V olk ovi c h [1, 2] studied the arithm etic read-once formula mo del. An arithmetic read-once form ula is giv en by a tree whose no des are tak en from { + , ×} , and whose lea v es are v ariables or field constants, sub j ect to the restriction that eac h v ariables x i is allo we d to app ear at most once. In their wo rk, efficien t blac k-b o x deterministic PIT algorithms are giv en for Σ k -R O-formulas, for “mo derate” k . W e remark that due to a construction by V aliant [19], giv en a RO-form ula F of size s computing f , one can express f as a “read-once” determinantal expression f = det ( M ), where M is a O ( s )- dimensional matrix, whose entries are v ariables or field elemen ts. In this, eac h v ariable x i app ears at most once in M . Identit y testing read-once d e terminan tal expressions, is an imp ortan t sp ecial case of th e P IT p roblem, as it is we ll-kno wn that the b ipartit e p erfect matc hin g p roblem ( BIP AR TITE- PM ) r ed uce s to that form. Giving a blac k-b o x algorithm for testing suc h expr e ssions has the p oten tial of pu tt ing BIP AR TITE-PM in NC, whic h is a p rominen t op en problem in complexit y theory regarding p arallelizabilit y [20, 21, 22, 23]. 1.1 Results W e consider a generalizati on of the ab o ve mentioned RO-form ulas, namely r e ad-onc e algebr aic br anching pr o gr ams ( RO-ABP ) 1 . An algebraic b ranc hing program (ABP) is a la yered directed acyclic graph w it h tw o sp ecial v ertices s and t . Eac h edge is assigned a weig h t, w hic h is an element of X ∪ F , where X is a set of v ariables. F or a p a th in the graph its w eigh t is tak en to b e the pro duct of the weig h t on its edges. T he ABP itself computes a p olynomial wh i c h is the su m of the w eights of all paths from s to t . The ABP is s a id to b e r e ad-onc e if eac h v ariable app ears on at most one edge. A p olynomial f ∈ F [ X ] is called a RO-ABP-p olynomial if there exists a R O-ABP whic h computes f . Due to [19], if f can b e computed b y a R O-formula of size s , then f can b e computed by a RO- ABP of s ize O ( s ). Ho wev er, RO-ABPs are strictly more p o we rful than R O-form u la s . App endix A sho w s a R O-ABP computing g = x 1 x 2 + x 2 x 3 + · · · + x 2 n − 1 x 2 n . Example 3.12 in [1] s h o ws that 1 See Section 2 for a formal defin ition. 2 g can not b e computed b y a R O - form u la , if n ≥ 2. W e remark that the R O - ABP mo del in not unive rsal, e.g. for n ≥ 3, Q 1 ≤ i k n 4 , we have that f ≡ 0 ⇐ ⇒ ∀ a ∈ W n 5 k + A k , f ( a ) = 0 , wher e W n k = { y ∈ { 0 , 1 } n | w t ( y ) ≤ k } and A k = G m ( V 2 m ) for the m th-or der SV-ge ner ator with m = ⌈ log n ⌉ + 1 , and V ⊂ F is a arbitr ary set of size k n 4 + 1 . In the ab o ve for V , W ⊆ F n , V + W denotes the s e t { v + w : v ∈ V , w ∈ W } . By Th e orem 1, w e obtain the follo wing blac k-b o x PIT for Σ k -R O-ABPs: Theorem 2. L et f = P i ∈ [ k ] f i b e a sum of k RO-ABP-p olynomials in n variables. L et F b e a field with | F | > k n 4 . Given black-b ox ac c ess to f , it c an b e de cide d deterministic al ly in time n O ( k +log n ) whether f ≡ 0 . This strengthens a main result of [2] (Theorem 3, for th e non-prepr ocessed 2 case), wh ic h p ro vid e s a deterministic n O ( k +log n ) time PIT algo rithm for Σ k -R O-formulas. Namely , we prov e a strict separation b et w een Σ k -R O-formula and Σ k -R O-ABP , for k ≤ ⌊ n/ 2 ⌋ . W e sh o w that Theorem 3. Q i ∈ [2 n ] ,i is o dd Q j ∈ [2 n ] ,j is even x i x j c an not b e written as a sum of ⌊ n/ 2 ⌋ RO-formulas. The p olynomial of Theorem 3 can b e computed b y a single RO-AB P of size O ( n 2 ) (see Section 3). In the non-black- b o x setting we will pro ve the follo wing r e sult: 2 A generalization of our theorems t o p rep rocessed Σ k -RO-ABPs will not b e pursued here. 3 Theorem 4. L e t { A i } i ∈ [ k ] b e a set of k RO-ABPs in n variables. L et F b e a field with | F | > k n 2 . Given { A i } i ∈ [ k ] on the input, it c an b e de c i d e d deterministic al ly in time O ( k 2 n 7 s ) + n O ( k ) whether P i ∈ [ k ] f i ≡ 0 , wher e f i is the RO-ABP-p olynomial c ompute d b y A i , for i ∈ [ k ] . Since the construction in [19 ] can b e computed efficien tly , this strengthens Theorem 2 in [2], for the case of non-prep r ocessed Σ k -R O-formulas. Finally , if blac k-b o x access is gran ted to the individual f i ’s, whic h w e call the semi-black-b ox setting, w e obtain the follo wing result: Theorem 5. L e t { f i } i ∈ [ k ] b e a set of k R O-ABP- p olynomials in n variables. L et F b e a field with | F | > k n 2 . Given black-b ox ac c ess to e ach i nd i v id ual f i , it c an de cide d deterministic al ly in time k 2 n O ( log n ) + n O ( k ) whether P i ∈ [ k ] f i ≡ 0 . 1.2 T ec hniques for Σ k -R O-ABP PIT The r e sults for Σ k -R O-ABP PIT are obtained thr o u gh the har dness of r epr esentation approac h of [1, 2]. There the PIT algorithm is d e riv ed f r o m a statemen t that x 1 x 2 . . . x n cannot b e expressed as a sum of k ≤ n/ 3 RO-form ula compu t able p olynomials { f i } i ∈ [ k ] , if the p olynomials f i satisfy some sp ecial pr o p ert y . W e do not need to define this sp ecial p roperty for the discussion here, except that w e sh o uld name it: ¯ 0-justification. Unfortunately , the prop ert y of ¯ 0-justification, do es n ot work for th e Σ k -R O-ABP mo del. With some though t it can b e seen that the monomial x 1 x 2 . . . x n is expressible as the sum of three ¯ 0-justified RO-ABP-polynomials. Our main tec hnical con tr i bution is the dev elopment of a new “sp ecia l prop ert y”, called alignment , for whic h a hardn e ss of repr e sen tation theorem can still b e pro ved, b u t which also can b e satisfied simultaneously for a collect ion of R O-ABP-p olynomia ls by means of an efficien tly computable coord inate shift. With regards to the latter, consider f = f 1 + f 2 + . . . + f k , where eac h f i is a R O-ABP-p olynomia l. Observe that ∀ v ∈ F n , f ≡ 0 ⇐ ⇒ f ( x 1 + v 1 , x 2 + v 2 , . . . , x n + v n ) ≡ 0. With some technical w ork, we will establish a sufficie nt condition f o r alignment. With it we s h o w that we can compute a co ordinate shift v suc h that all f i ( x + v ) are aligned. Suc h a sh ift v is called a simultane ous alignment . In the case of ha v in g only b lack- b o x access to f , w e w il l show w e ha v e a “small” s et of candidates con taining at least one sim u l taneous alig nmen t. Th e PIT algo rithms will follo w from this. The rest of th i s pap er is organized as follo w s . Section 2 con tains preliminaries. In Section 3 w e compare Σ k -R O-formulas and Σ k -R O-ABPs. In Section 4 we p ro ve Generator Lemm a 1 . In Section 5 w e d ev elop the to o ls regarding alignment. Then in S ec tion 6 w e show ho w to compute a sim u lt aneous alignmen t. Section 7 contai ns the h a r dness of representat ion theorem for R O-ABPs. F rom these deve lopmen ts, w e put the PIT algorithms together in Section 8. 2 Preliminaries Let X = { x 1 , x 2 , . . . , x n } b e a set of v ariables and let F b e a field. Let W n k = { y ∈ { 0 , 1 } n | wt ( y ) ≤ k } , where wt ( y ) counts the num b er of ones in y . Definition 1. (RO-ABPs) An algebr aic br anching pr o gr am (ABP ) is a 4-tu ple A = ( G, w , s, t ) , wher e G = ( V , E ) is an e dge-lab ele d dir e cte d acyclic gr aph for which the vertex set V c an b e p arti- 4 tione d into levels L 0 , L 1 , . . . , L d , wher e L 0 = s and L d = t . V ertic es s and t ar e c al le d the sour c e and si nk of B , r esp e ctively. Edges may only go b etwe en c onse c utive lev els L i and L i +1 . The lab el function w : E → X ∪ F assigns variables or field c onstants to the e dges of G . F or a p ath p in G , we extend the weig ht f unction b y w ( p ) = Q e ∈ p w ( e ) . L et P i,j denote the c ol le ction of al l dir e cte d p aths p fr om i to j in G . The pr o gr am A c omputes the p olynomial ˆ A := P p ∈ P s,t w ( p ) . The size of A i s define d to b e | V | . An ABP is said to b e r e ad-onc e if | w − 1 ( x i ) | ≤ 1, for eac h x i ∈ X . That is, ev ery v ariable is read at most once by the program. A p olynomial f ∈ F [ X ] is called a RO-ABP-p olynomial , if there exists a R O-ABP w h ic h computes f . W e use the follo wing notation: for x i present on arc ( v , w ) in a R O-ABP A : beg in ( x i ) = v and end ( x i ) = w . W e let s ource ( A ) and sink ( A ) stand for the source and sink of A . F or an y no des v , w in A , w e denote the sub program with sour c e v and sink w b y A v,w . A layer of a R O-ABP A is an y subgraph ind uced by tw o consecutiv e lev els L i and L i +1 in A . W e will assume RO-ABPs are in the form giv en by the follo wing s t raigh tforwardly pr o ven lemma: Lemma 2. If f ∈ F [ X ] is a R O-ABP- p olynomial, then f c an b e c ompute d by a R O -ABP A , wher e every layer c ontains at most one variable-lab ele d e dge. Let f b e a p olynomia l in the ring F [ X ]. F or α ∈ F , f | x i = α denotes the p olynomial f ( x 1 , x 2 , . . . x i − 1 , α, x i +1 , . . . , x n ). Extending this to sets of v ariables, for a subset I ⊆ [ n ] and an assignmen t a ∈ F n , f | x I = a I is the the p olynomial resulting from setting the v ariable x i to a i in f f o r ev ery i ∈ I . This is n o t to b e confused with the follo wing notation: for S ⊆ F n , we will write f | S ≡ 0 to denote that ∀ a ∈ S, f ( a ) = 0. The follo w ing t w o notions are tak en f rom [2]. W e sa y that a p olynomial f dep ends on a variable x i if there exists an a ∈ F n and b ∈ F , suc h that f ( a 1 , a 2 , a i − 1 , a i , a i +1 , . . . , a n ) 6 = f ( a 1 , a 2 , a i − 1 , b, a i +1 , . . . , a n ). The set of v ariables x i that f dep ends on is denoted by V ar ( f ). F or a p olynomial f ∈ F [ X ], th e p artial derivative with r esp e ct to x i , denoted by ∂ f ∂ x i , is d efi ned as f | x i =1 − f | x i =0 . W e will fr ee ly u se the prop erties listed for this notion in [2]. F or example, a mul- tilinear p olynomial f dep ends on x i if and only if ∂ f ∂ x i 6≡ 0. In addition, ∂ f ∂ x i do es n o t dep end on x i . P artial deriv ativ es comm ute, wh ic h w e express by sa ying that ∂ 2 f ∂ x i x j = ∂ 2 f ∂ x j x i . Setting v alues to v ari- ables comm utes with taking partial deriv ativ es in the follo wing w ay: ∀ i 6 = j , ∂ f ∂ x i | x j = a = ∂ ( f | x j = a ) ∂ x i . Lemma 3. L et f ∈ F [ X ] b e a RO-ABP-p olynomial, then ∂ f ∂ x i is a RO-ABP-p olynomial. Pr o of. Let p = | v ar ( f ) | . In case p = 0 it is trivial. Assume p > 0. If x i 6∈ v ar ( f ), then ∂ f ∂ x i ≡ 0, in wh ic h case th e p rop erty trivially holds. No w supp ose x i ∈ v ar ( f ). Hence x i m u st app ear somewhere in A . Sa y x i is on the arc ( v 1 , w 1 ) from leve l L j to L j +1 , where L j = { v 1 , v 2 , . . . , v m 1 } and L j +1 = { w 1 , w 2 , . . . , w m 2 } , for certain j, m 1 , m 2 . W e can write f = X a ∈ [ m 1 ] X b ∈ [ m 2 ] f s,v a w ( v a , w b ) f w b ,t , (1) 5 where f or any no des p and q in A , f p,q is the p olynomial computed by subp rogram A p,q . Then ∂ f ∂ x i = f | x i =1 − f | x i =0 = X a ∈ [ m 1 ] X b ∈ [ m 2 ] f s,v a w ( v a , w b ) | x i =1 f w b ,t − X a ∈ [ m 1 ] X b ∈ [ m 2 ] f s,v a w ( v a , w b ) | x i =0 f w b ,t = X a ∈ [ m 1 ] X b ∈ [ m 2 ] f s,v a  w ( v a , w b ) | x i =1 − w ( v a , w b ) | x i =0  f w b ,t = f s,v 1 f w 1 ,t . Hence w e obtain a v alid RO-AB P computing ∂ f ∂ x i from A b y setting the lab el of the wire ( v 1 , w 1 ) to 1, and remo ving all other wires b et wee n la yers L j and L j +1 . The p ro of of the ab o ve lemma pro vides the insigh t that a RO-ABP compu ting ∂ f ∂ x i can b e obtained f rom a R O -ABP compu ting f , by setting x i = 1 and remo ving all other edges in the la yer con taining x i . This fact w ill b e used at sev eral places in the p ap er. Finally , obs er ve the follo wing simple-but-useful factor-lemma: Lemma 4. If f ∈ F [ X ] is a R O-ABP- p olynomial such that f 6≡ 0 and f = g · ( β x i − α ) , then g is a RO-ABP-p olynomial. Pr o of. This follo ws f rom th e fact that for ev ery γ with β γ − α 6 = 0, g = 1 β γ − α · f | x i = γ . 2.1 Com binatorial Nullstellensatz and a Lemma by Gauss Lemma 5 (Lemma 2.1 in [25]) . L et f ∈ F [ X ] b e a nonzer o p olynomial such that the de gr e e of f in x i is b ounde d by r i , and let S i ⊆ F b e of size at le ast r i + 1 , for al l i ∈ [ n ] . Then ther e exists ( s 1 , s 2 , . . . , s n ) ∈ S 1 × S 2 × . . . × S n with f ( s 1 , s 2 , . . . , s n ) 6 = 0 . Lemma 6. (Gauss) L et P ∈ F [ X , y ] b e a nonzer o p olynomial, and let g ∈ F [ X ] b e such that P | y = g ( x ) ≡ 0 . Then y − g ( x ) is an irr e ducible factor of P in the ring F [ X ] . 3 Separation of R O-A BP and Σ ⌊ n/ 2 ⌋ -R O-form ulas F or n ≥ 2, let f n b e defined as f n ( x 1 , x 2 , . . . , x 2 n − 1 , x 2 n ) = Y i ∈ [2 n ] ,i is o dd Y j ∈ [2 n ] ,j is even x i x j . Prop osition 1. f n c an b e c ompute d by an R O-ABP of size O ( n 2 ) . Pr o of. The R O-ABP is sho w n in Figure 1. Note that b et ween the ( n + 1)th leve l and th e ( n + 2) th lev el there is an n by n complete bipartite graph. Prop osition 2. A p olynomial p ( x 1 , x 2 , . . . , x n ) that c ontains thr e e terms of form αx i x j + β x j x k + γ x k x l , wher e i, j, k , l ∈ [ n ] ar e p airwise differ ent, and α, β , γ ∈ F ar e nonzer o, c an not b e c ompute d by a RO-formula , for n ≥ 4 . 6 ...... x 1 x 3 x 2 n − 1 x 2 x 4 x 2 n s t ...... l 0 l 1 l n +1 l n +2 l n +3 l 2 n +2 l 2 n +3 Figure 1: A RO-ABP computing f n . Pr o of. F or the purp ose of con tradiction, supp ose th ere is a RO-form ula F computin g p . Setting all x m = 0, for m ∈ [ n ] \ { i, j, k, l } , wo uld result in an RO-form ula F ′ computing p ′ ( x i , x j , x k , x l ) = αx i x j + β x j x k + γ x k x l + ax i + bx j + cx k + dx l + e . How ev er, p ′ can n ot b e computed by an R O-formula. One argues this in a similar man n er as for x 1 x 2 + x 2 x 3 + x 3 x 4 (See example 3.12 in [1]). Consider the complete bipartite graph G n = ( V n , E n ) for f n , called the graph asso ciated with f n , sh o wn in Figure 2. Eve ry edge repr esen ts a term in f n . Th e term x i x j + x j x k + x k x l can b e view ed as a length-3 path in G n . Prop osition 3. L et n ≥ 2 . In G n , for an e dge set S ⊆ E n with | S | ≥ 2 n − 1 , S must c ontain a length-3 p ath. Pr o of. W e ju st need to pro ve that for G n , the maxim um “length-3 path free” edge set is of size at most 2( n − 1). This is pr ov ed by induction on n . F or n = 2, it is easy to see th at it h olds. Supp ose for n < l the claim holds. T h en for n = l , for any length-3 path free edge set S , consider the follo wing t wo cases: 1. If there exists an edge e = ( u, v ) ∈ S , for which u or v has no other outgoing edges, let S ′ = S \{ e } . S ′ is a length-3 path free set in G l − 1 . By induction, | S ′ | ≤ 2( l − 2). Thus S has at most 1 + 2( l − 2) < 2( l − 1) edges. 2. Otherw ise, partition the vertice s adjacent to edges in S in to t wo sets V 1 and V 2 , where V 1 con tains all v ertices of degree one, and V 2 con tains all vertic es of degree larger than one. 7 ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ x 1 x 3 x 5 x 2 n − 3 x 2 n − 1 x 2 x 4 x 6 x 2 n − 2 x 2 n Figure 2: Th e bip artite graph G n for f n . It is noted that since no length-3 paths exist, w e hav e that | S | = | V 1 | . If | V 2 | ≥ 2, then | V 1 | ≤ 2 l − 2 = 2( l − 1), s ince th ere are at most 2 l v ertices adjacen t to edges in S . In case | V 2 | = 1, then S is a star, i.e. a single v ertex u connected to a collection of v ertices v 1 , v 2 , . . . , v k . Then k ≤ l an d | S | = k ≤ l ≤ 2( l − 1), for l ≥ 2. Theorem 6. f n c an not b e r epr ese nte d as a sum of ⌊ n/ 2 ⌋ R O-formulas. Pr o of. F or th e purp ose of contradictio n, supp ose f n can b e rep r esen ted as a sum of ⌊ n/ 2 ⌋ R O- form u la-p olynomials q 1 , q 2 , . . . , q ⌊ n/ 2 ⌋ . Let G n = ( V n , E n ) b e th e graph asso ciated with f n . F or an y q i , let S i ⊆ E n b e the set of edges representing the terms app earing in q i of the f orm x a x b , where a ∈ [2 n ] is even, and b ∈ [2 n ] is o d d. Note that since f has n 2 man y terms, some q i should hav e | S i | ≥ 2 n . T h en by Claim 3, S i con tains a length-3 path. Ther efore αx i x j + β x j x k + γ x k x l app ears in q i , f or distin ct i, j, k and nonzero constants α, β , γ ∈ F . Du e to Claim 2, q i can n ot b e computed b y a R O-form ula, w hic h is a con tr adiction. 4 Pro of of Generator Lemma 1 Let p = | V ar ( f ) | . The pro of pro ceeds by induction on p . Th e b ases p = 0 and p = 1 trivially hold. Supp ose p > 1. Hence m ≥ 1. Cons id er arbitrary R O -ABP A computing f . L et s and t b e the source an d sink of A , r esp ectiv ely . Wlog . assume that only th e p v ariables in V ar ( f ) are present in A , and assume A satisfies the condition yielded by Lemma 2. Observe that for some v ariable x i there are at most p/ 2 v ariables in la yers b efore the la ye r con taining x i , and at most p/ 2 v ariables in lay ers after. (If p is o d d it splits (( p − 1) / 2) , ( p − 1) / 2) if p is eve n it splits ( p/ 2 − 1 , p/ 2)). Sa y x i is on the arc ( v 1 , w 1 ) from la yer L j to L j +1 , where L j = { v 1 , v 2 , . . . , v m 1 } and L j = { v 1 , v 2 , . . . , v m 2 } , for certain j, m 1 , m 2 . W e can write f = m 1 X a =1 f s,v a f v a ,t , (2) 8 where for an y no d es p and q in A , f p,q is the p olynomial computed by subprogram of A p,q . Consider f ′ = f ( G 1 m , . . . , G i − 1 m , x i , G i +1 m , . . . , G n m ). Claim 1. Write f ′ = x i · ∂ f ∂ x i ( G 1 m , , . . . , G i − 1 m , G i +1 m , . . . , G n m ) + f ( G 1 m , , . . . , G i − 1 m , 0 , G i +1 m , . . . , G n m ) . Then ∂ f ∂ x i ( G 1 m , , . . . , G i − 1 m , G i +1 m , . . . , G n m ) 6≡ 0 . Pr o of. Since f d ep ends on x i and f is m u ltilinear, ∂ f ∂ x i 6≡ 0. Let f ′′ = ∂ f ∂ x i . W e will sho w that f ′′ ( G m ) 6≡ 0. Obser ve that in the r .h.s. of (2) only f v 1 ,t dep end s on x i . Th is implies that f ′′ = ∂ f v 1 ,t ∂ x i · f s,v 1 . Observe that | V ar ( f s,v 1 ) | and | V ar ( ∂ f v 1 ,t ∂ x i ) | are b oth at most p/ 2. Since f ′′ 6≡ 0, b oth f s,v 1 and ∂ f v 1 ,t ∂ x i are not identic ally zero. Certainly f s,v 1 can b e computed by a R O-ABP . By Lemma 3, we know also ∂ f v 1 ,t ∂ x i can b e computed b y a R O-ABP . As p/ 2 < p , the induction h yp othesis applies. Since p/ 2 ≤ 2 m − 1 , it yields that f s,v 1 ( G m ) 6≡ 0 and ∂ f v 1 ,t ∂ x i ( G m ) 6≡ 0. Th er efore f ′′ ( G m ) 6≡ 0. This pro ves the claim. Recall the set A = { a 1 , . . . , a n } used for the construction of th e SV-generator. By Observ a- tion 5.2 in [2], f ( G m +1 ) | y m +1 = a i = f ′ | x i = G i m + z m +1 . Since z m +1 do es not app ear in G j m for an y j , we get b y Claim 1 th at f ( G m +1 ) | y m +1 = a i 6≡ 0. Hence f ( G m +1 ) 6≡ 0. 5 X-Aligned R O-ABP - p olynomials The f ollo wing lemma leads up to our central definition: Lemma 7. . F or al l i ∈ [ k ] , L et f ∈ F [ X ] b e a RO-ABP-p olynomial with | V ar ( f ) | ≥ 3 . Then for any x i ∈ V ar ( f ) , ther e exist distinct x j , x k ∈ X \{ x i } such that ∂ 2 f ∂ x j ∂ x k = g · ( β x i − α ) , wher e g is a RO-ABP-p olynomial that do es not dep end on x i , and α, β ∈ F . Pr o of. Let A b e a RO-ABP computing f . Wlog. assume all v ariables in X app ear in A . By Lemma 2 assume w log. that A has at most one v ariable p er la ye r . Let x r 1 , x r 2 , . . . , x r n b e the v ariables in X as they app ear la ye r-b y-la y er , when going from th e source to the sin k of A . Consider an arbitrary x i ∈ V ar ( f ). First, we handle the case th at i = r m , for some 1 < m < n . Let j = r m − 1 and k = r m +1 . So x j and x k are the v ariables right b efore and r igh t after x i in A , resp ectiv ely . Assume that x j and x k lab el the ed ges ( u, v ) and ( m, n ) r esp ectiv ely . Th en ∂ 2 f ∂ x j ∂ x k = f s,u f v,m f n,t , where f s,u f v,m , and f n,t are computed b y the su b programs A s,u , A v,m , and A n,t , resp ectiv ely . Ob s erv e that f v,m is of form β x i − α , for α, β ∈ F . T ake g = f s,u f v,m , whic h is easily seen to b e R O-ABP-computable by putting A s,u and A v,m in series, or by app ealing to Lemmas 3 and 4. The sp ecial case where i = r 1 ( i = r n ), i.e. x i is the first (last) v ariable in A , is handled similarly as ab o ve, by c h o osing x k ∈ X \{ x i , x j } arb itrarily and app ealing to Lemma 3. In the ab o ve lemma w e ha ve n o guarantee the α is nonzero, in case β 6 = 0. W e w ould lik e to consider p olynomials wh ic h are in general p osition in this regard. W e make the follo w in g definition: Definition 2. L et S ⊆ X . Every RO-ABP-p olynomial f ∈ F [ X ] with | V ar ( f ) | ≤ 2 is X -pr e- aligne d on S . A RO-ABP-p olynomial f ∈ F [ X ] with | V ar ( f ) | > 2 is X -pr e-aligne d on S , if the fol lowing c ondition is satisfie d: 9 1. for every x i ∈ S , ther e exist distinct x j , x k ∈ X \{ x i } such that ∂ 2 f ∂ x j ∂ x k = g · ( β x i − α ) , wher e g is a RO-ABP-p olynomial that do es not dep end on x i , and α, β ∈ F satisfy that α = 0 ⇒ β = 0 . If f is X -pr e- aligne d on V ar ( f ) , we simply say that f is X -pr e-aligne d. F or the X -pre-a lignmen t p rop erty to h old recursive ly w.r.t. setting v ariables to zero, is a particularly desirable prop ert y of a R O-ABP-p olynomial to ha ve, as w e will see. W e mak e the follo wing in ductiv e d efinition: Definition 3. Every RO-ABP-p olynomial f ∈ F [ X ] with | V ar ( f ) | ≤ 2 is X -aligne d. A R O-ABP- p olynomial f ∈ F [ X ] with | V ar ( f ) | > 2 is X -aligne d, if the fol lowing c onditions ar e satisfie d: 1. f is X -pr e -aligne d, and 2. for every x i ∈ V ar ( f ) , f | x i =0 is X \ { x i } -aligne d. Next we p ro ve some of the n eeded pr op erties of our notion, s tarting with the follo wing easily v erified statemen t: Prop osition 4. If f ∈ F [ X ] is X -pr e-aligne d, then ∀ µ ∈ F , µ · f i s X -pr e-aligne d. The same statement holds with aligne d inste ad of pr e- aligne d. The notion of X - pre-alignmen t is w ell-b eha ved w.r.t. taking p artial d eriv ative s. Th is will b e crucial f or obtaining the Hardness of Rep resen tation T heorem 8. W e ha ve the follo win g lemma: Lemma 8. F or any RO-ABP-p olynomial f ∈ F [ X ] and any x r ∈ X , the fol lowing hold: 1. If f is X -pr e-aligne d, then ∂ f ∂ x r is ( X \{ x r } ) -pr e-aligne d. 2. If f is X -aligne d, then ∂ f ∂ x r is ( X \{ x r } ) -aligne d. Pr o of. W e fi rst sh o w that Item 1 holds. Let f ′ = ∂ f ∂ x r and X ′ = X \{ x r } . By Lemma 3, we know that f ′ is a RO-AB P-p olynomial. Assume that | V ar ( f ′ ) | ≥ 3, since otherw ise the statemen t holds trivially . Consider arbitrary x i ∈ V ar ( f ′ ). Then x i ∈ V ar ( f ), so th er e exist distinct x j and x k in X \ { x i } , suc h that ∂ 2 f ∂ x j ∂ x k = g · ( β x i − α ), where g is a R O-ABP-p olynomial that d o es not dep end on x i , and α = 0 ⇒ β = 0. Consider the follo w ing tw o cases: Case I: r 6∈ { j, k } . Hence x j , x k ∈ X ′ \{ x i } . W e h a ve that ∂ 2 f ′ ∂ x j ∂ x k = ∂ 3 f ∂ x j ∂ x k ∂ x r = ∂ g ∂ x r · ( β x i − α ) . By Lemma 3 , ∂ g ∂ x r is a RO-A BP-p olynomial, and it clearly do es not dep end on x i , so we conclude th at f ′ is X ′ -pre-aligned on { x i } . Case I I: r ∈ { j, k } . Wlog. assume r = j . T h en x k ∈ X ′ \{ x i } . Since | V ar ( f ′ ) | ≥ 3, there must b e at least one more v ariable x l in V ar ( f ′ ) distinct f r om eac h of x k and x i . Th en x l ∈ X ′ \{ x i } . W e h a ve that ∂ f ′ ∂ x k = g · ( β x i − α ) . Hence ∂ 2 f ′ ∂ x k ∂ x l = ∂ g ∂ x l · ( β x i − α ) . W e again conclud e f ′ is X ′ -pre-aligned on { x i } . Since in th e ab o ve , x i w as take n arbitrarily from V ar ( f ′ ), we conclude f ′ is X ′ -pre-aligned. Item 2 is p r o ved by induction on | X | . Th e base case is wh en | X | ≤ 3. Th en | V ar ( f ′ ) | ≤ 2, and hence f ′ is X ′ -aligned. No w sup p ose | X | > 3. Assu me | V ar ( f ′ ) | > 2, since otherwise it is trivial. By Item 1, we kn ow f ′ is X ′ -pre-aligned. Consider an arbitrary x i ∈ V ar ( f ′ ). Then x i ∈ V ar ( f ). 10 W e ha v e that f ′ | x i =0 =  ∂ f ∂ x r  x i =0 = ∂ f | x i =0 ∂ x r . Since f | x i =0 is ( X \{ x i } )-alig ned, w e can apply the induction h yp othesis to conclude that ∂ f | x i =0 ∂ x r is ( X \ { x i } ) \{ x r } = ( X ′ \{ x i } )-alig ned. 5.1 A W ork able Sufficien t Condition Next w e establish a s u fficien t cond ition, so for a giv en RO-A BP-p olynomial f w e can mak e f ( x 1 + v 1 , x 2 + v 2 , . . . , x n + v n ) X -alig n ed, by means of compu ting s ome shift v ∈ F n . F or this, let us call a p olynomial f ∈ F [ X ] de c ent , if for all x a , x b ∈ V ar ( f ) with ∂ 2 f ∂ x a ∂ x b 6≡ 0, it h olds that the m onomial x a x b app ears in f with a nonzero constan t co efficient. Lemma 9. A RO-ABP- p olynomial f ∈ F [ X ] is X -aligne d, if | V ar ( f ) | ≤ 2 , or else for any I ⊆ V ar ( f ) with | I | ≤ | V ar ( f ) | − 3 , f | x I =0 is de c ent. Pr o of. W e use induction on | V ar ( f ) | . F or the base case | V ar ( f ) | ≤ 2 it is trivial. No w assum e | V ar ( f ) | > 2. T ak e I = ∅ . Th en we get that for any x a , x b ∈ V ar ( f ), if ∂ 2 f ∂ x a ∂ x b 6≡ 0 then the monomial x a x b app ears in f with a nonzero constan t co efficient. Let us fir st establish that f is X -pre-alig ned. C on s ider an arbitrary x i ∈ V ar ( f ). By Lemma 7, there exist d istinct x j , x k ∈ X \{ x i } such that ∂ 2 f ∂ x j ∂ x k = g · ( β x i − α ) , (3) where g is a R O -ABP-p olynomial that do es not dep end on x i , and α, β ∈ F . If β = 0, then f is X -pre-aligned on { x i } , so supp ose β 6 = 0. If (3) is identica lly zero, then we kno w g ≡ 0, so ∂ 2 f ∂ x j ∂ x k = g · ( β x i − α ′ ), f or an y arb itrary α ′ 6 = 0. If (3) is not iden tically zero, then w e kn o w x j x k is in f , wh ic h implies that α 6 = 0. W e conclud e that f is X -pre-al igned on { x i } . In the ab o ve, we find that f is X -pre-alig n ed on { x i } in an y of the considered cases. Since x i w as arb itrarily tak en from V ar ( f ), we conclude that f is X -pre-alig ned. Next, we sh o w Condition 2 of Definition 3 holds. Consider f ′ := f | x i =0 , for an arbitrary x i ∈ V ar ( f ). W e wan t to establish that the sufficient condition of Lemma 9 holds for f ′ ∈ F [ X \{ x i } ], since then we can b y apply the indu ction hypothesis and conclude that f ′ is ( X \{ x i } )-alig ned. If | V ar ( f ′ ) | ≤ 2 the sufficien t condition of the Lemma 9 clearly holds for f ′ . Otherwise, consid er I ′ ⊆ V ar ( f ′ ) of size at most | V ar ( f ′ ) | − 3. Let I = I ′ ∪ { x i } . T h en | I | ≤ | V ar ( f ) | − 3. No w consider x a , x b ∈ V ar ( f ′ x I ′ =0 ) = V ar ( f x I =0 ). Sup p ose ∂ 2 f ′ | x I ′ =0 ∂ x a ∂ x b 6≡ 0. Since the latter equals ∂ 2 f | x I =0 ∂ x a ∂ x b 6≡ 0, we kno w that x a x b app ears w ith a nonzero constant co efficien t in f | x I =0 . This im p lies x a x b app ears with a non zero constan t co efficient in f | x I ′ =0 . Hence f ′ x I ′ =0 is decen t. W e conclud e the sufficien t condition of the Lemma 9 holds for f ′ ∈ F [ X \{ x i } ]. Hence by the induction h yp othesis w e conclude that f ′ is ( X \ { x i } )-alig ned. Lemma 10. A ny de c ent RO-ABP-p olynomial f ∈ F [ X ] is X -aligne d. Pr o of. W e show that the condition of Lemma 9 is satisfied. If | V ar ( f ) | ≤ 2 this is clear. O th erwise, consider arb itrary I ⊆ V ar ( f ) with | I | ≤ | V ar ( f ) | − 3. L et x a , x b ∈ V ar ( f | x I =0 ), b e suc h that ∂ 2 f | x I =0 ∂ x a ∂ x b 6≡ 0 . W e ha ve that x a , x b ∈ V ar ( f ), and it must b e that ∂ 2 f ∂ x a ∂ x b 6≡ 0, s ince ∂ 2 f | x I =0 ∂ x a ∂ x b =  ∂ 2 f ∂ x a ∂ x b  | x I =0 . Hence x a x b is in f . This implies that x a x b is in f | x I =0 . 11 5.2 Nearly Unique N onalignmen t In addition to the ab ov e, w e cru cially need the follo wing “Nearly Unique Nonalignmen t Lemma”. Lemma 11. L et f ∈ F [ X ] b e an X -pr e-aligne d RO-ABP-p olynomial for which ∂ 2 f ∂ x p ∂ x q 6≡ 0 , for any distinct x p , x q ∈ X . Then ther e ar e at most two γ ∈ F such that f | x n = γ is not ( X \{ x n } ) -pr e-aligne d. Before giving th e pr o of, we need a lemma. Lemma 12. L et f ∈ F [ X ] b e a R O-ABP- p olynomial with | V ar ( f ) | ≥ 3 that is X -pr e - aligne d on S , for some S ⊆ V ar ( f ) . Assu me that for any distinct x p , x q ∈ X , ∂ 2 f ∂ x p ∂ x q 6≡ 0 . In any R O-ABP A c omputing f , for any x i ∈ S , 1. if ther e exi sts a non-c onstant layer with variable x a right b efor e the x i -layer, and ther e exists a non-c onstant layer with variable x b right after the x i -layer, then ∂ 2 f ∂ x a ∂ x b = g · ( β x i − α ) , wher e g is a RO-ABP-p olynomial that do es not dep end on x i , and α, β ∈ F satisfy that α = 0 ⇒ β = 0 . F u rthermor e, − α e qu als the sum of weights of al l p aths fr om en d ( x a ) to beg in ( x b ) that do not go over x i . Pr o of. Consider x i ∈ S . Since f is X -pre-al igned on S , we kno w there exist distinct x j , x k ∈ X \{ x i } with ∂ 2 f ∂ x j ∂ x k = h · ( β ′ x i − α ′ ) , where h is a RO-ABP-polynomial that d o es not dep end on x i , and α ′ , β ′ ∈ F satisfy that α ′ = 0 ⇒ β ′ = 0. Sin ce ∂ 2 f ∂ x j ∂ x k 6≡ 0, it must b e that α ′ 6 = 0. Case I: In A , the x i -la y er lies in b et w een the x j -la y er and x k la ye r . Wlog assume the x i la ye r lies b efore th e x k -la y er and after the x j -la y er (according to the order of the D A G u nderlying A ). W rite ∂ 2 f ∂ x j ∂ x k = p 1 p 2 · ( q 1 q 2 x i + q 3 ) , w here • p 1 is the s um of we igh ts ov er all paths in A from sour ce ( A ) to beg in ( x j ), and p 2 is the s um of w eights ov er all paths in A from en d ( x k ) to sin k ( A ). • q 3 is the su m of weig h ts ov er all paths from end ( x j ) to beg in ( x k ) that b yp ass the x i -edge, q 1 is the sum of w eigh ts ov er all paths f rom end ( x j ) to beg in ( x i ), and q 2 is the s um of w eigh ts o ve r all paths from end ( x i ) to beg in ( x k ). No w we h a ve th at p 1 p 2 · ( q 1 q 2 x i + q 3 ) = h · ( β ′ x i − α ′ ) . Since b oth p 1 p 2 and h d o not d ep end on x i , it must b e that ( β ′ x i − α ′ ) | ( q 1 q 2 x i + q 3 ) . Note that β ′ cannot equ al 0, since then one of q 1 , q 2 w ould b e zero. The latter implies that ∂ 2 f ∂ x i ∂ x j ≡ 0 or ∂ 2 f ∂ x i ∂ x k ≡ 0, whic h is a con tradiction. Since β ′ 6 = 0, we can conclude that q 3 = µq 1 q 2 for some µ ∈ F , µ 6 = 0. No w we need the follo wing claim: Claim 2. Given an RO-ABP A c omputing f ( x 1 , . . . , x n ) , if for any distinct x p , x q ∈ X , ∂ 2 f ∂ x p ∂ x q 6≡ 0 , then Q i ∈ [ n ] x i app e ars in f . F urthermor e, for two v ariables x i and x j , if x i is b efor e x j in A , if we let S b e the set of variables in b etwe en x i and x j , then Q x m ∈ S x m is a term in the p olynomial ˆ A ( end ( x i ) , beg in ( x j )) . Pr o of. Su p p ose the v ariable la y ers in A are arr anged according to th e p ermuta tion φ : [ n ] → [ n ], that is, x φ ( i ) lab els th e i th v ariable la yer. T hen we that 12 1. ˆ A ( s, beg in ( x φ (1) )) 6≡ 0 (Since otherwise ∂ 2 f ∂ x φ (1) ∂ x φ (2) ≡ 0), 2. Similarly ˆ A ( end ( x φ ( n ) ) , t ) 6≡ 0, and 3. F or i ∈ [ n − 1], ˆ A ( begin ( x φ ( i ) ) , end ( x φ ( i +1) )) 6≡ 0 (Since otherwise ∂ 2 f ∂ x φ ( i ) ∂ x φ ( i +1) ≡ 0). The co efficient of Q i ∈ [ n ] x i is just ˆ A ( s, beg in ( x φ (1) )) · ˆ A ( end ( x φ ( n ) ) , t ) Y i ∈ [ n − 1] ˆ A ( begin ( x φ ( i ) ) , end ( x φ ( i +1) )) , and hence Q i ∈ [ n ] x i app ears in f . A similar argumen t yields the statemen t for ˆ A ( end ( x i ) , beg in ( x j )). As in the pr o of of Lemma 7, w rite ∂ 2 f ∂ x a ∂ x b = g · ( β x i − α ), where g is a R O-ABP-p olynomial that do es not dep end on x i , and − α equals the sum of weigh ts ov er all paths from end ( x a ) to beg in ( x b ) not going o v er x i . W e ha ve th ree cases: 1. Neither x j nor x k is the m ost adjacen t v ariable to x i in A . By ab o ve claim, x a app ears in a monomial of q 1 , and x b app ears in a monomial q 2 . Hence, there is a monomial in q 1 q 2 with x a x b . As q 3 = µq 1 q 2 , for µ 6 = 0, the s ame can b e s aid for q 3 . But this implies α 6 = 0, as the co efficien t of x a x b is − α · ˆ A ( end ( x j ) , beg in ( x a )) ˆ A ( end ( x b ) , beg in ( x k )). 2. x j is not the most adjacent v ariable to x i in A , bu t x k = x b . Then similarly q 1 q 2 has a m on omial with x a in it, and th er efore the same h olds for q 3 . Th erefore α 6 = 0, as the co efficien t of x a in q 3 is − α · ˆ A ( end ( x j ) , beg in ( x a )). 3. x j = x a , but x k is not the most adjacen t v ariable to x i in A . This is argued similarly as the second item. This concludes the argument for this case. Case I I: In A , the x i -la y er lies b efore the x j -la y er and x k -la y er. Wlog. assume that the x j la ye r lies b efore the x k la ye r . Similarly as in C ase I, we write ∂ 2 f ∂ x j ∂ x k = p 1 p 2 · ( q 1 q 2 x i + q 3 ) , b ut where n o w we ha ve that • p 1 = ˆ A end ( x j ) ,beg in ( x k ) , and p 2 = ˆ A end ( x k ) ,sink ( A ) , • q 1 = ˆ A sour ce ( A ) ,beg in ( x i ) , • q 2 = ˆ A end ( x i ) ,beg in ( x j ) , • q 3 = ˆ A [ x i = 0] sour ce ( A ) ,beg in ( x j ) . Then p 1 p 2 · ( q 1 q 2 x i + q 3 ) = h · ( β ′ x i − α ′ ) . S ince b oth p 1 p 2 and h d o not dep end on x i , it must b e that ( β ′ x i − α ′ ) | ( q 1 q 2 x i + q 3 ) . S imilarly as b efore, w e get q 3 = µq 1 q 2 for some µ ∈ F , µ 6 = 0. The rest of th e pro of is similar to Case I. On e argues th at 1) wh en x j 6 = x b , q 1 q 2 con tains a monomial with x a x b . T o mak e x a x b app ear in a m onomial q 3 w e need α 6 = 0, and 2) wh en x j = x b , q 1 q 2 con tains a monomial with x a , and to mak e x a app ear in a monomial of q 3 , we need α 6 = 0. 13 Case I I I : In A , th e x i -la y er lies after the x j -la y er and x k -la y er. This case is symm etrical to Case I I. W e also need the follo wing p r op osition: Prop osition 5. L et f ∈ F [ X ] b e a RO-ABP-p olynomial with | V ar ( f ) | ≥ 3 , and let S ⊆ V ar ( f ) . Then f is X -pr e-aligne d on S if and only if f ′ := ( x n +1 + 1) f is X ∪ { x n +1 } -pr e-aligne d on S . Pr o of. Let X ′ = X ∪ { x n +1 } . It is easy to see that assuming f is X -pre-aligned on S , we ha ve that f is X ′ -pre-aligned on S . Con versely , assume f ′ is X ′ -pre-aligned on S . Let x i ∈ S . Then th ere exist x j , x k ∈ X ′ \{ x i } , suc h that ∂ 2 f ′ ∂ x j ∂ x k = g ( β x i + α ), where g is a RO-ABP-polynomial that do es n ot dep end on x i , and α = 0 implies β = 0. If x n +1 6∈ { x j , x k } , then ∂ 2 f ′ ∂ x j ∂ x k = ∂ 2 f ∂ x j ∂ x k ( x n +1 + 1). Setting x n +1 = 0, we ha ve that ∂ 2 f ∂ x j ∂ x k = ( g | x n +1 =0 )( β x i + α ). So we get the required X -pre-alignmen t of f on { x i } . O th erwise, sa y wlog. x j = x n +1 . W e ha ve that ∂ f ∂ x k = ∂ 2 f ′ ∂ x n +1 ∂ x k = g ( β x i + α ). One easily obtains the required X -pre-al ignmen t of f on { x i } , by taking one more ∂ x l , for some v ariable x l ∈ X \{ x i , x k } , and then using L emma 3. W e are no w r eady to giv e the pro of of Lemma 11. 5.3 Pro of W e pro ve the lemma by induction on | X | . F or the base case w e tak e | X | ≤ 3, in wh ic h case the statemen t clearly h olds. No w sup p ose | X | > 3. Let f ′ = f | x n = γ , for some γ . Let X ′ = X \{ x n } . Supp ose f ′ is not X ′ -pre-aligned. Hence | V ar ( f ′ ) | ≥ 3. W e w ant to sho w this can happ en for at most one γ . Consider an arbitrary RO-AB P A computing f . Let f e = f ( x n +1 + 1)( x n +2 + 1)( x n +3 + 1)( x n +4 + 1). L et X e := X ∪ { x n +1 , x n +2 , x n +3 , x n +4 } . By Prop osition 5, f e is X e -pre-aligned on V ar ( f ). Let f ′ e := ( f e ) | x n = γ and X ′ e := X ′ ∪ { x n +1 , x n +2 , x n +3 , x n +4 } . Note that f ′ e = f ′ ( x n +1 + 1)( x n +2 + 1)( x n +3 + 1)( x n +4 + 1). So also by Prop osition 5, f ′ e is n ot X ′ e -pre-aligned on V ar ( f ′ ) if and on ly if f ′ is not X ′ -pre-aligned on V ar ( f ′ ). W e will sho w the former happ ens for at most one γ . So let us assume that f ′ e is n ot X ′ e -pre-aligned on V ar ( f ′ ). W e can easily obtain a R O -ABP A e from A , whic h computes f e . In this, we mak e su re x n +1 and x n +2 are th e fir s t and second v ariable in A e , and x n +3 and x n +4 are the f ore-last and last v ariable in A e . F or eac h x i ∈ V ar ( f ′ ), let x j i b e the v ariable righ t after x i in A e , and let x k i b e the v ariable b efore x i in A e . Note that w e hav e made sure these alw a ys exist in A e . Since f e is X e -pre-aligned on V ar ( f ), by Lemma 12, ∂ 2 f e ∂ x j i ∂ x k i = g · ( β i x i − α i ), where g is a RO-ABP-polynomial that do es not dep end on x i , and α i = 0 ⇒ β i = 0. F u rthermore, w e hav e that α i is the s um of weigh ts of all paths fr om end ( x k i ) to begin ( x n ), whic h do not go o v er x i in A e . Consid er the follo wing tw o cases: Case I: n 6∈ { j i , k i } , for any x i ∈ V ar ( f ′ ). Then for an y i , ∂ 2 f ′ e ∂ x j i ∂ x k i = ( g i ) | x n = γ · ( β i x i − α i ) , wh ich con tradicts the assumption that f ′ e is not X ′ e -pre-aligned on V ar ( f ′ ). Case I I: n ∈ { j i , k i } , for some x i ∈ V ar ( f ′ ). By symm etry we can assu me w log. that j i = n (the case k i = n is handled similarly). Since ∂ 2 f ∂ x j i ∂ x k i 6≡ 0, and α i = 0 implies β i = 0, W e hav e that α i 6 = 0. 14 W e kno w that in A e there still exists a v ariables la yer, say with v ariables x l , right after the x j i -la y er. Let b i = beg in ( x i ) , e i = end ( x i ) , b n = beg in ( x n ), and e n = end ( x n ). Let s = end ( x k i ) and t = beg in ( x l ). T hen wr ite: ∂ 2 f e ∂ x l ∂ x k i = p 1 p 2 ( c s,b i c e i ,b n c e n ,t x i x n + c s,b i c e i ,t x i + c s,b n c e n ,t x n + c s,t ) , where in the ab ov e eac h constant c v,w is the sum of weigh ts o v er all p aths f rom v to w going o ve r constan t lab eled edges only . Note that c s,b n = α i 6 = 0. F u rthermore, p 1 is the sum of we igh ts of all paths from sour ce ( A e ) to beg in ( x k i ), and p 2 is th e sum of w eights o ve r all p aths from end ( x l ) to sink ( A e ). T hen ∂ 2 f ′ e ∂ x l ∂ x k i = p 1 p 2 (( c s,b i c e i ,b n c e n ,t γ + c s,b i c e i ,t ) x i + c s,b n c e n ,t γ + c s,t ) , W e ha ve that f ′ e can only n ot b e X ′ e -pre-aligned on { x i } if c s,b n c e n ,t γ + c s,t = 0. This can hap p en for more than one γ only if c s,b n c e n ,t = 0. Since c s,b n 6 = 0, this happ ens only if c e n ,t = 0, but the latter implies that ∂ 2 f e ∂ x l ∂ x n ≡ 0, which in turn imp lies that ∂ 2 f ∂ x l ∂ x n ≡ 0, which is a cont radiction. Finally , p utting together from what w e observed from the ab o ve t wo cases, note that, Case I I can app ly at most twice f or a v ariable x i ∈ V ar ( f ′ ). Namely , p ossibly once f or the v ariable righ t b efore x n , and p ossibly once for the v ariable after x n . W e conclude the lemma holds. Corollary 1. Supp ose | F | > 3 . L et h, g ∈ F [ X ] b e RO-ABP-p olynomials suc h that h = g · ( β x n − α ) , for β ∈ F \{ 0 } . If h is X -pr e- aligne d, then g is ( X \{ x n } ) -pr e-aligne d. Pr o of. If w e set x n to any v alue γ 6 = α/β , we get that h | x n = γ is a nonzero constan t multiple of g . By Lemma 11, there are at most t w o γ su c h that h | x n = γ is not ( X \{ x n } )-pre-aligned. No w use Prop osition 4 to conclude that g is ( X \{ x n } )-pre-aligned. 6 Sim ultaneous A lignmen t of R O-ABP-p olynomials Definition 4. A simultane ous X -alignment for a set of RO-ABP-p olynomials { f i ∈ F [ X ] } i ∈ [ k ] is a ve ctor v ∈ F n such that f i ( x 1 + v 1 , x 2 + v 2 , . . . , x n + v n ) is X -aligne d for every i ∈ [ k ] . W e present an algorithm for fi nding a simultaneo us X -alignment for a set of R O-ABP- p olynomials. W e assum e that we hav e a p olynomial iden tity testing algorithm PIT RO-ABP for testing a s in gle RO-ABP . W e prov e a corollary of Lemma 10 fir st. Corollary 2. L e t { f i } i ∈ [ k ] b e a set of RO-ABP-p olynomials in F [ X ] . Then v ∈ F n is a simultane ous X -alignment for { f i } i ∈ [ k ] , if it is a simultane ous nonzer o for { ∂ 2 f i ∂ x a ∂ x b | ∂ 2 f i ∂ x a ∂ x b 6≡ 0 } i ∈ [ k ] ,a,b ∈ [ n ] . Pr o of. Consider { f ′ i = f i ( x 1 + v 1 , x 2 + v 2 , . . . , x n + v n ) } i ∈ [ k ] . Due to Lemma 10, we only n eed to sho w that for every i , for ev ery x a , x b ∈ V ar ( f i ), if ∂ 2 f ′ i ∂ x a ∂ x b 6≡ 0 then the monomial x a x b app ears in f ′ i with a nonzero constan t co efficien t. Observe that the m on omial x a x b app ears in f ′ i with a nonzero constan t co efficien t ⇐ ⇒ ∂ 2 f ′ i ∂ x a ∂ x b ( ¯ 0) 6 = 0. Th e latter holds, as ∂ 2 f ′ i ∂ x a ∂ x b ( ¯ 0) = ∂ 2 f i ∂ x a ∂ x b ( v ) 6 = 0. 15 No w the argument is similar as for Lemma 4.3 in [2], bu t with first order partial d eriv ative s replaced b y second order ones. T his yields the follo w in g theorem: Theorem 7. L et F b e a field with | F | > k n 2 . Ther e exists an algorithm for finding a simultane ous X -alignment for a set of RO-ABP p olynomials { f i ∈ F [ X ] } i ∈ [ k ] . The algorithm makes or acle c al ls to the pr o c e dur e PI T RO-ABP . The f i s ar e only ac c esse d thr ough this subr outine. The running-time of the algorithm is O ( k 2 n 5 · t ) , wher e t is an upp er b ound on the time ne e de d f or any subr outine c al l to PIT RO-ABP . Pr o of. W e assum e that we ha ve a p olynomial identit y testing algorithm PIT RO-ABP for testing a single RO-AB P , suc h that PI T RO-ABP outputs T r ue if f ≡ 0 and F al se otherwise. W e ha ve the follo wing algorithm: Algorithm 1 Alignment Finding. Input: A set of R O -ABP-p olynomials { f i ∈ F [ X ] } i ∈ [ k ] . Output: A simultaneous alignment v for { f i } i ∈ [ k ] . Oracle: PIT algorithm PIT RO-ABP . 1: L = ∅ 2: for all f i and ( x a , x b ), a, b ∈ [ n ], a 6 = b do 3: If PIT RO-ABP ( ∂ 2 f i ∂ x a ∂ x b ) = F al se , add it to L 4: end for 5: for all j ∈ [ n ] do 6: Find c such that for every g ∈ L , P IT RO-ABP ( g | x j = c ) = F al se 7: v j ← c 8: F or ev ery g ∈ L , g ← g | x j = c 9: end for 10: re t urn v W e first m ake tw o r emarks, wh ic h p ertain to applyin g Algorithm 1 in th e setting where we only ha ve blac k-b o x access to eac h f i . Consider the s et L the algorithm constructs w ith the execution of th e fir st for -loop. Sin ce we only ha v e b lac k-b o x access to f i , the giv en p seudo co d e is int ended to mean L is constructed symbolically . Having b lack- b ox access to f i is enough to ha ve blac k-b o x access to any elemen t of L . Namely , by Lemma 3, f ′ := ∂ 2 f i ∂ x a ∂ x b is a RO-AB P . Note that blac k-b o x access to f i is s ufficien t for b eing able to compute f ′ ( a ) for an y a ∈ F n . This is all the blac k-b o x R O-ABP algorithm n eeds to decide whether f ′ ≡ 0. Similarly , on line 8 the substitution is not actually carried out, b ut done sym b olically . So it is just r emem b ered that x j is set to c . F or example, sup p ose that up to some p oin t in the execution the algorithm it has set x i = c i , for i ∈ [ m ]. Th en on line 6, for ev aluating PIT RO-ABP ( g | x j = c ), the blac k- b ox algorithm is granted access to a R O -ABP in n − m v ariables g ( c 1 , c 2 , . . . , c m , x m +1 , . . . , x n ). The qu eries it make s can b e answ ered with only blac k-b o x access to g . No w, b y C orollary 2 it suffices to fi nd a common nonzero of the set L . First how ev er, we need to explain ho w to fin d c such that g | x j = c 6≡ 0. Let V ⊂ F with | V | = k n 2 + 1 b e giv en . W e claim V alw a ys includes a go o d v alue. Th is is b ecause we ha v e at m ost k n 2 m u ltilinear p olynomials in L , and for a sp ecific one there is at most one bad v alue, d ue to Lemma 6. The algorithm can simply try all element s in V to get the required c . Th e correctness of the algorithm is no w eviden t, from the observ ation that it simply maintai ns the inv arian t th at all g ∈ L are n ot identic ally zero. 16 The run n ing time of th e algorithm is as follo ws: f or line 2 we need O ( k n 2 ) calls to PIT RO-ABP . F or lin e 7 we n eed O ( n · ( k n 2 + 1) · ( k n 2 )) = O ( k 2 n 5 ) calls to P I T RO-ABP . Thus th e total r unnin g time of th e algorithm is O ( k 2 n 5 · t ), wh ere t is an upp er b ound on the time needed for any sub routine call to PIT RO-ABP . By Lemma 1 and us in g Lemma 5, PIT RO-ABP can b e implemen ted in the blac k-b o x setting to run in time n O (log n ) , where n is the num b er of v ariables of the inpu t RO-ABP-polynomial. In the non-blac k-b o x setting, as is sho w in App endix C, PI T RO-ABP can b e implemente d to run in time O ( n 2 s ), when giv en an RO-AB P o v er n v ariables of size s . This yields the follo wing tw o corollarie s: Corollary 3. Pr ovide d | F | > k n 2 , ther e exists an non-black-b ox algorithm for finding a simultane ous X -alignment for a set { f i ∈ F [ X ] } i ∈ [ k ] , wher e f i is c ompute d by a RO-ABP A i , for i ∈ [ k ] . The algorithm r e c ei ves { A i } i ∈ [ k ] on the input, and it runs in time O ( k 2 n 7 s ) , wher e s is an upp er b ound on the size of any A i . Corollary 4. Pr ovide d | F | > k n 2 , ther e exists a black-b ox algorithm for finding a simultane ous X -alignment f or a set of R O -ABP- p olynomials { f i ∈ F [ X ] } i ∈ [ k ] . The algorithm queries individual f i s, and runs in time k 2 n O (log n ) . 6.1 Sim ultaneous Alignmen t Hit ting Set Here we present a blac k-b o x algorithm to find a candidate set A k of size ( k n ) O (log n ) , whic h is guaran teed to con tain a simultaneo us X -alignmen t f or any set of k RO-A BP-p olynomials { f i ∈ F [ X ] } i ∈ [ k ] . Lemma 13. L et F b e a field with | F | > k n 4 , and let V ⊆ F with | V | = k n 4 + 1 b e giv e n. L et { f i } i ∈ [ k ] b e a set of RO-ABP- p olynomials in F [ X ] . L et G m : F 2 m → F n b e the m th-or der SV-gener ator with m = ⌈ log n ⌉ + 1 . Then A k := G m ( V 2 m ) c ontains a simultane ous X - alignment for { f i } i ∈ [ k ] . Pr o of. let L = { ∂ 2 f i ∂ x a ∂ x b | ∂ 2 f i ∂ x a ∂ x b 6≡ 0 } i ∈ [ k ] ,a,b ∈ [ n ] . Let P ( x 1 , . . . , x n ) = Q g ∈ L g ( x 1 , . . . , x n ). By Lemma 3, eac h g ∈ L is a R O-ABP-p olynomial. Hence b y L emma 1, for m = ⌈ log n ⌉ + 1, the S V-generator ( G 1 m , G 2 m , . . . , G n m ), satisfies th at g ( G 1 m , G 2 m , . . . , G n m ) 6≡ 0, for all g ∈ L . So P ( G 1 m , G 2 m , . . . , G n m ) 6≡ 0. Note that th ere are 2 m v ariables in P ( G 1 m , . . . , G n m ), and the degree of ev ery v ariable is b ound ed b y k n 2 · n 2 = k n 4 . Thus b y Lemma 5, ∃ a ∈ V 2 m , P ( G 1 m ( a ) , . . . , G n m ( a )) 6 = 0. Hence A k = G n ( V 2 m ) is ensured to con tain a nonzero of P . An y nonzero of P is a sim ultaneous n onzero of all g ∈ L . By Corollary 2, A k con tains a simulta neous X -alignmen t for { f i } i ∈ [ k ] . 7 A Hardness of Represen tation Theorem for R O-ABPs The follo wing theorem is an adaption of T heorem 6.1 in [2] to th e notion of X -pre-alignmen t. O ne notable difference in the p ro of is that for the main case separation, w e distinguish b etw een whether there are 3rd-order partial deriv ativ es v anish in g or not (rather than 2nd-order partial as in [2]). Theorem 8. Assume | F | > 3 . L et P n = Q i ∈ [ n ] x i . If { f i ∈ F [ X ] } i ∈ [ k ] is a set of k X -pr e-aligne d R O-ABP- p olynomials for which P n = P i ∈ [ k ] f i , then n < 7 k . 17 Pr o of. The pro of p ro ceeds by indu ction on k . F or the b ase case k = 1, since f 1 = P n , and f 1 is X -pre-aligned, it must b e that n ≤ 2. Namely , if n > 2, then for x i ∈ V ar ( P n ), wh atev er distinct x j , x k ∈ X \{ x i } we select, ∂ 2 f 1 ∂ x j ∂ x k = x i · Q x r ∈ X \{ x i ,x j ,x k } . T his cannot b e of the form g · ( β x i + α ) with g b eing an R O-ABP not d ep endin g on x i , and α = 0 ⇒ β = 0, as Definition 2 requires. Namely , since g do es not dep end on x i , it m u st b e that β 6 = 0. Hence α 6 = 0, and th u s g · ( β x i + α ) is not homogeneous. Since x i · Q x r ∈ X \{ x i ,x j ,x k } is homogeneous, th is is a contradictio n. No w assume k > 1. Su pp ose we can w r ite P n = P i ∈ [ k ] f i . F or pu rp ose of contradict ion, assume that n ≥ 7 k . Hence n ≥ 14. Case I: ∃ distinct p, q , r ∈ [ n ] and s ∈ [ k ], su c h that ∂ 3 f s ∂ x p ∂ x q ∂ x r ≡ 0. Wlog. assume that p = n − 2 , q = n − 1 , r = n and s = k . T h en P i ∈ [ k − 1] ∂ 3 f i ∂ x n − 2 ∂ x n − 1 ∂ x n = P n − 3 . By Lemma 8, all of the terms ∂ 3 f i ∂ x n − 2 ∂ x n − 1 ∂ x n are ( X \{ x n − 2 , x n − 1 , x n } )-pre-aligned. By indu c- tion, it must b e that n − 3 < 5( k − 1). Hence n < 5 k − 2, whic h is a con tradiction. Case I I: 6 ∃ distinct p, q , r ∈ [ n ] and s ∈ [ k ], suc h that ∂ 3 f s ∂ x p ∂ x q ∂ x r ≡ 0. W e kno w ∀ i , | V ar ( f i ) | ≥ 3. Since f i is X -pre-aligned, there exist distinct x j i , x k i ∈ X \{ x i } suc h that ∂ 2 f ∂ x j i ∂ x k i = g i · ( β i x n − α i ) , where g i is a RO-ABP-polynomial that do es not dep end on x i , and α i = 0 ⇒ β i = 0. Note that in this case, g i 6≡ 0, since otherwise a second order partial v anish es. Hence b oth j i and k i are certainly not equal to x n . It m u st b e that β i 6 = 0, since otherwise ∂ 3 f ∂ x j i ∂ x k i ∂ x n ≡ 0. Hence also α i 6 = 0. Claim 3. A ny g i is ( X \{ x j i , x k i , x n } ) -pr e-aligne d. Pr o of. Assume that | V ar ( g i ) | ≥ 3, since otherwise the claim is trivial. Let h = g i · ( β i x n − α i ). By Lemma 8, h is ( X \{ x j i , x k i } )-pre-aligned. Since β i 6 = 0, applying Corollary 1 yields that g i is ( X \{ x j i , x k i , x n } )-pre-aligned. No w, let A = { α i β i : i ∈ [ k ] } . Define for γ ∈ A , E γ = { i ∈ [ k ] : γ = α i β i } and B γ = { i ∈ [ k ] : γ 6 = α i β i and ( f i ) | x n = γ is not ( X \{ x n } )-pre-aligned } . Note that P γ ∈ A | E γ | = k . By Nearly Unique Nonalignmen t Lemma 11, P γ ∈ A | B γ | ≤ 2 k . Hence there exists γ 0 ∈ A su c h that | B γ 0 | ≤ 2 | E γ 0 | . Let I = E γ 0 ∪ B γ 0 , and let J = { j i : i ∈ I } ∪ { k i : i ∈ I } . W e ha ve th at 2 ≤ | J | ≤ 2 | I | ≤ 6 | E γ 0 | . Observe th at x n 6∈ J . Defin e for any i , f ′ i = ∂ J f i . W e ha ve th e follo win g three prop erties: 1. Eac h f ′ i is an ( X \ J )-pre-aligned R O -ABP-p olynomial, due to Lemma 8. 2. F or every i ∈ I , f ′ i = ( β i x n − α i ) h i , where h i is a R O-ABP-p olynomial. Namel y , since j i , k i ∈ J , f ′ i = ∂ J \{ j i ,k i } [ g i ( β i x n − α i )] = ( β i x n − α i ) · ∂ J \{ j i ,k i } g i . 3. In the ab o ve, eac h h i is an ( X \ ( J ∪ { x n } ))-pre-alig ned R O-ABP-p olynomial. Namely , by Claim 3, g i is ( X \{ x j i , x k i , x n } )-pre-aligned. Hence, us in g Lemma 8, we get th at h i is an ( X \ ( J ∪ { x n } ))-pre-alig ned R O -ABP-p olynomial. F or an y i , d efine f ′′ i = ( f ′ i ) | x n = γ 0 . Then we ha ve the follo w ing three prop erties: 1. ∀ i ∈ E γ 0 , f ′′ i ≡ 0. 2. ∀ i ∈ B γ 0 , f ′′ i = ( β i γ 0 − α i ) h i , so f ′′ i is an ( X \ ( J ∪ { x n } ))-pre-alig ned R O-ABP-p olynomial, due to Pr op osition 4. 18 3. F or eve ry i ∈ [ k ] \ I , ( f i ) | x n = γ 0 is X \{ x n } -pre-aligned. Since n 6∈ J , f ′′ i = ( f ′ i ) | x n = γ 0 = ∂ J [ f | x n = γ 0 ]. S o by Lemma 8, f ′′ i is an ( X \ ( J ∪ { x n } ))-pre-alig ned RO-ABP-polynomial. Wlog. assume that J = { ˜ n + 1 , ˜ n + 2 , . . . , n − 2 , n − 1 } . Th en | J | = n − 1 − ˜ n . Th en P i ∈ [ k ] f ′′ i = ( ∂ J P n ) | x n = γ 0 = γ 0 · P ˜ n . Let ˜ X = { x 1 , . . . , x ˜ n } . W e hav e foun d a repr esen tation of P ˜ n as a sum of ˜ k ˜ X -pre-aligned RO-ABP- p olynomials, where 7 ˜ k ≤ 7( k − | E γ 0 | ) ≤ n − 7 | E γ 0 | = n − 1 − 6 | E γ 0 | + 1 − | E γ 0 | ≤ ˜ n + 1 − | E γ 0 | ≤ ˜ n . This con tradicts the indu ction hyp othesis, and hence n < 7 k . 8 A V anishing Theorem and the PIT Algorithms The f ollo wing theorem is analogous to Theorem 6.4 in [2]. Theorem 9. Supp ose | F | > 3 . L et { f i ∈ F [ X ] } i ∈ [ k ] b e a set of k X -aligne d RO-ABPs. L et f = P i ∈ [ k ] f i . Then f ≡ 0 ⇐ ⇒ f | W n 7 k ≡ 0 . W e need to argue only the “ ⇐ ”-direction. Assume that f | W n 7 k ≡ 0. W e use ind uction on the n u m b er of v ariables n . The base case is when n < 7 k . In this case it follo ws from Lemma 5 that f ≡ 0. F or the ind uction case assume n ≥ 7 k . W e restrict one v ariable at a time. Consid er a v ariable x ℓ , for ℓ ∈ [ n ]. Cons ider a restriction of the p olynomials f i ’s and f to the subspace x ℓ = 0. By condition 2 in the defin ition of aligne d , eac h of the restricted p olynomials f ′ i = f i | x ℓ =0 are ( X \ { x ℓ } )-alig ned. Let f ′ = P k i =1 f ′ i . Clearly , f ′ | W n − 1 7 k = f ′ | W n 7 k ≡ 0. Thus fr om the induction h y p othesis, f ′ = f | x ℓ =0 ≡ 0, whic h implies that x ℓ divides f . Since ℓ was arb itrarily c h osen, this implies that P n = Q k i =1 x i divides f . But since f is multilinea r, this giv es f = c · P n where c is a constan t an d P n = Q i ∈ [ n ] x i . Th us c · P n is the su m of k RO-AB Ps wh ic h are also X -aligned (and therefore certainly X -pre- aligned). Since n ≥ 7 k , by Theorem 8, w e can conclude that c = 0. Hence f ≡ 0. No w we are ready to giv e the iden tit y testing algorithms for Σ k -R O-ABP-p olynomials giv en b y { f i ∈ F [ X ] } i ∈ [ k ] . T he algorithm is s im p le. W e use the fact th at that ∀ v ∈ F n , f ≡ 0 ⇐ ⇒ f ( x 1 + v 1 , x 2 + v 2 , . . . , x n + v n ) ≡ 0. Assuming that we hav e some common alignment v for { f i } i ∈ [ k ] , w e kno w th at eac h f i ( x 1 + v 1 , x 2 + v 2 , . . . , x n + v n ) is X -aligned. In this case, Theorem 9 is applicable, and it suffices to test if the p olynomial ev aluates to zero on the set W n 7 k . Based on the three approac hes to get a common alignment, the algorithms are as f ollo ws: 1. ( Non-black-b ox setting ) By Corollary 3, we ob tain a simultaneous alignment in time O ( k 2 n 7 s ). Then it take s n O ( k ) to test all p oint s in W n 7 k , so the ru nning-time is O ( k 2 n 7 s ) + n O ( k ) . This pro ves Th eorem 4. In this case w e need | F | > k n 2 . 2. ( Semi-black-b ox setting ) By Corollary 4 , we obtain a simultaneous alignmen t in time k 2 n O (log n ) . Then it tak es n O ( k ) to test all p oints in W n 7 k , so the runn ing-time is k 2 n O (log n ) + n O ( k ) . This pr o ves Theorem 5. In this case we need | F | > k n 2 . 3. ( Black-b ox setting ) In this case we only hav e blac k-b o x access to f = P i ∈ [ k ] f i . Let f v ( x 1 , . . . , x n ) = f ( x 1 + v 1 , . . . , x n + v n ). Then it is easy to see that f ≡ 0 ⇐ ⇒ ∀ v ∈ A k , f v | W n 7 k ≡ 0. In th is case th e runn ing-time is n O (log n + k ) . This p ro ves Theorem 2. In th is case w e n eed | F | > k n 4 . 19 References [1] A. Shpilk a and I. V olk ovic h. Read-once p olynomial identit y testing. In Pr o c e e dings of the 40th Annual STOC , pages 507–516 , 2008. [2] A. Shpilk a and I. V olk o vich. Impro ved p olynomial iden tit y testing of r ead-once form u las. In Appr oximation, R andomization and Combinatorial Optimization. Algorith ms and T e chniqu e s, volume 5687 of LNCS , p ages 700–713, 2009. [3] M. Agra w al. Pro ving lo wer b ound s via pseudo-random generators. In Pr o c. 25th Annual Confer enc e on F oundations of Softwar e T e chnolo gy and The or etic al Computer Scienc e , pages 92–10 5, 2005. [4] N. Saxena. Progress of p olynomial iden tity testing. T ec h nical R ep ort ECCC TR09-101, Elec- tronic Colloqu ium in C omputational C omp lexit y , 2009. [5] J.T. Sch w artz. F ast pr ob ab ilistic algorithms for p olynomial iden tities. J . Assn. Comp. Mach. , 27:701 –717, 1980. [6] R. Zipp el. Pr obabilistic algorithms for sp arse p olynomials. In Pr o c e e dings of the International Symp osium on Symb olic and Algebr aic Manipulation (EUROSAM ’79) , v olume 72 of L e ct. Notes i n Comp. Sci. , p ages 216–226. S p ringer V erlag, 1979. [7] V. Kabanets an d R. I mpagliazzo. Derandomizing p olynomial id entit y testing means pr o ving circuit low er b ounds . Computational Complexity , 13(1–2) :1–44, 2004. [8] M. Ben-Or and P . Tiw ari. A d eterministic algorithm for sparse multiv ariate p olynomial in ter- p olation. In Pr o c . 20th Annual ACM Symp osium on the The ory of Computing , pages 301–309. A CM, 1988. [9] A.R. Kliv ans and D.A. Spielman. Rand omness efficient iden tity testing of multi v ariate p oly- nomials. In P r o c. 33r d Annual ACM Symp osium on the The ory of Computing , pages 216–22 3, 2001. [10] R. Lip ton and N. Vishnoi. Deterministic iden tit y testing for multiv ariate p olynomials. In Pr o c e e dings of the fourte enth annual ACM-SIAM symp osium on Discr e te algorithm s (SODA 2003) , pages 756–760 , 2003. [11] Z . Dvir and A. Shp ilk a. L o cally d eco dable co d es with tw o qu eries and p olynomial iden tit y testing for depth 3 circuits. SIAM J. Comput. , 36(5):1404 –1434, 2006. [12] N. Kay al and N. Saxena. Pol ynomial identi t y testing for depth 3 circuits. Computational Complexity , 16(2):115–1 38, 2007. [13] V. Arvind and P . Mukh opadhy ay . T he monomial ideal m em b ership pr oblem and p olynomial iden tity testing. In Pr o c e e dings of the 18th International Symp osium on Algorithms and Com- putation (ISAA C 2007) , v olume 4835 of L e ctur e Notes in Computer Scienc e , pages 800–811. Springer, 2007. 20 [14] Z .S . Karn in and A. Sh p ilk a. Deterministic blac k b o x p olynomial ident it y testing of depth- 3 arithm etic circuits with b ounded top fan-in. In Pr o c. 23r d An nual IEEE Confer enc e on Computation al Complexity , p ages 280–291 , 2008. [15] N. Kay al and S. S araf. Blac k b ox p olynomial identit y testing of depth-3 circuits. In Pr o c. 49th Annual IEEE Symp osium on F oundations of Computer Scienc e , 2009. [16] Z .S . Karnin, P . Mukh ppadhy a y , A. Shp ilk a, and Ily a V olk o vic h . Deterministic ident it y testing of depth 4 multilinear circu its with b ou n ded top fan-in. T ec h nical Rep ort TR09–116, E lectronic Collo q u ium on Comp utational Comp lexit y (EC C C), Nov emb er 2009. [17] R. Raz and A. Shpilk a. Deterministic p olynomial iden tity testing in n on commutat iv e mo dels. Computation al Complexity , 14(1):1–19 , 2005. [18] M. Agra w al an d V. Vina y . Arithmetic circuits: A c hasm at dep th four. In Pr o c. 49th Annual IEEE Symp osium on F oundations of Computer Sci enc e , pages 67–75, 2008. [19] L. V alian t. Completeness classes in algebra. T ec h n ical Rep ort C SR-40-79, Dept. of C omputer Science, Universit y of Edinburgh, April 1979. [20] L. Lo v´ asz. On determinants, matc hing, and random algorithms. In FCT’79 : F undamentals of Computation The ory , pages 565–574 , 1979 . [21] K. Mu lm u ley , U. V azirani, and V. V azirani. Matc hing is as easy as matrix inv ersion. Combi- natoric a , 7:105–113, 1987. [22] E. Allender, K. Reinhardt, and S. Zhou. Isolation, matc hing and coun ting uniform and non u ni- form u pp er b ound s. J. Comput. Syst. Sci. , 59(2):16 4–181 , 199 9. [23] S . Datta, R. Kulk arni, and S. Ro y . Deterministically isolating a p er f ect matc hin g in b ip artite planar graphs. In Pr o c. 25th Annual Symp osium on The or e tic al Asp e c ts of Computer Scienc e , v olume 08001 of L eib niz Int. Pr o c. in Informatics , pages 229–24 0, 2008. [24] M. Jan s en. W eak ening assum ptions for deterministic sub exp onen tial time non-singu lar ma- trix completion, 2010. T o Ap p ear, 27th Inte rnational Symp osium on Theoretica l Asp ects of Computer Science (S T ACS 2010). [25] N. Alon. Combinatorial nullste llensatz. Combinatorics, Pr ob ability and Computing , 8(1–2):7– 29, 1999. A Figure 3 Figure 3 sho w s an R O -ABP computing x 1 x 2 + x 2 x 3 + x n − 1 x n , when n is ev en. The case w hen n is o dd is d ealt with similarly . Unlab eled ed ges are lab eled with 1. B Example : R O-AB Ps Are Not Univ ersal Prop osition 6. The de gr e e-2 elementary symmetric p olynomial e n ( x 1 , x 2 , . . . , x n ) = Q 1 ≤ i

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