Resistant Polynomials and Stronger Lower Bounds for Depth-Three Arithmetical Formulas

Transcription

1 Resistant Polynomials an Stronger Lower Bouns for Depth-Three Arithmetical Formulas Maurice J. Jansen University at Buffalo Kenneth W.Regan University at Buffalo Abstract We erive quaratic lower bouns on the -complexity of sum-of-proucts-of-sums (ΣΠΣ) formulas for classes of polynomials f that have too few partial erivatives for the (near-)quaratic lower boun techniques of Shpilka an Wigerson [SW99, Shp01] (after [NW96]). Our techniques introuce a notion of resistance which connotes full-egree behavior of f uner any projection to an affine space of sufficiently high imension. They also show stronger lower bouns over the reals than the complex numbers or over arbitrary fiels. Separately, by applying a special form of the Baur-Strassen Derivative Lemma tailore to ΣΠΣ formulas, we obtain sharper bouns on +, -complexity than those shown for -complexity in [SW99], most notably for the lowest-egree cases of the polynomials they consier. 1 Introuction In contrast to the exponential size lower bouns on constant-epth Boolean circuits for majority an relate functions [FSS81, Yao85, Hås86], Shpilka an Wigerson [SW99] observe that in arithmetic complexity, over fiels of characteristic zero, super-quaratic lower bouns are not even known for constant-epth formulas. Inee they are unknown for unboune fan-in, epth 3 formulas that are sums of proucts of affine linear functions, which they call ΣΠΣ formulas. These formulas have notable upper-boun power because they can carry out forms of Lagrange interpolation. As they ascribe to M. Ben-Or, ΣΠΣ formulas can compute the elementary symmetric polynomials Sn k (efine as the sum of all egree-k monomials in n variables, an analogous to majority an thresholk Boolean functions) in size O(n 2 ) inepenent of k. Thus ΣΠΣ formulas present a substantial challenge for lower bouns, as well as being a nice small-scale moel to stuy. Shpilka an Wigerson efine the multiplicative size of an arithmetical (circuit or) formula φ to be the total fan-in to multiplication gates. We enote this by l (φ), an write l(φ) for the total fan-in to all gates, i.e. + gates as well. The best known lower boun for general arithmetical circuits has remaine for thirty years the Ω(n log n) lower boun on l by the Degree Metho of Strassen [Str73] (see also [BS82, BCS97]). However, this comes nowhere near the exponential lower bouns conjecture by [Val79] for the permanent an expecte by many for other NP-har arithmetical functions. For polynomials f of total egree n O(1), the metho is not even capable of Ω(n 1+ɛ ) circuit lower bouns, not for any ɛ > 0. Hence it is notable that [SW99] achieve lower bouns on l 3 (f), where the subscript-3 refers to ΣΠΣ formulas, of Ω(n 2 ) for f = Sn k when k = Θ(n), n 2 ɛ k for Sn k with small values of k, an Ω(N 2 / polylog(n)) for the eterminant, with N = n 2. However, Part of this work by both authors was supporte by NSF Grant CCR Corresponing author: Dept. of CSE at UB, 201 Bell Hall, Buffalo, NY ; (716) x114, fax ; regan@cse.buffalo.eu. 1

2 their methos have a similar limitation of Ω(n 2 ) for ΣΠΣ formulas. Shpilka [Shp01] got past this only in some further-restricte cases, an also consiers a epth-2 moel consisting of an arbitrary symmetric function of sums. This barrier provies another reason to stuy the ΣΠΣ moel, in orer to unerstan the obstacles an what might be neee to surpass them. The techniques in [NW96, SW99, Shp01] all epen on the set of th-orer partial erivatives of f being large. This conition fails for functions such as f(x 1,..., x n ) = x n xn n, which has only n th-orer partials for any. We refine the analysis to show the sufficiency of f behaving like a egree-r polynomial on any affine subspace A of sufficiently high imension (for this f, r = n an any affine line suffices). Our technical conition is that for every polynomial g of total egree at most r 1 an every such A, there exists a -th orer partial of f g that is non-constant on A. This enables us to prove an absolutely sharp n 2 boun on l 3 (f) for this f compute over the real or rational numbers, an a lower boun of n 2 /2 over any fiel of characteristic zero. Note the absence of O, Ω notation. We prove similar tight bouns for sums of powere monomial blocks, powers of inner-proucts, an functions epening on l p -norm istance from the origin, an also replicate the bouns of [SW99, Shp01] for symmetric polynomials. Even in the last case, we give an example where our simple existential conition may work eeper than the main question highlighte in [Shp01] on the maximum imension of subspaces A on which Sn k vanishes. In the secon half, we prove lower bouns on +, complexity l 3 (f) that are significantly higher (but still sub-quaratic) than those given for l 3 (f) in [SW99] when the egree r of f is small. This is one intuitively by exploiting a close-form application of the Baur-Strassen Derivative Lemma [BS82] to ΣΠΣ formulas, showing that f an all of its n first partial erivatives can be compute with only a constant-factor increase in l an l over ΣΠΣ formulas for f. 2 Preliminaries A ΣΠΣ-formula is an arithmetic formula consisting of four consecutive layers: a layer of inputs, next a layer of aition gates, then a layer of multiplication gates, an finally the output sum gate. The gates have unboune fan-in from the previous layer (only), an iniviual wires may carry arbitrary constants from the unerlying fiel. Given a ΣΠΣ-formula for a polynomial p, we can write p = s i=1 M i, where M i = Π i j=1 an = c i,j,1 x 1 + c i,j,2 x c i,j,n x n + c i,j,0. Here i is the in-egree of the ith multiplication gate (fix any orer on the multiplication gates), an c i,j,k is nonzero iff there is a wire from x k to the aition gate computing. Note that is homogeneous of egree 1, i.e. strictly linear, if c i,j,0 = 0, an is affine linear otherwise. 2.1 Affine Linear Subspaces an Derivatives An affine linear subspace A of F n is a set of the form A = V + w = { v + w : v V }, where V is a linear subspace of F n, an w is a vector in F n. The imension of A is efine to be the vector space imension of V. Let X = (x 1,..., x n ) be an n-tuple of variables. For any affine subspace A, we can always fin a set of variables B X, an affine linear forms l b in the variables X \ B, for each b B, such that A is the set of solutions of {x b = l b : b B}. This representation is not unique. The set B is calle a base of A. The size B always equals the co-imension of A. 2

3 In the following, whenever we consier an affine linear subspace A, we assume we have fixe some base B of A. Any of our numerical progress measures use to prove lower bouns will not epen on the choice of a base. The following notion oes epen on the choice of a base: Definition 2.1 ([SW99]). Let A be an affine linear subspace of F n, an let f F [x 1,..., x n ]. Then the restriction of f to A is the polynomial obtaine from f by substituting l b for the variable x b for each b B, is enote by f A. If W is a set of polynomials, efine W A = {f A f W }. For linear polynomials l = c 1 x c n x n + c 0, we enote l h = c 1 x c n x n. For a set S of linear polynomials, S h = {l h : l S}. Observe that if S h is an inepenent set, then the set of common zeroes of S is affine linear of imension n S. 3 Resistance of polynomials We state our new efinition in the weakest an simplest form that suffices for the lower bouns, although the functions in our applications all meet the stronger conition of Lemma 3.3 below. Definition 3.1. A polynomial f in variables x 1, x 2,..., x n is (, r, k)-resistant if for any polynomial g(x 1, x 2,..., x n ) of egree at most r 1, for any affine linear subspace A of co-imension k, there exists a th orer partial erivative of f g that is non-constant on A. For a multiset X of size with elements taken from {x 1, x 2,..., x n }, we will use the notation f to inicate the th-orer erivative with respect to the variables in X. As our applications all have r = eg(f), we call f simply (, k)-resistant in this case. Then the case = 0 says that f itself has full egree on any affine A of co-imension k, an in most cases correspons to the nonvanishing conition in [SW99]. We separate our notion from [SW99] in applications an notably in the important case of the elementary symmetric polynomials in Section 3.2 below. The conclusion of Definition 3.1 is not equivalent to saying that some ( + 1)st-orer partial of f g is non-vanishing on A, because the restriction of this partial on A nee not be the same as a first-partial of the restriction of the th-orer partial to A. Moreover, (, k)-resistance nee not imply ( 1, k)-resistance, even for, k = 1: consier f(x, y) = xy an A efine by x = 0. We have the following theorem: Theorem 3.1 Suppose f(x 1, x 2,..., x n ) is (, r, k)-resistant, then l 3(f) r k Proof. Consier a ΣΠΣ-formula that computes f. Remove all multiplication gates that have egree at most r 1. Doing so we obtain a ΣΠΣ formula F computing f g, where g is some polynomial of egree at most r 1. Say F has s multiplication gates. Write: f g = s i=1 M i, where M i = Π i j=1 an = c i,j,1 x 1 + c i,j,2 x c i,j,n x n + c i,j,0. The egree of each multiplication gate in F is at least r, i.e. i r, for each 1 i s. Now select a set S of input linear forms using the following algorithm: 3

4 S = for i = 1 to s o repeat + 1 times: if( j {1, 2,..., i }) such that S h {li,j h } is a set of inepenent vectors then S = S { } Let A be the set of common zeroes of the linear forms in S. Since S h is an inepenent set, A is affine linear of co-imension S ( + 1)s. Claim 3.2 If at a multiplication gate M i we picke strictly fewer than + 1 linear forms, then any linear form that was not picke is constant on A. Proof. Each linear form l that was not picke ha l h alreay in the span of S h, for the set S built up so far. Hence we can write l = c + l h = c + g S c gg h, for certain scalars c g. Since each g h is constant on A, we conclue l is constant on A. We conclue that for each multiplication gate at least one of the following hols: 1. ( + 1) input linear forms vanish on A, or 2. fewer than ( + 1) linear form vanishes on A, an all others are constant on A. For each multiset X of size with elements from { x 1, x 2,..., x n }, the th orer partial erivative is in the linear span of the set i (f g) { l ij : 1 i s, J {1, 2,..., i }, J = } j=1 j / J This follows from the sum an prouct rules for erivatives an the fact that a first orer erivative of an iniviual linear form l ij is a constant. Consier 1 i s an J {1, 2,..., i } with J =. If item 1. hols for the multiplication gate M i, then i j=1 j / J (1) l ij (2) vanishes on A, since there must be one l ij that vanishes on A that was not selecte, given that J =. If item 2 hols for M i, then (2) is constant on A. Hence, we conclue that (1) is constant on A. Since f is (, r, k)-resistant, we must have that the co-imension of A is at least k + 1. Hence ( + 1)s k + 1. Since each gate in F is of egree at least r, we obtain l 3(F) r k Since F was obtaine by removing zero or more multiplication gates from a ΣΠΣ-formula computing f, we have proven the statement of the theorem. To prove lower bouns on resistance, we supply the following lemma that uses the syntactic notion of affine restriction. In certain cases this will be convenient. 4

5 Lemma 3.3 Over fiels of characteristic zero, for any r, k > 0, an any polynomial f(x 1, x 2,..., x n ), if for every affine linear subspace A of co-imension k, there exists some th orer partial erivative of f such that ( ) f eg( ) r + 1 then f is (, r + 1, k)-resistant. Proof. Assume for every affine linear subspace A of co-imension k, there exists some th orer partial erivative erivative of f such that ( ) f eg( ) r + 1. Let g be an arbitrary polynomial of egree r. Then ( ) f g = ( ) f g = ( ) f ( ) The term f has egree at least r + 1, whereas the term ( ) r. Hence eg( f g ( ) g. ( ) g can have egree at most ) r Since over fiels of characteristic zero, syntactically ifferent polynomials efine ifferent mappings, we conclue f g must be non-constant on A. The main ifference between Lemma 3.3 an the original Definition 3.1 appears to be the orer of quantifying the polynomial g of egree r 1 out front in the former, whereas analogous consierations in the lemma universally quantify it later (making a stronger conition). We have not foun a neat way to exploit this ifference in any prominent application, however. 3.1 Applications We will now prove lower bouns on the ΣΠΣ-formula size of several explicit polynomials Sum of Nth Powers Polynomial Consier f = n i=1 x n i. For this polynomial we have ΣΠ-circuits of size O(n log n). This can be shown to be optimal using Strassen s egree metho. By that metho we know any circuit for f has size Ω(n log n). The obvious ΣΠΣ-formula has aitive size n 2 wires in the top linear layer, an has n multiplication gates of egree n. We prove that this is essentially optimal. has multiplica- Theorem 3.4 Over fiels of characteristic zero, any ΣΠΣ-formula for f = n i=1 x n i tive size at least n 2 /2. Proof. We will show that f is (1, n 1)-resistant. The result then follows from Theorem 3.1. Let g be an arbitrary polynomial of egree eg(f) 1 = n 1. Letting g 1,..., g n enote the first orer partial erivatives of g, we get that the ith partial erivative of f g equals nx n 1 i g i (x 1,..., x n ). 5

6 Note that the g i s are of total egree at most n 2. We claim there is no affine linear subspace of imension greater than zero on which all f/ x i are constant. Consier an arbitrary affine line, parameterize by a variable t: x i = c i + i t, where c i an i are constants for all i [n], an with at least one i nonzero. Then (f g) x i restricte to the line is given by n(c i + i t) n 1 h i (t), for some univariate polynomials h i (t) of egree n 2. Since there must exist some i such that i is nonzero, we know some partial erivative restricte to the affine line is parameterize by a univariate polynomial of egree n 1, an thus, given that the fiel is of characteristic zero, is not constant for all t. In case the unerlying fiel is the real numbers R an n is even, we can improve the above result to prove an absolutely tight n 2 lower boun. We start with the following lemma: Lemma 3.5 Let f = n i=1 x n i. Over the real numbers, if n is even, we have that for any affine linear subspace A of imension k 1, eg(f A ) = n. Proof. Since f is symmetric we can assume without loss of generality that the following is a base representation of A: Then x k+1 = l 1 (x 1,..., x k ) x k+2 = l 2 (x 1,..., x k ). x n = l n k (x 1,..., x k ). f A = x n x n k + l n l n n k. We conclue that f A must inclue the term x n 1, since each ln j has a non-negative coefficient for the term x n 1, since n is even. has multi- Theorem 3.6 Over the real numbers, for even n, any ΣΠΣ-formula for f = n i=1 x n i plicative size at least n 2. Proof. Using Lemma s 3.3 an 3.5 we conclue that over the real numbers f is (0, n 1)-resistant. Hence, by Theorem 3.1 we get that l 3 (f) eg(f)n 1 = n2. Let us note that f = n i=1 x n i is an example of a polynomial that, even for large, has relatively few, namely only n, partial erivatives. This makes application of the partial erivatives technique of [SW99], which we will escribe an exten in the next section, problematic. 6

7 3.1.2 Blocks of Powers Let the unerlying fiel have characteristic zero, an suppose n = m 2 for some m. Consier the m blocks of m powers polynomial m im f = x m j. i=1 j=(i 1)m+1 The straightforwar ΣΠΣ-formula for f, that computes each term/block using a multiplication gate of egree n, is of multiplicative size n 3/2. We will show this is tight. Proposition 3.7 The blocks of powers polynomial f efine above is (0, m 1)-resistant. Proof. Consier an affine linear space of co-imension m 1. For any base B of A, restriction to A consists of substitution of the m 1 variables in B by linear forms in the remaining variables X/B. This means there is at least one term/block B i := im j=(i 1)m+1 x m j of f whose variables are isjoint from B. This block B i remains the same uner restriction to A. Also, for every other term/block there is at least one variable that is not assigne to. As a consequence, B i cannot be cancele against terms resulting from restriction to A of other blocks. Hence eg(f A ) = eg(f). Hence by Lemma 3.3 we have that f is (0, m 1)-resistant. Corollary 3.8 For the blocks of powers polynomial f efine above, l 3 (f) nm = n3/2. Proof. Follows immeiately from Theorem 3.1 an Proposition 3.7. Alternatively, one can observe that by substitution of a variable y i for each variable appearing in the ith block one obtains from a ΣΠΣ-formula F for f a formula for f = m i=1 yi n of the same size as F. Theorem 3.4 generalizes to show that l 3 (f ) 1 2 n3/2, which implies l 3 (f) 1 2 n3/ Polynomials epening on istance to the origin Over the real numbers, 2 (x) = x x x2 n is the square of the Eucliean istance of the point x to the origin. Polynomials f of the form q( 2 (x)) where q is a single-variable polynomial can be reaily seen to have high resistance. Only the leaing term of q matters. For example, consier f = (x x x2 n) m. On any affine line L in R n, eg(f L ) = 2m. Therefore, by Lemma 3.3, over the reals, f is (0, n 1)-resistant. Hence by Theorem 3.1 we get that Proposition 3.9 Over the real numbers, l 3 ((x2 1 + x x2 n) m ) 2mn. Observe that by reuction this means that the mth-power of an inner prouct polynomial, efine by g = (x 1 y 1 + x 2 y x n y n ) m, must also have ΣΠΣ-size at least 2mn over the reals numbers. Results for l p norms, p 2, are similar. 3.2 Symmetric Polynomials The special case of (0, k)-resistance implicitly appears in [Shp01], at least insofar as the sufficient conition of Lemma 3.3 is use for the special case = 0 in which no erivatives are taken. For the elementary symmetric polynomial S r n of egree r 2 in n variables, Theorem 4.3 of [Shp01] implies (via Lemma 3.3) that S r n is (0, n n+r 2 )-resistant. Shpilka proves for r 2, l 3(S r n) = Ω(r(n r)), 7

8 which can be verifie using Theorem 3.1: l 3 (Sn) r (r + 1)(n n+r 2 ) = Ω(r(n r)). For r = Ω(n) this yiels a tight Ω(n 2 ) boun as observe in [Shp01]. The symmetric polynomials Sn k collectively have the telescoping property that every th-orer partial is (zero or) the symmetric polynomial Sn k on an (n )-subset of the variables. Shpilka [Shp01] evolves the analysis into the question, What is the maximum imension of a linear subspace of C n on which Sn r vanishes? In Shpilka s answer, ivisibility properties of r come into play as is witnesse by Theorem 5.9 of [Shp01]. To give an example case of this theorem, one can check that S9 2 vanishes on the 3-imensional linear space given by {(x 1, ωx 1, ω 2 x 1, x 2, ωx 2, ω 2 x 2, x 3, ωx 3, ω 2 x 3 ) : x 1, x 2, x 3 C}, where ω can be selecte to be any primitive 3r root of unity. Let ρ 0 (f) = max{k : for any linear space A of coimension k, f A 0} Shpilka prove for r > n/2, that ρ 0 (Sn) r = n r, an for r 2, that n r 2 < ρ 0 (Sn) r n r. For S9 2 we see via ivisibility properties of that the value for ρ 0 can get less than the optimum value, although the n r 2 lower boun suffices for obtaining the above mentione l 3 (Sn) r = Ω(r(n r)) lower boun. We have some inication from computer runs using the polynomial algebra package Singular [GPS05] that the unruly behaviour seen for ρ 0 because of ivisibility properties for r n/2 can be mae to go away by consiering the following notion: ( ) f ρ 1 (f) = max{k : for any linear space A of coimension k, there exists i, 0} x i One can still see from the fact that Sn r is homogeneous an using Lemma 3.3 an Theorem 3.1 that l 3 (Sr n) r (ρ 1(Sn r )+1) 2. Establishing the exact value of ρ 1 (Sn), r which we conjecture to be n + 1 r at least over the rationals, seems at least to simplify obtaining the l 3 (Sn) r = Ω((n )) lower boun. In any case we can prove the following relation between the two notions: Lemma 3.10 For r 2, ρ 1 (S r+1 n+1 ) ρ 0(S r 1 n ). Proof. Suppose A is a linear space of coimension k = ρ 1 (Sn+1 r+1 ) + 1 over {x 1, x 2,..., x n+1 } such that all partials of Sn+1 r+1 vanish on A, i.e. for every i, S r n(x 1,..., x i,..., x n+1 ) A = 0. (3) Here we enote x i to inicate that the variable x i is exclue. If k = n + 1, then ρ 1 (Sn+1 r+1 ) = n, so the statement of the lemma is trivial. Otherwise, some variable is not being substitute for by the restriction. By symmetry we can assume wlog. that this is x n+1. Hence we can write Equation (3) as: for all i, an x n+1 S r 1 n 1 (x 1,..., x i,..., x n ) A + S r n 1(x 1,..., x i,..., x n ) A = 0, S r n(x 1,..., x n ) A = 0. Aing the first n equations an subtracting the last n r times one obtains 0 = x n+1 n i=1 S r 1 n 1 (x 1,..., x i,..., x n ) A = (n r)x n+1 S r 1 n (x 1,..., x n ) A 8

9 In other wors Sn r 1 (x 1,..., x n ) A = 0. Simplify the substitution corresponing to A by taking x n+1 = 0 in the efining k equations. We obtain a set of k substitutions on k variables from {x 1, x 2,..., x n }. This efines a linear space of coimension k on which S r 1 n k = ρ 1 (S r+1 n+1 ) + 1. vanishes. So ρ 0 (S r 1 n ) < For another example, S6 3 is mae to vanish at imension 3 not by any subspace that zeroes out 3 co-orinates but rather by A = { (u, u, w, w, y, y) : u, w, y C }. Now a a new variable t in efining f = S7 4. The notable fact is that f 1-resists the imension-3 subspace A obtaine by ajoining t = 0 to the equations for A, upon existentially choosing to erive by a variable other than t, such as u. All terms of f/ u that inclue t vanish, leaving 10 terms in the variables v, w, x, y, z. Of these, 4 pairs cancel uner the equations x = w, z = y, but the leftover vwx + vyz part equates to uw 2 + uy 2, which not only oesn t cancel but also ominates any contribution from the lower-egree g. Gröbner basis runs using Singular imply that S7 4 is (1, 4)-resistant over C as well as the rationals an reals, though we have not yet mae this a consequence of a general resistance theorem for all Sn. r Hence our (1, k)-resistance analysis for S7 4 is not impacte by the achieve upper boun of 3 represente by A. Amittely the symmetric polynomials f have O(n 2 ) upper bouns on l 3 (f), so our istinction in this case oes not irectly help surmount the quaratic barrier. But it oes show promise of making progress in our algebraic unerstaning of polynomials in general. 4 Bouns for +,*-Complexity The partial erivatives technique of [SW99] ignores the wires of the formula present in the first layer. In the following we show how to account for them. As a result we get a sharpening of several lower bouns, though not on l 3 but on total formula size. We refine the [SW99] result for *-complexity: Theorem 4.1 ([SW99]) Let f F [x 1,..., x n ]. Suppose for integers, D, κ it hols that for every affine subspace A of co-imension κ, im( (f) A ) > D. Then to our result for +,*-complexity: l 3(f) min( κ2, D )); Theorem 4.2 (new) Let f F [x 1,..., x n ]. Suppose for integers, D, κ it hols that for every affine subspace A of co-imension κ, n i=1 im[ ( f x i ) A ] > D. Then l 3 (f) min( κ2 + 2, (κ+ D )). Comparing the two theorems, we see that the result by Shpilka an Wigerson provies a lower boun on multiplicative complexity, while our result gives a lower boun on the total number of wires. We o get an extra factor n of aitions with the n i=1 im[ ( f x i ) A ] > D conition compare to just im( (f) A ) > D. Potentially this can lea to improve lower bouns on the total size of the formula, better than one woul be able to infer from the lower boun on multiplicative complexity of Theorem 4.1 alone. We shall see that we can inee get such kins of improvements in the applications section below. 9 (κ+

10 We employ the following suite of concepts an lemmas from [SW99] irectly. Definition 4.1 ([SW99]). For f F [x 1,..., x n ], let (f) be the set of all th orer formal partial erivatives of f w.r.t. variables from {x 1,..., x n }. For a set of polynomials A = {f 1,..., f t }, let span(a) = { t i=1 c i f i c i F }, i.e., span(a) is the linear span of A. We write im[a] as shorthan for im[span(a)]. We have the following elementary sub-aitivity property for the measure im[ (f)]. Proposition 4.3 ([SW99]) For f 1, f 2 F [x 1,..., x n ] an constants c 1, c 2 F, im[ (c 1 f 1 + c 2 f 2 )] im[ (f 1 )] + im[ (f 2 )]. One also nees to boun the growth of im[ (f)] in case of multiplication. For multiplication of affine linear forms, we have the following two bouns. Proposition 4.4 ([SW99]) Let M = Π m i=1 l i, where each l i is affine linear. Then im[ (M)] ( ) m. Proposition 4.5 ([SW99]) Let M be a prouct gate with im[m h ] = m, then for any, ( ) m + im[ (M)]. Note that for polynomials f 1,..., f s, span(f 1,..., f s ) A = span(f 1 A,..., f s A ), an that im[w A ] im[w ]. Now we moify Proposition 4.3 a little to get a result implicitly use by Shpilka an Wigerson in their arguments. Proposition 4.6 (cf. [SW99]) For f 1, f 2 F [x 1,..., x n ] an constants c 1, c 2 F, an affine linear subspace A, we have that im[ (c 1 f 1 + c 2 f 2 ) A ] im[ (f 1 ) A ] + im[ (f 2 ) A ]. Finally, we require: Lemma 4.7 ([SW99]) For every n, κ,, an every affine subspace A of co-imension κ, we have that ( ) n κ im[ (Sn 2 ) A ]. Now we can prove our sieways improvement of Shpilka an Wigerson s main Theorem 3.1 [SW99]. Proof of Theorem 4.2. Consier a minimum-size ΣΠΣ-formula for f with multiplication gates M 1,..., M s. We have that f = s i=1 M i, where for 1 i s, M i = Π i j=1 an = c i,j,1 x 1 + c i,j,2 x c i,j,n x n + c i,j,0, 10

11 for certain constants c i,j,k F. Computing the partial erivative of f w.r.t. variable x k we get Let f x k = s i i=1 j=1 S = {i : im[m h i ] κ}. c i,j,k M i. (4) If S κ +2, then l 3(f) κ2 κ +2. Suppose S < +2. If S =, then let A be an arbitrary affine subspace of co-imension κ. Otherwise, construct an affine space A as follows. Since S ( + 2) < κ, an since for each j S, im[mi h] κ, it is possible to pick + 2 input linear forms l j,1,..., l j,+2 of each multiplication gate M j with j S, such that {lj,1 h,..., lh j,+2 j S} is a set of S ( + 2) < κ inepenent homogeneous linear forms. Define A = {x : (x) = 0, for any i S, j [ + 2]}. We have that the co-imension of A is at most κ. W.l.o.g. assume the co-imension of A equals κ. For each i S, +2 linear forms of M i vanish on A. This implies that im[ ( M i ) A ] = 0. for any i S. For any i / S, by Proposition 4.5, im[ ( M i ) A ] < ( ) κ +. Let D k = im[ ( f x k ) A ]. By Proposition 4.6 an equation (4), Hence there must be at least D k i/ S D k ( κ+ ) j c i,j,k 0 im[ ( M i ) A ]. terms on the r.h.s., i.e. there are at least that many wires from x k to gates in the first layer. Hence in total the number of wires to the first layer is at least ni=1 D i ( κ+ ) > ( D κ+ ). We can apply a similar iea to aapt the other main theorem from [SW99]: Theorem 4.8 ([SW99]) Let f F [x 1,..., x n ]. Suppose for integers, D, κ it hols that for every affine subspace A of co-imension κ, im( (f A )) > D. Then for every m 2, l 3(f) min(κm, D ( m )). Theorem 4.9 (new) Let f F [x 1,..., x n ]. Suppose for integers, D, κ with 1, it hols that for every affine subspace A of co-imension κ, n i=1 im[ ( f x i A )] > D. Then for every m 2, l 3 (f) min( 1 2 κm, D ( m 1 )). 11

12 Proof. Consier a minimum size ΣΠΣ-formula for f with multiplication gates M 1,..., M s. We have that f = s i=1 M i, where for 1 i s, M i = Π i j=1, with = c i,j,1 x 1 + c i,j,2 x c i,j,n x n + c i,j,0. If there are κ 2 multiplication gates M i of egree greater than m then alreay l 3 (f) > 1 2κm. So suppose the number t of multiplication gates of egree greater than m is less than κ 2. Wlog. assume these gates are given by M 1, M 2,..., M t. For i = 1, 2,..., t, pick two input linear forms l i,1, l i,2 of M i, such that for the total collection l 1,1, l 1,2,..., l i,1, l i,2 we have that the strictly linear parts l1,1 h, lh 1,2,..., lh i,1, lh i,2 are inepenent. It might be that at some i t, we cannot fin any l i,1 or l i,2 with li,1 h or lh i,2 inepenent from the previously collecte linear forms. In this case, we just pick l i,1 if that one is still inepenent, an skip to the next inex i. If we can t even fin l i,1 for which l i,1 is inepenent, we pick no linear form an procee to the next i. Let A be the zero set of all the collecte input linear forms. Then A has co-imension at most κ. Wlog. we may assume that the co-imension of A equals κ. Observe that f x k A = s i i=1 j=1 c i,j,k ( M i ) A. (5) Now for a multiplication gate M i of egree m, there are three cases: either we picke two input linear forms of M i, or we picke just one, or none at all. In the first case, ( M i ) A = 0 in the r.h.s. of (5), for all i, j. In the secon an thir case, we know that for every input l of M i that was not picke, l h is a linear combination of l h i s for l i s that were picke. Hence r l h A = c i (li h ) = constant. i=1 As a consequence, ( M i ) A = constant in the r.h.s. of (5), for all i, j. Since 1, in either three cases, we obtain that ( M i ) = 0. For multiplication gates M i of egree at most m, Proposition 4.4 gives A us that im[ (( M i ) A )] ( m 1). Let Dk = im[ ( f x k )]. By Proposition 4.3, we see there are at A least D k / ( m 1) terms in (5). This implies that there are at least that many wires fanning out of xk. Aing up for all variables, we conclue that l 3 (f) D/ ( m 1). 4.1 Some Applications In [SW99] it was prove that for log n, l 3 (S2 n ) = Ω( n +2 2 ). Note for = 2, this lower boun is only Ω(n). We can apply Theorem 4.2 to prove the following stronger lower boun on the total formula size of Sn 2. In particular for = 2, we get an Ω(n 4 3 ) boun. 12

13 Theorem 4.10 For 1 log n, l 3 (Sn 2 ) = Ω( n +1 2 ). Proof. For any affine subspace A of co-imension κ an 2 we have that ( ) n im[ 1 ( S2 n n κ ) x A ] im[ (Sn 2 ) A ]. i i=1 The latter inequality follows from Lemma 4.7. Applying Theorem 4.2 we get that l 3 (S 2 n ) min( κ2 + 1, ( n κ ) ( κ+ 1 1 )) = min( κ2 + 1, ( n κ ) ( κ+ ) κ + ). (6) Set κ = 1 9 n +1. Then we have that ( n κ ) ( κ+ ) κ + ( n κ κ + κ + ) ( 8/9n 2/9n +1 ) κ + = 4 n κ n n +1. Hence (2) is at least min( n (+1), n 2 +1 ) = Ω( n +1 2 ). Corollary 4.11 l 3 (S 4 n) = Ω(n 4/3 ). Shpilka an Wigerson efine the prouct-of-inner-proucts polynomial over 2 variable sets of size n (superscript inicate ifferent variables, each variable has egree one) by n P IPn = x i jyj. i i=1 j=1 Theorem 4.12 For any constant > 0, l 3 (P IP n) = Ω(n 2 +1 ). Proof. Let f = P IP n. Essentially we have that f x i j = y i jp IP 1 n, where the P IPn 1 must be chosen on the appropriate variable set. Let A be an arbitrary affine linear subspace of co-imension κ. Then n i=1 j=1 im[ 1 ( f x i )] = j n i=1 j=1 im[ 1 (y i jp IP 1 n )] (n κ) im[ 1 (P IP 1 n )] The last inequality follows because at least n κ of the y-variables are not assigne to with the restriction to A. From Lemma 4.9 in [SW99] one gets im[ 1 (P IP 1 n ) n κn 2. 13

14 Using Theorem 4.9 we get Taking κ = n +1, one gets for constant that l 3 (f) min( κ2 2, (n κ)(n κn 2 ) ( κ 1 ) ). 1 l 3 (P IP n) = Ω(n 2 +1 ). For comparison, in [SW99] one gets l 3 (P IP n) = Ω(n 2 +2 ). 5 Conclusion Possible Further Tools We have taken some further steps after [SW99], obtaining absolutely tight (rather than asymptotically so) multiplicative size lower bouns for some natural functions, an obtaining somewhat improve bouns on +, -size for low-egree symmetric an prouct-of-inner-prouct polynomials. However, these may if anything enhance the feeling from [SW99, Shp01] that the concepts being employe may go no further than quaratic for lower bouns. One cannot after all say that a function f(x 1,..., x n ) is non-vanishing on an affine-linear space of co-imension more than n. The quest then is for a mathematical invariant that scales beyon linear with the number of egree--or-higher multiplication gates in the formula. Notably absent in current lower boun techniques for ΣΠΣ-formulas are ranom restriction type arguments, whereas many of the results for Boolean constant epth circuits of [Ajt83, FSS81, Yao85, Hås89] procee using ranom restrictions. Note that Raz manage to use ranom restrictions in conjunction with a partial erivatives base technique in his work on multilinear arithmetical formulas [Raz04a, Raz04b]. We speculate that the weaker existential requirement in our resistance notion may help it aapt to ranom-restriction scenarios, although plenitue of kth-partials may still be neee to ensure the function oes not collapse too far. In any event, the search for stronger mathematical techniques to prove exponential lower bouns, even in the self-containe ΣΠΣ formula case, continues. Acknowlegments We thank Avi Wigerson for comments on a very early version of this work, an referees of a later version for very helpful criticism. References [Ajt83] M. Ajtai. Σ formulae on finite structures. Annals of Pure an Applie Logic, 24:1 48, [BCS97] P. Bürgisser, M. Clausen, an M.A. Shokrollahi. Algebraic Complexity Theory. Springer Verlag, [BS82] W. Baur an V. Strassen. The complexity of partial erivatives. Theor. Comp. Sci., 22: , [FSS81] M. Furst, J. Saxe, an M. Sipser. Parity, circuits, an the polynomial-time hierarchy. In Proc. 22n Annual IEEE Symposium on Founations of Computer Science, pages ,

15 [GPS05] G.-M. Greuel, G. Pfister, an H. Schönemann. Singular 3.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, [Hås86] [Hås89] [NW96] J. Håsta. Almost optimal lower bouns for small-epth circuits. In Proc. 18th Annual ACM Symposium on the Theory of Computing, pages 6 20, J. Håsta. Almost optimal lower bouns for small-epth circuits. In S. Micali, eitor, Ranomness an Computation, volume 5 of Avances in Computing Research, pages JAI Press, Greenwich, CT, USA, N. Nisan an A. Wigerson. Lower bouns on arithmetic circuits via partial erivatives. Computational Complexity, 6: , [Raz04a] R. Raz. Multilinear formulas for permanent an eterminant are of super-polynomial size. In Proc. 36th Annual ACM Symposium on the Theory of Computing, to appear; also ECCC TR [Raz04b] R. Raz. Multilinear NC 1 multilinear NC 2. In Proc. 45th Annual IEEE Symposium on Founations of Computer Science, pages , [Shp01] A. Shpilka. Affine projections of symmetric polynomials. In Proc. 16th Annual IEEE Conference on Computational Complexity, pages , [Str73] V. Strassen. Berechnung un Programm II. Acta Informatica, 2:64 79, [SW99] A. Shpilka an A. Wigerson. Depth-3 arithmetic formulae over fiels of characteristic zero. Technical Report 23, ECCC, [Val79] L. Valiant. The complexity of computing the permanent. Theor. Comp. Sci., 8: , [Yao85] A. Yao. Separating the polynomial-time hierarchy by oracles. In Proc. 26th Annual IEEE Symposium on Founations of Computer Science, pages 1 10,