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 an Kenneth W.Regan University at Buffalo (SUNY) 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 techniques of Shpilka an Wigerson [10, 9]. This involves 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 by Shpilka an Wigerson [10], most notably for the lowest-egree cases of the polynomials they consier. Keywors. Computational complexity, arithmetical circuits, lower bouns, constant epth formulas, partial erivatives. 1 Introuction In contrast to the presence of exponential size lower bouns on constant-epth Boolean circuits for majority an relate functions [3, 13, 7], an epth-3 arithmetical circuits over finite fiels [5, 6], Shpilka an Wigerson [10] observe that over fiels of characteristic zero (which are infinite), super-quaratic lower bouns are not known even 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 S k n (efine as the sum of all egree-k monomials in n variables, an analogous to majority an threshol-k 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 [11] (see also [1, 2]). However, this comes nowhere near the exponential lower bouns conjecture by Valiant [12] for the permanent an expecte by many for 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.

2 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 [10] achieve better lower bouns on l 3(f), where the subscript-3 refers to ΣΠΣ formulas. These were 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, Ω(n 2 ) is the best this can o for ΣΠΣ formulas. Shpilka [9] got past this only in some further-restricte cases, an also consiere 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 [8, 10, 9] 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 x n 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 [10, 9] 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 [9] on the maximum imension of subspaces A on which Sn k vanishes. In Section 5 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 [10] when the egree r of f is small. This is one intuitively by exploiting a close-form application of the Baur-Strassen Derivative Lemma [1] 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 we can write p = s i=1 M i, where M i = Π i j=1 l i,j, an l i,j = 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, an c i,j,k is nonzero iff there is a wire from x k to the aition gate computing l i,j. Let X = (x 1,..., x n ) be an n-tuple of variables. For any affine linear subspace A F n, we can always fin a set of variables B X, an affine linear forms

3 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. In the following, we always assume some base B of A to be fixe. Any of our numerical progress measures use to prove lower bouns will not epen on the choice of a base. Following Shpilka an Wigerson [10], for polynomial f F [x 1,..., x n ], the restriction of f to A is efine to be the polynomial obtaine by substitution of l b for variable x b for each b B, an is enote by f A. For a set of polynomials W, efine W A = {f A f W }. For a linear form 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 forms, S h = {l h : l 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 1 below. Definition 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 non-vanishing conition in [10]. We separate our notion from [10] in applications an notably in the important case of the elementary symmetric polynomials in Section 4.4 below. The conclusion of Definition 1 is not equivalent to saying that some (+1)storer 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. Theorem 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 l i,j an l i,j = 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:

4 S = for i = 1 to s o repeat + 1 times: if ( j {1, 2,..., i }): S h {li,j h } is a set of inepenent vectors then S = S {l i,j } 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. We claim that 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. Namely, 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. This settles the claim, an yiels that for each multiplication gate either 1. ( + 1) input linear forms vanish on A, or 2. fewer than ( + 1) linear forms vanish on A, with all others constant on A. For each multiset X of size with elements from { x 1, x 2,..., x n }, the th orer partial erivative (f g)/ is in the linear span of the set i { 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 l ij vanishes on A, since there must be one l ij that vanishes j / J on A that was not selecte, given that J =. If item 2 hols for M i, then this prouct is constant on A. Hence, we conclue that (f g)/ 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: Lemma 1. 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.

5 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 ) g =. ( ) ( ) The term f has egree at least r + 1, whereas the term g can ( ) have egree at most r. Hence eg( f g ) 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 1 an the original Definition 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. 4 Applications 4.1 Sum of nth powers polynomial Consier f = n i=1 xn i. By repeate squaring for each xn i, one obtains ΣΠ circuits (not formulas) of size O(n log n). All arithmetical circuits require size Ω(n log n) for f [1]. The expression for f yiels a ΣΠΣ formula φ with n multiplication gates of egree n, with n 2 wires in the top linear layer fanning in to them. This works over any fiel, but makes l(φ) = l (φ) = n 2. We prove that this is close to optimal. Theorem 2. Over fiels of characteristic zero, any ΣΠΣ-formula for f = n i=1 xn i has multiplicative size at least n 2 /2. Proof. By Theorem 1 it suffices to show f is (1, n 1)-resistant. Let g be an arbitrary polynomial of egree 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 ). 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 x i = c i + i t parameterize by a variable 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.

6 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. Theorem 3. Over the real numbers, for even n, any ΣΠΣ-formula for f = n i=1 xn i has multiplicative size at least n 2. Proof. Since f is symmetric we can assume without loss of generality that A has the base representation x k+1 = l 1 (x 1,..., x k ),..., x n = l n k (x 1,..., x k ). Then f A = x n x n k + l n l n n k. Hence f A must inclue the term x n 1, since each lj n has a non-negative coefficient for the term x n 1 an n is even. Thus via Lemma 1 we conclue that over the real numbers f is (0, n 1)-resistant. Hence, by Theorem 1 we get that l 3(f) eg(f) n 1 = n2. Let us note that f = n i=1 xn i is an example of a polynomial that has few -th partial erivatives, namely only n of them regarless of. This reners the partial erivatives technique of Shpilka an Wigerson [10] which we will escribe an exten in the next section not irectly applicable. 4.2 Blocks of powers polynomials Let the unerlying fiel have characteristic zero, an suppose n = m 2 for some m. Consier the m blocks of m powers polynomial f = m im i=1 j=(i 1)m+1 xm j. 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 1. The blocks of powers polynomial f 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 xm 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 1 we have that f is (0, m 1)-resistant. Corollary 1. For the blocks of powers polynomial f, l 3(f) nm = n 3/2. 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 yn i of the same size as F. Theorem 2 generalizes to show that l 3(f ) 1 2 n3/2, which implies l 3(f) 1 2 n3/2.

7 4.3 Polynomials epening on istance to the origin Over the real numbers, 2 (x) = x x x 2 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 x 2 n) m. On any affine line L in R n, eg(f L ) = 2m. Therefore, by Lemma 1, over the reals, f is (0, n 1)-resistant. Hence by Theorem 1 we get that Proposition 2. Over the real numbers, l 3((x x x 2 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 ΣΠΣsize at least 2mn over the reals numbers. Results for l p norms, p 2, are similar. 4.4 The case of symmetric polynomials The special case of (0, k)-resistance is implicitly given by Shpilka [9], at least insofar as the sufficient conition of Lemma 1 is use for the special case = 0 in which no erivatives are taken. For the elementary symmetric polynomial Sn r of egree r 2 in n variables, Theorem 4.3 of [9] implies (via Lemma 1) that Sn r is (0, n n+r 2 )-resistant. Shpilka proves for r 2, l 3(Sn) r = Ω(r(n r)), which can be verifie using Theorem 1: l 3(Sn) r (r + 1)(n n+r 2 ) = Ω(r(n r)). The symmetric polynomials Sn k collectively have the telescoping property that every th-orer partial is (zero or) the symmetric polynomial S k n on an (n )-subset of the variables. Shpilka [9] 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 [9]. 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 either primitive 3r root of unity. Let ρ 0 (f) = max{k : for any linear A of coim. 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 [4] that the unruly behavior 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 A of coim. k, there exists i, 0} x i

8 One can still see from the fact that Sn r is homogeneous an using Lemma 1 an Theorem 1 that l 3(Sn) r r (ρ1(sr n )+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 the full version we prove that for r 2, ρ 1 (Sn+1 r+1 ) ρ 0(Sn r 1 ). 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. 5 Bouns for +,*-Complexity The partial erivatives technique use by Shpilka an Wigerson [10] 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 employ the concepts an lemmas from [10]. 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 } span(a) = { t i=1 c if i c i F }. Write im[a] as shorthan for im[span(a)]. Note span(f 1,..., f t ) A = span(f 1 A,..., f t A ), an that im[w A ] im[w ]. The basic inequality from [10] then becomes: Proposition 3. im[ (c 1 f 1 + c 2 f 2 ) A ] im[ (f 1 ) A ] + im[ (f 2 ) A ]. We refine two main results in [10] -complexity into results with tighter bouns but for +, -complexity. In each case we compare ol an new versions. Theorem 4 ([10]). 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 l 3(f) min( κ2, D ( κ+ ) ); Theorem 5 (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, D ( κ+ ) ).

9 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 l i,j an l i,j = c i,j,1 x 1 +c i,j,2 x c i,j,n x n +c i,j,0, for certain constants c i,j,k F. Computing the partial erivative of f w.r.t. variable x k we get f x k = s i i=1 j=1 c i,j,k M i l i,j. (1) Let S = {i : im[mi h ] κ}. 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 : l i,j (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 coimension of A equals κ. For each i S, +2 linear forms of M i vanish on A. This implies that im[ ( Mi l i,j ) A ] = 0. for any i S. For any i / S, by Proposition 2.3 in [10], im[ ( Mi l i,j ) A ] < ( ) κ+. Let Dk = im[ ( f x k ) A ]. By Proposition 3 an equation (1), D k i / S j c i,j,k 0 D k ( κ+ im[ ( M i l i,j ) A ]. Hence there must be at least 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 n D i i=1 ( κ+ ) > ( D κ+ ). Theorem 6 ([10]). 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 D 3(f) min(κm, ( m ) ). Theorem 7 (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 ) ). The proof of Theorem 7 is analogous to above an appears in the full version. In [10] it was prove that for log n, l 3(Sn 2 ) = Ω( n +2 2 ). Note that for = 2, this lower boun is only Ω(n). We can apply Theorem 5 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.

10 Theorem 8. 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 ) A ] im[ (Sn 2 ) A ]. x i i=1 The latter inequality follows from Lemma 4.4 in [10]. Applying Theorem 5 we get that ( n κ ) ( n κ ) l 3 (Sn 2 ) min( κ2 + 1, )) = min( κ2 + 1, ) κ + ). (2) ( κ+ 1 1 Set κ = 1 9 n +1. Then we have that ) ( n κ ( κ+ ) κ+ ( κ+ (n κ κ+ ) κ+ ( 8/9n ) κ+ 2/9n +1 = 4 n κ n n +1. Hence (2) is at least min( n (+1), n 2 +1 ) = Ω( n +1 2 ). Corollary 2. 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 P IP n = i=1 n j=1 xi j yi j. Theorem 9. For any constant > 0, l 3 (P IP n) = Ω(n 2 +1 ). Proof. Let f = P IPn. Essentially we have that f = y i 1 x i jp IPn, where the j 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 [10] one gets Using Theorem 7 we get im[ 1 (P IP 1 n ) n κn 2. l 3 (f) min( κ2 2, (n κ)(n κn 2 ) ( κ 1 ) ). Taking κ = n +1, one gets for constant that l3 (P IPn) = Ω(n 2 +1 ). For comparison, in [10] one gets l 3(P IPn) = Ω(n 2 +2 ). 1

11 6 Conclusion We have taken some further steps after Shpilka an Wigerson [10, 9], 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 [10, 9] 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. Acknowlegments We thank Avi Wigerson for comments on a very early version of this work, an referees of later versions for very helpful criticism. References 1. W. Baur an V. Strassen. The complexity of partial erivatives. Theor. Comp. Sci., 22: , P. Bürgisser, M. Clausen, an M.A. Shokrollahi. Algebraic Complexity Theory. Springer Verlag, M. Furst, J. Saxe, an M. Sipser. Parity, circuits, an the polynomial-time hierarchy. Math. Sys. Thy., 17:13 27, 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, D. Grigoriev an M. Karpinski. An exponential lower boun for epth 3 arithmetic circuits. In Proc. 30th Annual ACM Symposium on the Theory of Computing, pages , D. Grigoriev an A.A. Razborov. Exponential lower bouns for epth 3 algebraic circuits in algebras of functions over finite fiels. Applicable Algebra in Engineering, Communication, an Computing, 10: , (preliminary version FOCS 1998). 7. 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: , A. Shpilka. Affine projections of symmetric polynomials. J. Comp. Sys. Sci., 65: , A. Shpilka an A. Wigerson. Depth-3 arithmetic formulae over fiels of characteristic zero. Computational Complexity, 10:1 27, V. Strassen. Berechnung un Programm II. Acta Informatica, 2:64 79, L. Valiant. The complexity of computing the permanent. Theor. Comp. Sci., 8: , A. Yao. Separating the polynomial-time hierarchy by oracles. In Proc. 26th Annual IEEE Symposium on Founations of Computer Science, pages 1 10, 1985.