Arithmetic complexity in algebraic extensions

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1 Arithmetic complexity in algebraic extensions Pavel Hrubeš Amir Yehudayoff Abstract Given a polynomial f with coefficients from a field F, is it easier to compute f over an extension R of F than over F? We address this question, and show the following. For every polynomial f, there is a noncommutative extension R so that over R, f has polynomial size formulas. On the other hand, if F is algebraically closed, no commutative extension R can decrease formula or circuit complexity of f. To complete the picture, we prove that over any field, there exist hard polynomials with zero-one coefficients (this is a basic theorem, but we could not find it written explicitly). Finally, we show that low dimensional extensions are not very helpful in computing polynomials. As a corollary, we obtain that elementary symmetric polynomials have formulas of size n O(log log n) over any field, and that division gates can be efficiently eliminated from circuits, over any field. 1 Introduction We investigate the following general question: given a field F and a polynomial f with coefficients from F, is it easier to compute f over a ring extension R F than over F? As our model of computation we take the standard model for computing polynomials, arithmetic circuits. In principle, the circuit complexity of a polynomial can decrease when working in a larger field or ring. This is related to the fact that in the circuit model, we assume that addition and multiplication of elements of the underlying ring can be performed at unit cost, no matter how complicated the ring is. One example, where the size of the underlying field is important, is the computation of the symmetric polynomials; the symmetric polynomials have small formulas over large fields, but we do not know whether they have polynomial size formulas when the field is small. This is School of Mathematics, Institute for Advanced Study, Princeton NJ. s: and Partially supported by NSF grant CCF

2 essentially due to the powerful interpolation argument (first suggested by Ben-Or, see [8]). In the converse direction, we know how to prove exponential lower bounds on the size of depth three circuits over finite fields [5, 6], but we do not know how to do that over infinite fields. Nevertheless, it is not known whether circuits over larger fields are more powerful than over smaller ones. In a different perspective, it was shown in [9] that permanent is computable by polynomial size formulas when working over a large Clifford algebra. In this approach, it is assumed that addition and multiplication of elements of the Clifford algebra can be performed at unit cost. In fact, the algebra has an exponential dimension and if we wanted to represent the computations as, say, matrix addition and multiplication, we would require exponentially large matrices. We can view this as a computation over a noncommutative ring extension of the real numbers. We start with a mildly surprising observation (that generalizes the Clifford based formulas for permanent) any polynomial of degree d in n variables has a formula of size O(dn) over some noncommutative ring extension. We then show that the situation is quite different in the case of commutative extensions if a field F is algebraically closed and f is a polynomial over F, then for any commutative ring extension R of F, computing f over R is not easier than over F. In this sense, commutative extension are weaker than noncommutative ones. This, however, does not guarantee that for any field F, there exist polynomials which are hard in any commutative extension of F. To complete the picture, we show that over any field, there exist polynomials with zero-one coefficients which require large circuits over the field. Thus, when the field F is algebraically closed, the hard polynomials are also hard over any commutative extension of F. This theorem seems to be a folklore result; it is a theorem without which proving lower bounds for algebraic circuits would be virtually impossible. However, we could not find an explicit statement of it. We note that a similar argument shows that over any field, there exist zero-one tensors of high tensor rank (or zero-one matrices that are rigid). Finally, we consider ring extensions of a bounded linear dimension. We show that if R F has, as a vector space, small dimension, then any computation over R can be efficiently simulated by a computation over F (see Theorem 10 for more details). As two applications, we show that elementary symmetric polynomials have formulas of size n O(log log n) over any field, and that divisions can be efficiently eliminated from circuits over any field (the latter was known for infinite fields). 2

3 2 Preliminaries We denote the family of multisets of [n] = {1,..., n} of size d by {n} d, and we denote by [n] d the family of d-tuples with entries in [n]. Logarithms are always of base two. Rings and polynomials We assume that a ring is always a unital ring, a ring with multiplicative identity element 1 R. Let X = {x 1, x 2,..., x n } be a finite set of variables. A monomial is a product x i1 x i2 x ik with i 1 i 2 i k. We shall write it as x I, where I = {i 1,..., i k } is a multiset. A polynomial over a ring R is a finite formal sum f = I f I x I, where f I R is the coefficient of the monomial x I in f. Addition of polynomials is defined by (f + g) I = f I + g I. A product f g is defined as I,J f Ig J x I J, where I J is the union of the multisets I and J. We denote by R[X] the ring of polynomials in the variables X with coefficients from R. Note that the variables multiplicatively commute with each other, as well as with every element of R, regardless to whether the ring R is commutative or not. If R is a ring extending the ring R, the dimension of R over R, dim R (R ), is defined as the smallest natural number k such that there exist e 0, e 1,..., e k R with the following properties: 1. e 0 = 1 and every a R commutes with every e i, i = 1,..., d, and 2. every b R can be uniquely written as i=0,...,k b ie i, with b i R (if no such k exists, the dimension is infinite). When R is a field, dim R (R ) is the dimension of R as a vector space over R. Arithmetic circuits and formulas We are interested in the arithmetic complexity of a polynomial f over a field F, and how can this complexity change when computing f over a ring R extending F. The ring R, however, is no longer required to be a field (or even commutative). An arithmetic circuit Φ over the ring R and the variables X is a directed acyclic graph with every node of indegree either two or zero, labelled in the following manner: every vertex of indegree 0 is labelled by either a variable in X or an element of R. Every other node in Φ has indegree two and is labelled by either or +. A circuit Φ computes a polynomial f R[X] 3

4 in the obvious manner. An arithmetic circuit is called a formula, if the outdegree of each node in it is one (and so the underlying graph is a directed tree). The size of a circuit is the number of nodes in it, and the depth of a circuit is the length of the longest directed path in it. If f is a polynomial with coefficients from R, we denote by C R (f) the size of a smallest circuit over R computing f. We denote by L R (f) the size of a smallest formula over R computing f. Finally, D R (f) denotes the smallest depth of a circuit computing f in R. 3 General extensions We start with an elementary property of arithmetic complexity measures. Let H : R R be a ring homomorphism. If f = I f Ix I is a polynomial in R[X], then f H R [X] denotes the polynomial I H(f I)x I. The following simple proposition tells us that allowing a circuit to use elements of a larger ring can not make it weaker. Proposition 1 Let f R[X]. Let H : R R be a ring homomorphism. Then C R (f H ) C R (f) and L R (f H ) L R (f). In particular, if R R, then C R (f) C R (f) and L R (f) L R (f). Proof. Let Φ be a smallest circuit computing f over R. Let Ψ be the circuit Φ after substituting each constant a R by H(a) R. Thus Ψ computes f H, and so C R (f H ) C R (f). The proof of L R (f H ) L R (f) is similar. Let us now show that if one allows noncommutative ring extensions of arbitrary dimension, every polynomial can be computed by a polynomial size formula. Theorem 2 Let R be a ring. Let f R[X] be a polynomial of degree d (recall X = n). Then there exists R R such that L R (f) = O(dn). The ring R is noncommutative of large dimension (as we will see it must be). Proof. We can assume that f is homogeneous of degree d (otherwise introduce a new variable t and consider polynomial j td j H j (f) with H j (f) the j th homogeneous part of f). Let us introduce dn new variables Z = {zj i : i = 1,..., d and j = 1,..., n}. Let S be the ring of noncommutative polynomials in the variables Z with coefficients from R 4

5 (defined similarly to a ring of polynomials except that the variables do not commute among themselves). Consider the formula (z1x i 1 + z2x i znx i n ), (1) i=1,...,d which defines a polynomial F S[X]. The size of this formula is O(dn). Recall that for J {n} d, F J S is the coefficient of x J in F. Let I S be the ideal generated by the polynomials F J f J, J {n} d. It is sufficient to prove that I R = {0}. For then R = S/I extends R and (1), taken as a formula over R, computes the polynomial f (in this case, F J = f J in the ring R ). Our goal now is to prove that I R = {0}. For K = (k 1,..., k d ) [n] d, let z K denote the monomial zk 1 1 zk 2 2 zk d d. For the rest of this proof, we call such a monomial an ordered monomial. For J = {j 1 j 2 j d } {n} d, let J = (j 1, j 2,..., j d ) [n] d, the d-tuple that is the ordering of J. For K [n] d, let us define a K R by a K = { f J if K = J, 0 if K J for every J {n} d. Since the variables Z do not commute, every monomial h S can be uniquely written as h = h 1 z K1 h 2 z K2 h s z Ks h s+1, where each z Kl define is an ordered monomial (and s is a natural number). For h of such a form, h = a K1 a Ks h 1 h 2 h s h s+1. For g = j b jh j, where b j R and h j S are monomials, set g = j b jh j. Define the ideal I = {g S : g = 0}. Then I I, and for every nonzero a R, we have a I. Corollary 3 Let R be a ring. Then there exists a ring R R such that for every f R[X] of degree d, L R (f) = O(dn). Proof. Let us first show that for every ring R there exists a ring R such that every polynomial in f R[X] of degree d has a formula of size O(nd) in R. Let F d R[X] be the set of all polynomials of degree d over R. For every f F d, let R f R be the extension of R with L Rf (f) = O(dn), given by Theorem 2. Let R be direct sum of R f,f F d. Each 5

6 R f can be canonically embedded into R. We can also assume that R R. This gives L R (f) = O(dn) for every f F d. Next, define a sequence of rings R 0 = R, and R i+1 = R i R i (given by the above argument). Define R = i 0 R i. Every f R[X] has a finite number of coefficients, and hence there exists k such that f R k [X]. If f has degree d, then L Rk+1 (f) = O(dn). Finally, L R (f) = O(dn), since R k R. The situation is entirely different in the case of commutative extensions. We now show that if F is algebraically closed, then circuit size and formula size cannot be decreased by taking a commutative extension of F (a similar statement appears in [1]). In other words, given a field F and a polynomial f over F, the complexity of f in any commutative extension of F is at least the complexity of f in the algebraic closure of F. Theoretically, this would still allow the alternative that all polynomials from F would have polynomial size formulas in the algebraic closure. We shall dispense with this alternative in a latter theorem. Theorem 4 Assume that F is an algebraically closed field. Let R be a subring of F and let R R be a commutative ring. Then for every f R[X], we have C F (f) C R (f) and L F (f) L R (f). Theorem 4 appears in Chapter 4.3 in [1], we prove it here for completeness. Proof. Let us argue about circuit size, formula size is similar. Let Φ be a circuit over R computing f. Assume that Φ contains a 1,..., a k R \ R. Let us introduce new variables z 1,..., z k. Let Φ be the circuit obtained from Φ by replacing every a i by z i. Thus Φ defines a polynomial F with coefficient in S = R[z 1,..., z k ]. Let I S be the ideal generated by the polynomials F J f J. The ideal I does not contain 1. Indeed, for every J, the equations F J (z 1,..., z k ) f J = 0 have a solution a 1,..., a k in R, and so every polynomial h in I admits h(a 1,..., a k ) = 0. Hilbert s weak nullstellensatz tells us that there exist v 1,..., v k F such that F J (v 1,..., v k ) f J = 0, for every J. Let Φ be the circuit obtained by replacing every z i in Φ by v i F. Thus Φ computes the polynomial f, and the size of Φ is the same as the size of Φ. Our next goal is to show that for every field (and hence every commutative ring), there exist hard polynomials with zero-one coefficients. Lemma 5 Let F be a field. Let F : F n F m be a polynomial map of degree d > 0, that is, F = (F 1,..., F m ), each F i is of degree d. Then F (F n ) {0, 1} m (2d) n. 6

7 Proof. Without loss of generality assume that F is algebraically closed. We use a few notion from algebraic geometry, for formal definitions see, for example, [4]. We start by proving the following stronger claim: Claim 6 Let V F n be an irreducible variety of dimension k and degree r > 0. F (V ) {0, 1} m r(2d) k. Then Proof. The proof of the claim is by induction on k. If k = 0, the variety V is a single point and so F (V ) 1. Let k > 0, and assume that there exists some i = 1,..., m such that both of the varieties V 0 = V {F i = 0} and V 1 = V {F i = 1} are nonempty (if no such i exists, F (V ) 1). Since V 0 and V 1 are proper subvarieties of V, the dimension of both V 0 and V 1 is less than k, the dimension of V. Let ɛ {0, 1}, and consider V ɛ. Since F i has degree at most d, Bezout s Theorem (see Section 2.2 in [3]) tells us that the irreducible components of V ɛ, say Vɛ 1,..., Vɛ tɛ, satisfy j [t ɛ] rj ɛ rd, where rɛ j = deg(vɛ j ). By the inductive assumption, F (Vɛ j ) {0, 1} m rɛ(2d) j k 1 for every j [t ɛ ]. Thus, F (V ) {0, 1} m F (Vɛ j ) {0, 1} m 2rd(2d) k 1 = r(2d) k. ɛ {0,1} j [t ɛ] Since F is algebraically closed, then F n is an irreducible variety of dimension n and degree 1, and Claim 6 implies the lemma. Since a polynomial computed by a circuit is an image of a polynomial map, we have the following theorem. Theorem 7 Let d, n N and let F be a field. Let m = ( ) n+d 1 d. Then there exists a homogeneous polynomial f of degree d in the variables x 1,..., x n with zero-one coefficients such that C F (f) Ω ( m ) ( (n ) ) + d 1 d/2 Ω and d ( ) ( ( ) ) m n + d 1 d L F (f) Ω Ω (d log(n + d)) 1. log m d Proof. Let us start with the first inequality. We first consider a type of circuits we call skeletons. Let z 1,..., z s be new variables. A circuit Γ in the variables x 1,..., x n, z 1,..., z s 7

8 with no field elements in it is called a skeleton. A skeleton Γ computes a polynomial g = J g Jx J S[x 1,..., x n ] with S = F[z 1,..., z s ]. We say that a skeleton Γ defines a polynomial f F[x 1,..., x n ], if there exist v 1,..., v s F such that f(x 1,..., x n ) = g(x 1,..., x n, v 1,..., v s ). Assume that the size of Γ is at most s. Thus the degree of every g J, as a polynomial in z 1,..., z s, is at most 2 s. Let F : F s F m be the map F I (v 1,..., v s ) = g I (v 1,..., v s ) for every I = 1,..., m (we think of I as determining a monomial of total degree exactly d). The map F is a polynomial map of degree at most 2 s. Moreover, if Γ defines a homogeneous polynomial f, then the vector of coefficients of f is in the image of F. By Lemma 5, the skeleton Γ computes at most (2 s+1 ) s polynomials with zero-one coefficients. Every skeleton of size at most s contains at most n + s variables, symbols +, and no constants. Thus, there are at most (n + s + 2) s s 2s skeletons of size at most s (the indegree of each node is at most two). Hence, skeletons of size at most s define at most 2 s(s+1) (n + s + 2) s s 2s 2 cs2 polynomials with zero-one coefficients, where c > 0 is a constant (we can assume s > n). Back to considering general circuits. Let Φ be a circuit over F in the variables x 1,..., x n of size at most s. The circuit Φ contains at most s elements of F, say a 1,..., a s (the ordering is arbitrary but fixed). Let Γ be the skeleton obtained from Φ by replacing every a i by z i. The size of Γ is at most s. In addition, if Φ computes a polynomial f F[x 1,..., x n ], then Γ defines f. Since there exist 2 m homogeneous polynomials of degree d with zeroone coefficients, in order to compute all such polynomials, we must have 2 cs2 2 m and so s Ω(m 1/2 ). For the latter inequality in the statement of the theorem, we use the estimate ( n+d 1 ) ( d n+d 1 ) d. d To lower bound L F (f), we use a similar argument. Consider formula skeletons instead of circuit skeletons. The difference is that for formulas g J has degree at most s. Hence F : F s F m is a map of degree at most s, and a formula skeleton defines at most (2s) s polynomials with zero-one coefficients. We upper bound the number of formula skeletons of size s by the number of circuit skeletons above, and conclude that formula skeletons define at most (2s) s (n+s+2) s s 2s 2 cs log s polynomials with zero-one coefficients (c > 0 is a constant). This gives 2 cs log s 2 m and hence s Ω(m log 1 m). For the latter inequality in the statement of the theorem, we estimate log ( ) ( ) d n+d 1 d log e(n+d 1) = O(d log(n + d)). To match the statement of the theorem, we must show that that the lower bounds on C F and L F can be achieved simultaneously. This follows from the fact that the above arguments give that majority of the polynomials with zero-one coefficients have this complexity. d 8

9 Corollary 8 The statement of Theorem 7 holds for any non-trivial commutative ring R (instead of F). Proof. Let R be a non-trivial commutative ring (i.e., 0 1), and let I be a maximal ideal in R. Thus F = R/I is a field. Let f be the polynomial given by Theorem 7 with the field F. Let H be the canonical homomorfism H : R F. Proposition 1 tells us that C F (f H ) C R (f). Finally, the polynomial f has zero-one coefficients, and so f = f H. Existence of hard polynomials with real coefficients was proved, for example, in [2]. The argument of [2] does not apply to polynomials with zero-one coefficients the considered polynomials have algebraically independent coefficients (so-called generic polynomials). On the other hand, the circuit lower bound on generic polynomials essentially matches the formula lower bound from Theorem 7, whereas the circuit lower bound in Theorem 7 is roughly a square root of the expected value. Tensor rank and matrix rigidity Lemma 5 can be applied to several other problems; for example, it implies the existence of zero-one tensors of high tensor rank, and zero-one matrices of high rigidity. Valiant defined the concept of matrix rigidity [10], and proved the existence of matrices of high rigidity over any field. In the case of a finite field, the proof is by a counting argument, and in the infinite case, a dimension consideration. The matrices take as entries all possible field elements. Existence of rigid zero-one matrices was proved in [7], in the case of the field of real numbers. The authors use a real number version of Lemma 5 which is due to Warren [11]. Warren s theorem is a stronger version of Lemma 5, but it applies exclusively to R. Here is a sketch of a proof of the existence of tensors of high rank. Consider a three dimensional tensor, say T : [n] 3 F. Recall that a rank one tensor is a tensor such that t(x 1, x 2, x 3 ) = t 1 (x 1 )t 2 (x 2 )t 3 (x 3 ) for every x 1, x 2, x 3 [n]. Also recall that the tensor rank of a tensor T is defined as the minimal integer r such that T = i [r] ti with t i of rank one. Let us count the number of zero-one tensors of rank at most r. Each rank one tensor t i = t i 1 ti 2 ti 3 is defined by 3n variables, n variables for each t i j. Think of ti as a polynomial map from F 3n to F n3 ; as such it has degree three. Similarly, a rank r tensor is a polynomial map from F 3nr to F n3. Lemma 5 tells us that the number of zero-one tensors of rank at most r is at most 6 3nr. On the other hand, the number of zero-one tensors is 2 n3. Thus, there exist zero-one tensors of tensor rank at least Ω(n 2 ). 9

10 4 Extensions of small dimension We start by observing that formulas could be thought of as balanced (this is standard). Lemma 9 Let R be a ring. Then for every polynomial f, c 1 D R (f) log L R (f) D R (f), where c 1 is a universal constant. Proof. Since the number of nodes in a tree with indegree at most two of depth l is at most 2 l, log L R (f) D R (f). The proof of the other inequality is by induction on the size of the formula. Let Φ be a smallest formula for f, that is, L R (f) = s where s is the size of Φ. Since the indegree is at most two, let u be a node in Φ so that Φ u, the subformula of Φ rooted at u, is of size between s/3 and 2s/3. Denote by g the polynomial computed by u, and denote by f a, a R, the polynomial computed at the output of Φ after deleting the edges going into u and labelling it by a. Thus, f = (f 1 f 0 )g + f 0 (this follows by induction on the structure of Φ). All the polynomials g, f 0 and f 1 have formulas of size at most 2s/3. By induction, every h {g, f 0, f 1 } admits D R (h) c log L R (h); let Φ h be a formula for h of depth D R (h) c log(2s/3). Set Ψ = (Φ f 1 + ( 1) Φ f 0 ) Φ g + Φ f 0. Thus, Ψ computes f and its depth is at most c log(2s/3) + 4 c log s with c 1 a constant. The following theorem shows that extensions of low dimensions are not extremely helpful when computing polynomials. Theorem 10 Let R and R R be rings such that dim R (R ) = k. Let f R[X]. Then C R (f) O(k 3 )C R (f) and L R (f) (L R (f)) O(log k). Proof. Let 1 = e 0 and e 1,..., e k be elements of R such that every a R can be uniquely written as i=0,...,k a ie i. We denote a = a 0,..., a k. Addition and multiplication in R can be performed as (a + b) i = a i + b i and (a b) i = λ i (a, b), where λ i is a bilinear map over R. Every λ i is computable by a circuit φ i over R of size at most ck 2 and depth at most c log k with c > 0 a constant. 10

11 For f = f J x J R [X] and i = 0,... k, define f i R[X] as J f J,ix I, where f J,i = (f J ) i. Denote f = f 0,..., f k. For every i = 0,..., k, (f + g) i = f i + g i and (f g) i = λ i (f, g); for example, (f g) i = I,J (f I g J ) i x I x J = I,J ( λ i (f I, g J )x I x J = λ i f I x I, I J g J x J ) = λ i (f, g). We start by considering circuit size. Let Φ be an arithmetic circuit over R of size s and depth d computing f. We simulate the computations in R by the computation in R. Let us define a new circuit Ψ over R as follows. For every node u in Φ that computes g u, the circuit Ψ contains k + 1 nodes u 0,..., u k computing g0 u,..., gu k. We define Ψ inductively as follows. If u is a leaf, then g u is either a variable or an element of R. In this case, label each u i by (g u ) i. If u = v + w is sum node, set u i = v i + w i. If u = v w is a product node, set u i = φ i (v, w). The size of Ψ is at most O(k 3 )s and its depth is at most d O(log k). If u is the output node of Φ, then u 0 computes the polynomial f 0. Since f R[x 1,..., x n ], we have f = f 0, and hence u 0 computes f. This shows that C R (f) O(k 3 )C R (f). For formula size, we use Lemma 9. Let Φ be a circuit computing f over R of depth D = D R (f) c log L R (f). The above argument tells us that f has a circuit over R of depth at most D O(log k) computing f. Lemma 9 now tells us that L R (f) 2 D O(log k) L R (f) O(log k). One consequence of Theorem 10 is that formulas over R can polynomially simulate formulas over C (the same holds for circuits). More exactly, L C (f) (L R (f)) c 1 and C C (f) c 2 C R (f) for appropriate constants c 1, c 2 > 0. Here are two more applications of Theorem 10. Symmetric polynomials over finite fields The k-symmetric polynomial is the polynomial x i1 x i2 x ik. i 1 <i 2 < <i k [n] It is known that over large fields the symmetric polynomials have formulas of size roughly n 2, see [8]. This construction is by the so-called interpolation method, which requires the existence of n + 1 distinct field elements. Nevertheless, Theorem 10 implies that we can find relatively small formulas for the symmetric polynomials over any finite 11

12 field as well. Indeed, let F be a finite field and let E be an extension of F of dimension roughly log n, so that the size of E is at least n + 1. Over E we can use interpolation to construct a formula of size O(n 2 ) for the symmetric polynomial. Now, Theorem 10 tells us that we can simulate this formula by a formula over F of size n O(log log n). Eliminating division nodes in finite fields A circuit with divisions is a circuit that has one more type of nodes, divisions gates, that are labelled by /. Every node of such a circuit computes a formal rational function, an element of the field F(X). We require that for any node u/v in the circuit, v computes a nonzero rational function g (though it may happen that g = 0 on every input from F). Over an infinite field, if a polynomial f F[X] of degree d is computable by a circuit Φ of size s with divisions, it is also computable by a circuit Ψ of size O(d 2 s) without divisions (see [1], Chapter 7.1). The argument works over any field which is large enough. Here is how we can make it work over small fields as well. Without loss of generality, we can assume that Φ contains exactly one division node u/v that computes g 1 /g 2 with polynomials g 1, g 2 F[X]. For the argument to work, it is sufficient to find a 1,..., a n F such that g 2 (a 1,..., a n ) 0. Since the degree of g 2 is at most 2 s, it is enough to have F 2 s + 1 (as the Schwartz- Zippel Lemma tells us). If F is not large enough, we can take a field extension E of dimension more than s, and compute f over E efficiently. Now, Theorem 10 implies that we can, in fact, compute f over F by a circuit of size O(s 3 d 2 s) = poly(s, d). Acknowledgement We wish to thank Avi Wigderson for helpful discussions. References [1] P. Burgisser, M. Clausen and M. A. Shokrollahi. Algebraic complexity theory. A series of comprehensive studies in mathematics, Vol. 315, 1997, Springer. [2] P. Burgisser, T. Lickteig and M. Shub. Test complexity of generic polynomials, J. Compl. 8, pages , [3] V. Danilov. Algebraic varieties and schemes. In I. R. Shafarevich (Ed.), Algebraic Geometry I, Encyclopedia of Mathematical Sciences, Vol. 23, pages , Berlin, Springer. [4] Z. Dvir. Extractors for varieties. In the Proceedings of Computational Complexity Conference,

13 [5] D. Grigoriev and M. Karpinski. An exponential lower bound for depth 3 arithmetic circuits. In Proceedings of the 30th Annual STOC, pages , [6] D. Grigoriev and A. A. Razborov. Exponential complexity lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields. Applicable Algebra in Engineering, Communication and Computing 10(6), pages , [7] P. Pudlak and V. Rödl. Some combinatorial-algebraic problems from complexity theory. Discrete Mathematics 136 n.1-3, pages , [8] A. Shpilka and A. Wigderson. Depth-3 arithmetic formulae over fields of characteristic zero. Journal of Computational Complexity 10(1), pages 1 27, [9] R. Schott and S. Staples. Reductions in computational complexity using Clifford algebras. Advances in Applied Clifford Algebras, [10] L. G. Valiant. Graph-theoretic properties in computational complexity. J. Comput. Syst. Sci. 13(3), pages , [11] H. E. Warren. Lower bounds for approximations by nonlinear manifolds. Trans. AMS 133, pages ,

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