A Polynomial Degree Bound on Equations of Non-rigid Matrices and Small Linear Circuits
aa r X i v : . [ c s . CC ] N ov A Polynomial Degree Bound on Equations for Non-rigid Matricesand Small Linear Circuits
Mrinal Kumar ∗ Ben Lee Volk † Abstract
We show that there is an equation of degree at most poly ( n ) for the (Zariski closure of the) setof the non-rigid matrices: that is, we show that for every large enough field F , there is a non-zero n -variate polynomial P ∈ F [ x , , . . . , x n,n ] of degree at most poly ( n ) such that every matrix M which can be written as a sum of a matrix of rank at most n/ and a matrix of sparsity at most n / satisfies P ( M ) = 0 . This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyerand Landsberg [GHIL16] and improves the best upper bound known for this problem down from exp( n ) [KLPS14, GHIL16] to poly ( n ) .We also show a similar polynomial degree bound for the (Zariski closure of the) set of allmatrices M such that the linear transformation represented by M can be computed by analgebraic circuit with at most n / edges (without any restriction on the depth). As far aswe are aware, no such bound was known prior to this work when the depth of the circuits isunbounded.Our methods are elementary and short and rely on a polynomial map of Shpilka andVolkovich [SV15] to construct low degree “universal” maps for non-rigid matrices and smalllinear circuits. Combining this construction with a simple dimension counting argument toshow that any such polynomial map has a low degree annihilating polynomial completes theproof.As a corollary, we show that any derandomization of the polynomial identity testing problemwill imply new circuit lower bounds. A similar (but incomparable) theorem was proved byKabanets and Impagliazzo [KI04]. Let V ⊆ F n be a (not necessarily irreducible) affine variety and let I ( V ) denote its ideal. . Anon-zero polynomial P ∈ I ( V ) is called an equation for V . An equation for V may serve as a“proof” that a point x ∈ F n is not in V , by showing that P ( x ) = 0 .A fundamental observation of the Geometric Complexity Theory program is that manyimportant circuit lower bounds problems in algebraic complexity theory fit naturally into thesetting of showing that a point x lies outside a variety V [MS01, BIL + V to be the closure of a class of polynomials of low complexity, and x is thecoefficient vector of the candidate hard polynomial. ∗ Department of Computer Science & Engineering, IIT Bombay. Email: [email protected] † Center for the Mathematics of Information, California Institute of Technology, USA. Email: [email protected] For completeness, we provide the formal (standard) definitions for these notions in Section 1.4. et ∆( V ) := min = P ∈ I ( V ) { deg( P ) } . The quantity ∆( V ) can be thought of as a measureof complexity for the geometry of the variety V . The quantity ∆( V ) is a very coarse com-plexity measure. A recent line of work regarding algebraic natural proofs [FSV18, GKSS17]suggests to study the arithmetic circuit complexity of equations for varieties V that correspondto polynomials with small circuit complexity. Having ∆( V ) growing like a polynomial in n is anecessary (but not a sufficient) condition for a variety V to have an algebraic natural proof fornon-containment. A matrix M is ( r, s ) -rigid if M cannot be written as a sum R + S where rank ( R ) ≤ r and S contains at most s non-zero entries. Valiant [Val77] proved that if A is ( εn, n δ ) -rigid for someconstants ε, δ > then A cannot be computed by arithmetic circuits of size O ( n ) and depth O (log n ) , and posed the problem of explicitly constructing rigid matrices with these parameters,which is still open. It is easy to prove that most matrices have much stronger rigidity parameters:over algebraically closed fields a generic matrix is ( r, ( n − r ) ) -rigid for any target rank r .Let F be an algebraically closed field. Let A r,s ⊆ F n × n denote the set of matrices which arenot ( r, s ) -rigid. Let V r,s = A r,s denote the Zariski closure of A r,s . A geometric study of V r,s wasinitiated by Kumar, Lokam, Patankar and Sarma [KLPS14]. Among other results, they provethat for every s < ( n − r ) , ∆( V r,s ) ≤ n n . A slightly improved (but still exponential) upperbound was obtained by Gesmundo, Hauenstein, Ikenmeyer and Landsberg [GHIL16], who alsoconjectured that for some ε, δ > , ∆( V εn,n δ ) grows like a polynomial function in n . Thefollowing theorem which we prove in this note confirms this conjecture. Theorem 1.1.
Let ε < / , and let F be a field of size at least n . For every large enough n , there exists a non-zero polynomial Q ∈ F [ x , , . . . , x n,n ] , of degree at most n , which is anon-trivial equation for matrices which are not ( εn , εn )-rigid. That is, for every such matrix M , Q ( M ) = 0 . In fact, the conjecture of [GHIL16] was slightly weaker: they conjectured that ∆( U ) ispolynomial in n for every irreducible component U of V εn,n δ . As shown by [KLPS14], theirreducible components are in one-to-one correspondence with subsets of [ n ] × [ n ] of size n δ corresponding to possible supports of the sparse matrix S .As we observe in Remark 2.3, it is somewhat simpler to show that each of these irreduciblecomponents has an equation with a polynomial degree bound. However, since the number ofsuch irreducible components is exponentially large, it is not clear if there is a single equationfor the whole variety which is of polynomially bounded degree. We do manage to reverse theorder of quantifiers and prove such an upper bound in Theorem 1.1. This suggests that theset of non-rigid matrices is much less complex than what one may suspect given the results of[KLPS14, GHIL16]. The original motivation for defining rigidity was in the context of proving lower bounds foralgebraic circuits [Val77]. If A ∈ F n × n is an ( εn, n δ ) -rigid matrix, for any ε, δ > , then thelinear transformation represented by A cannot be computed by an algebraic circuit of depth O (log n ) and size O ( n ) .Every algebraic circuit computing a linear transformation is without loss of generality a linear circuit. A linear circuit is a directed acyclic graph that has n inputs labeled X , . . . , X n and n output nodes. Each edge is labeled by a scalar α ∈ F . Each node computes a linear function in X , . . . , X n defined inductively. An internal node u with children, v , . . . , v k , connected to it byedges labeled α , . . . , α k , computes the linear function P i α i ℓ v i , where ℓ v i is the linear functioncomputed by v i , ≤ i ≤ k . The size of the circuit is the number of edges in the circuit. t is possible to use similar techniques to those used in the proof of Theorem 1.1 in order toprove a polynomial upper bound on an equation for a variety containing all matrices A ∈ F n × n whose corresponding linear transformation can be computed by an algebraic circuit of size atmost n / (even without restriction on the depth). Note that this is nearly optimal as anysuch linear transformation can be computed by a circuit of size n . More formally, we show thefollowing. Theorem 1.2.
Let F be a field of size at least n . For every large enough n , there exists anon-zero polynomial Q ∈ F [ x , , . . . , x n,n ] , of degree at most n , which is a non-trivial equationfor matrices which are computed by algebraic circuit of size at most n / . Our proofs are based on a dimension counting arguments, and are therefore non-constructiveand do not give explicit equations for the relevant varieties. It thus remains a very interestingopen problem to provide explicit low-degree equations for any of the varieties considered inthis paper. Here “explicit” means a polynomial which has arithmetic circuits of size poly ( n ) . The question of whether such equations exists has a win-win flavor: if they do, this can aidin explicit constructions of rigid matrices, and on the other hand, if all equations are hard,we have identified a family of polynomials which requires super-polynomial arithmetic circuits.Assuming the existence of a polynomial time algorithm for polynomial identity testing, we areable to make this connection formal.Let
PIT denote the set of strings which describe arithmetic circuits (say, over C ) whichcompute the zero polynomial. It is well known that PIT ∈ coRP . Kabanets and Impagliazzo[KI04] proved that certain circuit lower bounds follow from the assumption that PIT ∈ P . As acorollary to Theorem 1.2, we are able to prove theorem of a similar kind. Corollary 1.3.
Suppose
PIT ∈ P . Then at least one of the following is true:1. There exists a family of n -variate polynomials of degree poly ( n ) over C , which can becomputed (as its list of coefficients, given the input n ) in PSPACE , which does not havepolynomial size constant free arithmetic circuits.2. there exists a family of matrices, constructible in polynomial time with an NP oracle (giventhe input n ), which requires linear circuits of size Ω( n ) . A constant free arithmetic circuit is an arithmetic circuit which is only allowed to use theconstants { , ± } .A different way to interpret Corollary 1.3 is as saying that at least one of the following threelower bound results hold: either PIT P , or (at least) one of the two circuit lower boundsstated in the corollary. We emphasize that the result holds under any (even so-called white box )derandomization of PIT .Our statement is similar to, but incomparable with the result of Kabanets and Impagliazzo[KI04] who proved that if
PIT ∈ P then either the permanent does not have polynomial sizeconstant free arithmetic circuits, or NEXP P / poly .Since ( εn, εn ) -rigid matrices have linear circuit of size εn , the last item of Corollary 1.3 inparticular implies a conditional construction of (Ω( n ) , Ω( n )) -rigid matrices (it is also possibleto directly use Theorem 1.1 instead of Theorem 1.2 to deduce this result). Unconditionalconstructions of rigid matrices in polynomial time with an NP oracle were recently given in[AC19, BHPT20]. However, the rigidity parameters in these papers are not enough to implycircuit lower bounds (furthermore, even optimal rigidity parameters are not enough to imply Ω( n ) lower bounds for general linear circuits).Since it is widely believed that PIT ∈ P , the answer to which of the last two items ofCorollary 1.3 holds boils down to the question of whether there exists an equation for non-rigid Although one may consider other, informal notions of explicitness which could nevertheless be helpful. atrices of degree poly ( n ) and circuit size poly ( n ) . If determining if a matrix is rigid is coNP -hard (as is known for some restricted ranges of parameters [MS10]), it is tempting to also believethat the equations should not be easily computable, as they provide “proof” for rigidity whichcan be verified in randomized polynomial time. However, it could still be the case that thoseequations that have polynomial size circuits only prove the rigidity of “easy” instances. For completeness, in this section we define some basic notions in algebraic geometry. A readerwho is familiar with this topic may skip to the next section.Let F be an algebraically closed field. A set V ⊆ F n is called an affine variety if there existpolynomials f , . . . , f t ∈ F [ x , . . . , x n ] such that V = { x : f ( x ) = f ( x ) = · · · = f t ( x ) = 0 } .For convenience, in this paper we often refer to affine varieties simply as varieties.For each variety V there is a corresponding ideal I ( V ) ⊆ F [ x , . . . , x n ] which is defined as I ( V ) := { f ∈ F [ x , . . . , x n ] : f ( x ) = 0 for all x ∈ V } . Conversely, for an ideal I ⊆ F [ x , . . . , x n ] we may define the variety V ( I ) = { x : f ( x ) = 0 for all f ∈ I } . Given a set A ⊆ F n we may similarly define the ideal I ( A ) . The (Zariski) closure of a set A , denoted A , is the set V ( I ( A )) . In words, the closure of A is the set of common zeros of allthe polynomials that vanish on A . It is also the smallest variety with respect to inclusion whichcontains A . By construction, A is a variety, and a polynomial which vanishes everywhere on A is also vanishes on A .Over C , it is instructive to think of the Zariski closure of A as the usual Euclidean closure.In fact, for the various sets A we consider in this paper (which correspond to sets of “lowcomplexity” objects, e.g., non-rigid matrices or matrices which can be computed with a smallcircuit), it can be shown that these two notions of closure coincide (see, e.g., Section 4.2 of[BI17]).A variety V is called irreducible if it cannot be written as a union V = V ∪ V of varieties V , V that are properly contained in V . Every variety can be uniquely written as a union V = V ∪ V ∪ · · · ∪ V m of irreducible varieties. The varieties V , . . . , V m are then called the irreducible components of V . In this section, we prove Theorem 1.1. A key component of the proof is the use of the followingconstruction, due to Shpilka and Volkovich, which provides an explicit low-degree polynomialmap on a small number of variables, which contains all sparse matrices in its image. Forcompleteness, we provide the construction and prove its basic property.
Lemma 2.1 ([SV15]) . Let F be a field such that | F | > n . Then for all k ∈ N , there exists anexplicit polynomial map SV n,k ( x , y ) : F k → F n of degree at most n such that for any subset T = { i , . . . , i k } ⊆ [ n ] of size k , there exists a setting y = α such that SV( x , α ) is identicallyzero on every coordinate j T , and equals x j in coordinate i j for all j ∈ [ k ] .Proof. Arbitrarily pick distinct α , . . . α n ∈ F , and let u , . . . , u n be their corresponding La-grange’s interpolation polynomials, i.e., polynomials of degree at most n − such that u i ( α j ) = 1 if j = i and otherwise (more explicitly, u i ( z ) = Q j = i ( z − α j ) Q j = i ( α i − α j ) ). et P i ( x , . . . , x k , y , . . . , y k ) = P kj =1 u i ( y j ) · x j , and finally let SV n,k ( x , y ) = ( P ( x , y ) , . . . , P n ( x , y )) . It readily follows that given T = { i , . . . , i k } as in the statement of the lemma, we can set y j = α i j for j ∈ [ k ] to derive the desired conclusion. The upper bound on the degree follows byinspection.As a step toward the proof of Theorem 1.1, we show there is a polynomial map on muchfewer than n variables with degree polynomially bounded in n such that its image containsevery non-rigid matrix. In the next step, we show that the image of every such polynomial maphas an equation of degree poly ( n ) . Lemma 2.2.
There exists an explicit polynomial map P : F εn → F n × n , of degree at most n ,such that every matrix M which is not ( εn, εn ) rigid lies in its image.Proof. Let k = εn and let u , v , x , y denote disjoint tuples of k variables each.Let U be a symbolic n × εn matrix whose entries are labeled by the variables u , and similarlylet V be a symbolic εn × n matrix labeled by v . Let UV( u , v ) : F k → F n × n be the degree 2polynomial map defined by the matrix multiplication U V .Finally, let P : F k → F n × n be defined as P ( u , v , x , y ) = UV( u , v ) + SV n ,k ( x , y ) , where SV n ,k is as defined in Lemma 2.1.Suppose now M is a non-rigid matrix, i.e., M = R + S for R of rank εn and S which is εn -sparse. Decompose R = U V for n × εn matrix U and εn × n matrix V . Let T denote thesupport of S . For convenience we may assume | T | = k (otherwise, pad with zeros arbitrarily).Let α ∈ F k denote the setting for y in SV n ,k which maps x , . . . , x k to T , and let s = ( s , . . . , s k ) denote the non-zero entries of S . Then P ( U , V , s , α ) = U V + S = R + S = M. To complete the proof of Theorem 1.1, we now argue that the image of any polynomial mapwith parameters as in Lemma 2.2 has an equation of degree at most n . Proof of Theorem 1.1.
Let V denote the subspace of polynomials over F in n variables ofdegree at most n . Let V denote the subspace of polynomials over F in εn variables of degreeat most n . Let P be as in Lemma 2.2, and consider the linear transformation T : V → V given by Q Q ◦ P , where Q ◦ P denotes the composition of the polynomial Q with the map P ,i.e., ( Q ◦ P )( x ) = Q ( P ( x )) (indeed, observe that since deg( Q ) ≤ n and deg( P ) ≤ n , it followsthat deg Q ◦ P ≤ n ).We have that dim( V ) = (cid:0) n + n n (cid:1) ≥ n n , whereas dim( V ) = (cid:0) εn + n εn (cid:1) ≤ (2 n ) εn < dim( V ) by the choice of ε , so that there exists a non-zero polynomial in the kernel of T , thatis, = Q ∈ V such that Q ◦ P ≡ .It remains to be shown that for any non-rigid matrix M , Q ( M ) = 0 . Indeed, let M be a non-rigid matrix. By Lemma 2.2, there exist β ∈ F εn such that P ( β ) = M . Thus, Q ( M ) = Q ( P ( β )) = Q ◦ P ( β ) = 0 , as Q ◦ P ≡ . Remark 2.3.
If the support of the sparse matrix is fixed a-priori to some set S ⊆ [ n ] × [ n ] ofcardinality at most ǫn , then it is easier to come up with a universal map ˜ P from F ǫn F n × n such that every matrix M whose rank can be reduced to at most ǫn by changing entries in theset S is contained in the image of ˜ P . Just consider ˜ P ( w , x , y ) = UV( u , v ) + W , where W is a matrix such that for all ( i, j ) ∈ [ n ] × [ n ] , if ( i, j ) ∈ S , then W ( i, j ) = w i,j and W ( i, j ) s zero otherwise. Here, each w i,j is a distinct formal variable. Combined with the dimensioncomparison argument we used in the proof of Theorem 1.1, it can be seen that there is a non-zerolow degree polynomial ˜ Q such that ˜ Q ◦ ˜ P ≡ . This argument provides a (different) equation ofpolynomial degree for each irreducible component of the variety of non-rigid matrices. ♦ Remark 2.4.
It is possible to use the equation given in Theorem 1.1, and using the methodsof [KLPS14], to construct “semi-explicit” ( εn, εn ) -rigid matrices. These are matrices whoseentries are algebraic numbers (over Q ) with short description, which are non-explicit from thecomputational complexity point of view. However, such constructions are also known usingdifferent methods (see Section 2.4 of [Lok09]). ♦ In this section, we prove Theorem 1.2. Our strategy, as before, is to observe that all matriceswith a small circuit lie in the image of a polynomial map P on a small number of variables andsmall degree. Circuits of size s can have many different topologies and thus we first constructa “universal” linear circuit, of size s ′ ≤ s , that contains as subcircuits all linear circuits of size s . Importantly, s ′ will affect the degree of P but not its number of variables. We note that thisconstruction of universal circuits is slightly different from similar constructions in earlier work,e.g., in [Raz10]; the key difference being that a naive use of ideas in [Raz10] to obtain the map P seems to incur an asymptotic increase in the number of variables of P , which is unacceptablein our current setting. We now define a map U ( x , y ) which is “universal” for size s linear circuits, i.e., it contains inits image all n × n matrices A whose corresponding linear transformation can be computed bya linear circuit of size at most s .Let s ≥ n . We first define a universal graph G for size s . G has a set V of n input nodeslabeled X , . . . X n and a set V s +1 of n designated output nodes. In addition, G is composed of s disjoint sets of vertices V , . . . , V s , each contains s vertices.Each vertex v ∈ V i , for ≤ i ≤ s + 1 , has as its children all vertices u ∈ V j for all ≤ j < i .It is clear than every directed acyclic graph with s edges (and hence at most s vertices, anddepth at most s ) can be (perhaps non-uniquely) embedded in G as a subgraph.We now describe the edge labeling. Let s ′ ≤ s be the number of edges in V and let e i denotethe i -th edge, ≤ i ≤ s ′ . The edge e i is labeled by the i -th coordinate of the map SV s ′ ,s ( x , y ) given in Lemma 2.1.Thus, the graph G with this labeling computes a linear transformation (over the field F ( x , y ) )in the variables X , . . . , X n . More explicitly, the ( i, j ) -th entry of the matrix U ( x , y ) representingthis linear transformation is given by the sum, over all paths from X i to the j -th output node,of the product of the edge labels on that path. This entry is a polynomial in x , y , so that wecan think of U as a polynomial map from F s to F n . Lemma 3.1.
The map U ( x , y ) defined above contains in its image all n × n matrices A whosecorresponding linear transformation can be computed by a linear circuit of size at most s . Thedegree of U is at most s ′ · ( s + 1) .Proof. Let A be a matrix whose linear transformation is computed by a size s circuit C . Thegraph of C can be embedded as a subgraph in the graph G constructed above (if the embeddingis not unique, pick one arbitrarily). Let e i , . . . , e i s be the edges of this subgraph, and let β = ( β , . . . , β s ) be their corresponding labels in C . By the properties of the map SV s ′ ,s ( x , y ) given in Lemma 2.1, it is possible to set the tuple of variables y to field elements α , . . . , α s such hat the j -th coordinate of SV( β , α ) equals β i if j = i k for some ≤ k ≤ s the otherwise.Observe that under this labeling of the edges, the circuit G computes the same transformationas the circuit C . Hence U ( β , α ) = A .To upper bound the degree of U , note that each edge label in G is a polynomial of degree s ′ , and each path is of length at most s + 1 . Analogous to the proof of Theorem 1.1, we now observe via a dimension counting argumentthat the image of the polynomial map U ( x , y ) has a equation of degree at most n . This wouldcomplete the proof of Theorem 1.2. Proof of Theorem 1.2.
As before, let V denote the subspace of polynomials over F in n vari-ables of degree at most n . Let V denote the subspace of polynomials over F in n / variablesof degree at most n . Let U be the map given by Lemma 3.1 for s = n / so that s ′ ≤ n , andthe degree of U is at most s ′ ( s + 1) ≤ n . Now, consider the linear transformation T : V → V given by Q Q ◦ U .Once again, we compute that dim( V ) = (cid:0) n + n n (cid:1) ≥ n n , whereas dim( V ) = (cid:0) n / n n / (cid:1) ≤ (2 n ) n / < dim( V ) , so that there exists a non-zero polynomial in the kernel of T , that is, = Q ∈ V such that Q ◦ U ≡ .By Lemma 3.1, if A has a circuit of size n / , it is in the image of U , so that Q ( A ) = 0 . Another algebraic object which is closely related to proving circuit lower bounds is the set ofthree dimensional tensors of high rank. A three dimensional tensor of rank at least r implies alower bound of r on an arithmetic circuit computing the bi-linear function associated with thetensor. Our arguments also provide polynomial degree upper bounds for the set of tensors of(border) rank at most n / . Lemma 4.1.
Let F be any field. There is a polynomial map P : F n / → F n of degree atmost such that for every dimensional tensor τ : [ n ] → F of rank at most n / lies in itsimage.Proof. This follows immediately from the definition.Indeed, let r = n / . Let u , . . . , u r , v , . . . , v r , w , . . . , w r be disjoint n tuples of vari-ables. Let U be a tensor of rank at most r over the ring F [ u , . . . , u r , v , . . . , v r , w , . . . , w r ] defined as follows. U ( u , v , w ) = r X i =1 u i ⊗ v i ⊗ w i . From the definition of U , it can be readily observed that for every tensor τ : F [ n ] → F ofrank at most r , there is a setting α , β , γ of the variables in u , v , w respectively such that U ( α , β , γ ) = τ . Moreover, each of the coordinates of U is a polynomial of degree equal to threein the variables in u , v , w . Let P be the degree three polynomial map which maps the variables u , . . . , u r , v , . . . , v r and w , . . . , w r to the coordinates of U .We now argue that for every polynomial map P given by Lemma 4.1 has an equation of nottoo large degree. Theorem 4.2.
Let F be any field. There exists a non-zero polynomial Q ∈ F [ x , , , . . . , x n,n,n ] ,of degree at most n , which is a non-trivial equation for three dimensional tensors τ : [ n ] × [ n ] × [ n ] F of rank at most n / . roof. As before, let V denote the subspace of polynomials over F in n variables of degreeat most n and let V denote the subspace of polynomials over F in n / variables of degreeat most n . Let P be the map given by Lemma 4.1. Now, consider the linear transformation T : V → V given by Q Q ◦ P .Observe that dim( V ) = (cid:0) n + n n (cid:1) ≥ n n , whereas dim( V ) = (cid:0) n / n n / (cid:1) ≤ (2 n ) n / < dim( V ) , so that there exists a non-zero polynomial in the kernel of T , that is, = Q ∈ V suchthat Q ◦ P ≡ .By Lemma 4.1, if τ is a tensor of rank at most n / , then it is in the image of P , and thus Q ( τ ) = 0 .The arguments here also generalize to tensors in higher dimensions. In particular, the fol-lowing analog of Lemma 4.1 is true. Lemma 4.3.
Let F be any field. Then, for all n, d ∈ N , there is a polynomial map P : F n d − / → F n d of degree at most d such that for every d dimensional tensor τ : [ n ] ⊗ d → F ofrank at most n d − / d lies in its image. Combining this lemma with a dimension comparison argument analogous to that in the proofof Theorem 4.2 gives the following theorem. We skip the details of the proof.
Theorem 4.4.
For every field F and for all n, d ∈ N , there exists a non-zero polynomial Q on n d variables and degree at most n d , which is a non-trivial equation for d dimensional tensors τ : [ n ] ⊗ d → F of rank at most n d − / d . We remark that a similar methods can be used to prove the existence of an equation of degree poly ( n ) for three dimensional tensors of slice rank (see, e.g., [BIL + n/ .The existence of such an equations was proved (using different techniques) in [BIL + In this section we prove Corollary 1.3. The strategy of the proof is simple: the proof of The-orem 1.2 implies a
PSPACE algorithm which produces a sequence of polynomials which areequations for the set of matrices with small linear circuits. If those equations require large cir-cuits, we are done, and if not, then there exists an equation with small circuits which (assuming
PIT ∈ P ) can be found using an NP -oracle. Using, once again, the assumption that PIT ∈ P ,we can also find deterministically a matrix on which the equation evaluates to non-zero, whichimplies the matrix requires large linear circuits.There are some technical difficulties involved with this plan which we now describe. The firstproblem is that even arithmetic circuits of small size can have large description as bit strings,due to the field constants appearing in the circuits. To prevent this issue, we only consider constant free arithmetic circuits, which are only allowed inputs labeled by { , ± } (but can stillcompute other constants in the circuit using arithmetic operations).The second problem is that, in order to be able to find a non-zero of the equation in the laststep of the algorithm (using the mere assumption that PIT ∈ P ), we need not only the size of thecircuit but also its degree to be bounded by poly ( n ) . Of course, by Theorem 1.2 the exists sucha circuit, but we need to be able to prevent a malicious prover from providing us with a poly ( n ) size circuit of exponential degree, and it is not known how to compute the degree of a circuitin deterministic polynomial time, even assuming PIT ∈ P . To solve this issue, we use an ideaof Malod and Portier [MP08], who showed that any polynomial with circuit of size poly ( n ) anddegree d also has a multiplicatively disjoint (MD) circuit of size poly ( n, d ) . An MD circuit is acircuit in which any multiplication gates multiplies two disjoint subcircuits. This is a syntacticnotion which is easy to verify efficiently and deterministically, and an MD circuit of size s isguaranteed to compute a polynomial of degree at most s . final technical issue is that the notion of MD circuits does not fit perfectly within theframework of constant free circuits. Therefore we use the notion of “almost MD” circuits, whichallow for the case which the inputs to a multiplciation gates are not disjoint, as long as at leastone of them is the root of a subcircuit in which only constants appear. Definition 5.1.
We say a gate v in a circuit is constant producing (CP) if in the subcircuitrooted at v , all input nodes are field constants.An almost-MD circuit is a circuit where every multiplication gate either multiplies two dis-joint subcircuits, or at least one of its children is constant producing. ♦ Lemma 5.2.
Suppose f is an n -variate polynomial of degree poly ( n ) which has a constantfree arithmetic circuit of degree poly ( n ) . Then f has a constant free almost-MD circuit of size poly ( n ) .Proof. Let C be a constant free arithmetic circuit for f . We first homogenize the circuit C to obtain a circuit C (a homogeneous circuit is a circuit in which every gate computesa homogeneous polynomial [SY10]). Since C is homogeneous, all the gates which computenon-zero field constants are CP gates. We then eliminate all gates which compute constants byallowing the edges entering sum gates to be labeled by field scalars, and interpreting a sum gateas computing a linear combination whose coefficients are given by the edge labels. We call thiscircuit C . This step does not maintain constant-freeness. However, every label appearing onthe edges of C was computed in C , so it can be computed by a constant-free arithmetic circuitof polynomial size.We now do the transformation detailed in [MP08] to C to obtain an MD circuit C , whichhas labels on the edges. This step does not produce new constants. Finally, we convert C to analmost-MD constant free circuit C , by re-computing every label appearing on the edge using afresh subcircuit for each label, and rewiring the circuit (which will convert the circuit from anMD circuit to an almost MD circuit). These subcircuits are guaranteed to have polynomial sizeconstant free circuits since these constant were all computed in C , which keeps the total size poly ( n ) .For circuits which compute low-degree polynomials, the mere existence of an algorithm forthe decision version of PIT allows one to construct an algorithm for the search version. Lemma 5.3.
Suppose
PIT ∈ P . Then there is a polynomial time algorithm that given a non-zero almost-MD arithmetic circuit C of size s computing an n -variate polynomial, finds in time poly ( n, s ) an element a ∈ C n such that C ( a ) = 0 .Proof. We abuse notation by denoting by C also the polynomial computed by the circuit C .Note that since C is almost-MD, the degree of C is at most s . Thus, there exists a ∈ { , , . . . , s } such that C ( a , x , . . . , x n ) is a non-zero polynomial in x , . . . , x n . By iterating over those s + 1 values from to s and using the assumption that PIT ∈ P , we can find such a value for a intime poly ( n, s ) . We then continue in the same manner with the rest of the variables.As we noted above, the assumption that C is almost-MD was used in Lemma 5.3 to boundthe degree of the circuit. It is also useful because it is easy to decide in deterministic polynomialtime whether a circuit is almost-MD. We now complete the proof of Corollary 1.3. Proof of Corollary 1.3.
For every n , the proof of Theorem 1.2 provides an equation Q n for theset of n × n matrices with small linear circuits. This polynomial can be found by solving alinear system of equations in a linear space whose dimension is exp( poly ( n )) . Using standard,small space algorithm for linear algebra [BvzGH82, ABO99], this implies that there exists afixed PSPACE algorithm which, on input n , outputs the list of coefficients of the polynomial Q n . onsider now the family { Q n } n ∈ N . If for any constant k ∈ N there exist infinitely many n ∈ N such that Q n requires circuits of size at least n k , it follows (by definition) that the PSPACE algorithm above outputs a family of polynomials with super-polynomial constant-freearithmetic circuits.We are thus left to consider the case that there exists a constant k ∈ N such that for alllarge enough n ∈ N , Q n can be computed by circuits of size n k . By Lemma 5.2, we may assumewithout loss of generality that these circuits are almost-MD circuits. Further suppose PIT ∈ P .We will show how to construct a matrix in polynomial time with an NP oracle which requireslarge linear circuits.Consider the language L of pairs (1 n , x ) such that there exists a string y of length at most n k such that xy describes an almost-MD circuit C such that C is non-zero, and C ◦ U ≡ ,where U is the polynomial map given in the proof of Theorem 1.2.Assuming PIT ∈ P , the language L is in NP , and by assumption for every large enough n there exists such a circuit. Thus, we can use the NP oracle to construct such a circuit C bit bybit. Finally, using Lemma 5.3 we can output a matrix M such that C ( M ) = 0 .By the properties of the circuit C and the map U , M does not have linear circuits of sizeless than n / .Many variations of Corollary 1.3 can be proved as well, with virtually the same proof. Byslightly modifying the language L used in the proof, it is possible to prove the same result evenunder the assumption PIT ∈ NP (recall that PIT ∈ coRP ). A similar statements also holds overfinite fields of size poly ( n ) , in which case the proof is simpler since there are no issues relatedto the bit complexity of the first constants. Finally, an analog of Corollary 1.3 also holds fortensor rank, by using Theorem 4.2 instead of Theorem 1.2: that is, assuming PIT ∈ P , eitherthere exists a construction of a hard polynomial in PSPACE , or an efficient construction withan NP oracle of a 3-dimensional tensor of rank Ω( n ) . We remark that for tensors of large rankthere are no analogs of [AC19, BHPT20], i.e., there do not exist even constructions with an NP oracle of tensors with slightly super-linear rank. References [ABO99] Eric Allender, Robert Beals, and Mitsunori Ogihara. The Complexity of MatrixRank and Feasible Systems of Linear Equations.
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