Algebraic Dependencies and PSPACE Algorithms in Approximative Complexity over Any Field

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S

: CCC 2018

Algebraic Dependencies and

PSPACE

Algorithms in Approximative Complexity

over Any Field

Zeyu Guo

Nitin Saxena

Amit Sinhababu

Received June 30, 2018; Revised June 16, 2019; Published December 14, 2019

Abstract: Testing whether a setfof polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. The complexity of algebraic independence testing is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS’07). Previously, the best complexity bound known wasNP#P(Mittmann, Saxena, Scheiblechner, Trans. AMS 2014). In this article we put the problem inAM∩coAM. In particular, dependence testing is unlikely to be NP-hard. Our proof uses methods of algebraic geometry. We estimate the size of the image and the sizes of the preimages of the polynomial map f over the finite field. Agap between the corresponding sizes for independent and for dependent sets of polynomials is utilized in theAMprotocols.

A conference version of this paper appeared in the Proceedings of the 33rd Computational Complexity Conference (CCC’18) [24] and as a poster in STOC 2018: TheoryFest.

∗_{Supported by DST and Research I Foundation of CSE, IITK.}

†_{Supported by DST/SJF/MSA-01/2013-14.} ‡_{Supported by DFG grant TH 472/5-1.}

ACM Classification:I.1, F.2.1, F.1.3, G.1.2

AMS Classification:03D15, 14Q20

Next, we study the open question of testing whether every annihilator off has zero constant term (Kayal, CCC’09). We introduce a new problem calledapproximate polynomial satisfiability(APS), which is equivalent to the preceding question by a classical character-ization in terms of the Zariski closure of the image of f. We show that APS is NP-hard and, using ideas from algebraic geometry, we put APS inPSPACE. (The best previous bound was EXPSPACE via Gröbner basis computation.) As an unexpected application of this to approximative complexity theory we obtain that, overanyfield, hitting sets for

VP can be constructed in PSPACE. This solves an open problem posed in (Mulmuley, FOCS’12, J. AMS 2017), greatly mitigating the GCT Chasm (exponentially in terms of space complexity).

1

Introduction

Algebraic dependence is a generalization of linear dependence. Polynomials f₁, . . . ,fm∈F[x1, . . . ,xn]are calledalgebraically dependentover the fieldFif there exists a nonzero polynomial (calledannihilator) A(y1, . . . ,ym)∈F[y1, . . . ,ym]such thatA(f1, . . . ,fm) =0. If noAexists, then the given polynomials are calledalgebraically independentoverF. Thetranscendence degree(trdeg) of a set of polynomials is the analog of rank in linear algebra. It is defined as the maximum number of algebraically independent polynomials in the set. Both algebraic dependence and linear dependence define matroid structures [20]. There exist algebraic matroids overFpthat are not linear [28].

The simplest example of algebraically independent polynomials isx1, . . . ,xn∈F[x1, . . .,xn]. As an example of algebraically dependent polynomials, we can take f1=x, f2=yand f3=x2+y2. Then, y2₁+y2₂−y₃is an annihilator. The underlying field is crucial in this concept. For example, for a given prime numberp, the polynomialsx+yandxp+ypare algebraically dependent over fields of characteristicp(as

(x+y)p=xp+ypin characteristicp). But they are algebraically independent over fields of characteristic zero (e. g.,Q) and over fields of positive characteristic`6=p.1

Thus, the following computational question, AD(F), is natural and it is the first problem we consider in this paper: Given polynomials f1, . . . ,fm∈F[x1, . . . ,xn], represented as algebraic circuits, test if they are algebraically dependent. It can be solved inPSPACEusing a classical result due to Perron [48,49]. We sketch the proof of this result here: Perron proved that given a set of algebraically dependent polynomials, there exists an annihilator whose degree is at most the product of the degrees of the polynomials in the set. (This exponential degree bound is tight [30].) It is not hard to see that testing the existence of such an annihilator reduces to solving a system of linear equations in exponentially many variables where the coefficients of the annihilator are regarded as the variables of the system. It is also known that solving a system of linear equations is in logspace-uniformNC[15,9,43], which is contained in

PolyL(polylogarithmic space). Combining these results we see that testing algebraic dependence (and computing an annihilator polynomial) is inPSPACE.

Computing the annihilator may be quite hard, but it turns out that the decision version is easy over zero (or large) characteristic using a classical result known as the Jacobian criterion [29,7]. The Jacobian efficiently reduces algebraic dependence testing of f1, . . . ,fmoverFto linear dependence testing of the

differentialsd f1, . . . ,d fmoverF(x1, . . . ,xn), where we viewd fias the vector

∂fi

∂x1

, . . . ,∂fi ∂xn

.

Placingd fias thei-th row gives us the Jacobian matrixJof f1, . . . ,fm. If the characteristic of the field Fis zero (or larger than the product of the degrees deg(fi)) then the transcendence degree of f1, . . . ,fm equals rank(J). It follows from the Polynomial Identity Lemma2that, with high probability, rank(J)is equal to the rank ofJevaluated at a random point inFn. This gives a simple randomized polynomial-time algorithm solving AD(F), over the fieldsFsatisfying the condition stated above.

For fields of positive characteristic, if the polynomials are algebraically dependent, then their Jacobian matrix is not full rank. But the converse is not true. There are infinitely many input instances (set of algebraically independent polynomials) for which the Jacobian criterion fails. The failure can be characterized by the notion of “inseparable extension” [47]. For example, xp,yp are algebraically independent overFp, yet their Jacobian determinant vanishes. Another example is,xp−1y,xyp−1 over Fpfor prime p>2. Mittmann, Saxena, and Scheiblechner [41] gave a criterion, called Witt-Jacobian, that works over fields of positive characteristic, improving the complexity of algebraic independence testing fromPSPACEtoNP#P. [47] gave another generalization of the Jacobian criterion that is efficient in special cases.

Given that an efficient algorithm to tackle positive characteristic is not in close sight, one could speculate the problem to beNP-hard or even outside the polynomial hierarchyPH. In the present article we show thatfor finite fields,AD(F)is inAM∩coAM(Theorem 1.1). This rules out the possibility ofNP-hardness, unless the polynomial-time hierarchy collapses [4]. Thus, AD(F)joins the league of problems of “intermediate” complexity, such as graph isomorphism and integer factoring.

Constant term of the annihilators. We come to the second problem AnnAtZerothat we discuss in this paper: Testing if the constant term ofeveryannihilator of a set of polynomials (given as algebraic circuits),f={f1, . . . ,fm}, is zero. The annihilators offconstitute a prime ideal of the polynomial ring F[y1, . . . ,ym]. This ideal is principal when the transcendence degree offism−1. This is a classical result in commutative algebra [39, Theorem 47]. See also [30, Lemma 7] for an exposition. In this case, we can decide inPSPACEif the constant term of the minimal annihilator is zero, as theuniqueannihilator (up to scaling) can be computed inPSPACE.

If the transcendence degree off is less thanm−1, the ideal of the annihilators off is no longer principal. Although the ideal is finitely generated, finding the generators of this ideal is computationally very hard. (E. g., using Gröbner basis techniques, we can do it inEXPSPACE[17, Section 1.2.1].) In this case, can we decide if all the annihilators off have constant term zero? We give two equivalent characterizations ofAnnAtZero—one geometric and the other algebraic—and we use them to devise a PSPACEalgorithm to solve it in all cases(Theorem 1.3).

Interestingly, there is an application of the above algorithm to algebraic complexity. We give a PSPACE-explicit construction of a hitting set of the classVP

Fq (Theorem 1.4). VPFq consists ofn-variate

polynomials of degreed=nO(1)over the fieldFq, that can be “infinitesimally approximated” by algebraic circuits of sizes=nO(1). A polynomial p(x1, . . . ,xn)over an algebraically closed fieldFis said to be infinitesimally approximatedby a circuit of sizes, if the circuit computes a polynomial of the form

p+εp1+ε2p2+· · ·+εmpm, where the circuit uses constants fromF(ε)(for example, 1/ε can be used

as a constant) andp1, . . . ,pm∈F[x1, . . . ,xn].

A setHof points is called ahitting setfor a setCof polynomials, if any nonzero polynomial inC

evaluates to a nonzero value in at least one point inH. It was shown by Heintz and Schnorr [27] that hitting sets of size poly(n,s)and bit length poly(n,s,logd)existfor the class ofn-variate polynomials of degree at mostdcomputed by sizesarithmetic circuits. The same result extends to the class ofn-variate polynomials of degree at mostd infinitesimally approximatedby sizesarithmetic circuits. Now the problem of constructing explicit hitting sets forVPasks to deterministically compute a set of points that is a hitting set for the class ofn-variate polynomials of degree at mostd, infinitesimally approximated by sizesarithmetic circuits.

The above problem is interesting as natural questions like explicit construction of the normalization map (in Noether’s Normalization Lemma NNL) reduce to the construction of a hitting set ofVP (Mul-muley [45]), which was previously known to be only inEXPSPACE[45,44]. This was recently put in

PSPACE, over the fieldC, by Forbes and Shpilka [21]. Their proof uses real analysis and does not apply to finite fields. We need to develop purely algebraic concepts to solve the finite field case (namely through AnnAtZero), which apply toanyfield.

To further motivate the concept of algebraic dependence, we list a few recent problems in computer science. The first problem is about constructing an explicit randomness extractor for sources which are polynomial maps over finite fields. Using the Jacobian criterion, [19,18] solved the problem for fields with large characteristic. The second application is to the famous polynomial identity testing (PIT) problem. To efficiently design hitting sets, for some interesting models, [7,3,34] constructed a family of trdeg-preserving maps. For more background and applications of algebraic dependence testing, see [47]. The annihilator has been a key concept to prove the connection between hitting sets and lower bounds [27], and in bootstrapping “weak” hitting sets [2].

Finally, we remark that a common thread among the problems we study in this paper is studying the closure of the polynomial map. Let f:Fn→Fnbe a given polynomial map. Then, we study two computational questions: (1) (Algebraic Dependence Testing) whether the dimension of the Zariski closure of the image of f is<n? (2) (Origin in closure) whether the origin0is in the Zariski closure of the image of f?

1.1 Our results

In this paper, we give Arthur-Merlin protocols and algorithms, with proofs using basic tools from algebraic geometry. The first theorem we prove is about AD(Fq).

Theorem 1.1. Testing algebraic independence of multivariate polynomials over a finite field is in AM∩coAM.

(namelyAM∩coAM). This rules out the possibility of AD(Fq)beingNP-hard (unless the polynomial hierarchy collapses to the second level [4]). Recall that, for a fieldFof characteristic zero or of large characteristic, AD(F)is incoRP(Section 2). We conjecture such a result for AD(Fq)too.

Our second result is about the problem AnnAtZero (i. e., testing whether all the annihilators of given fhave constant term zero). A priori it is unclear why it should have complexity better thanEXPSPACE. (Note: ideal membership isEXPSPACE-complete [40].)

First, we relate the problem AnnAtZero to a (new) version of polynomial system satisfiability, over the algebraic closureF.

Problem 1.2(Approximate polynomials satisfiability (APS)). Given polynomials

f1, . . . ,fm∈F[x1, . . . ,xn],

represented as algebraic circuits, does there existβ ∈F(ε)nsuch that for alli, fi(β)is in the idealεF[ε]

ofF[ε]? If yes, then we say thatf:={f1, . . . ,fm}is in APS.

It is easy to show that the function fieldF(ε)here can be equivalently replaced by the ringF[ε,ε−1] ofLaurent polynomials, or, the fieldF((ε))offormal Laurent series(use modεF[ε]). A reason why

these objects appear in algebraic complexity can be found in [11, Section 5.2] and [37, Section 5]. They help algebrize the notion of “infinitesimal approximation” (in real analysis think ofε→0 & 1/ε→∞).

A notable computational issue involved is that the degree bound ofεrequired forβ is exponential in the

input size [37, Proposition 3]; this may again be a “justification” why APS may require that much space.

Classically, theexactversion of APS has been well-studied—Does there existβ ∈Fnsuch that for alli, fi(β) =0? This is what Hilbert’s Nullstellensatz (HN) characterizes. HN also yields an impressive PSPACEalgorithm [32,33]. Note that if systemfhas an exact solution, then it is trivially in APS. But the converse is not true. For example,{x,xy−1}is in APS, but there is no exact solution inF. To see that it is in APS, assignx=εandy=1/ε. Also, the instance{x,x+1}is neither in APS nor does it have an

exact solution. Finally, note that if we restrictβ to come fromF[ε]n then APS becomes equivalent to exact satisfiability and HN applies. This can be seen by going moduloεF[ε], as the quotientF[ε]/εF[ε]

isF.

Coming back to AnnAtZero, we show that it is equivalent both to a geometric question and to deciding APS. This gives us, with more work, the following surprising consequence.

Theorem 1.3. APSisNP-hard and is inPSPACE.

We apply this to designing hitting sets and solving NNL. (We refer to [45] for the background.)

Theorem 1.4. There is aPSPACEalgorithm that (given input n,s,r in unary and suitably large q in binary) computes a set of points fromFnqof sizepoly(n,s)that hits all n-variate degree-r polynomials overFqthat can be infinitesimally approximated by size-s circuits.

Σ[det,m]be the set of points [f]such that f is a symbolic determinant of a matrixM∈_Fm×m _whose entries are linear forms inx1, . . . ,xr. Finally, let∆[det,m]⊆PV be the Zariski closure ofΣ[det,m].

Mulmuley [45] considers the problem of constructing a homogeneous linear map that mapsΣ[det,m]⊆

PV to a much smaller projective spacePk (anormalizing map). In particular, we want to compute a setS={p₀, . . . ,pk} in poly(m)-time, where k=poly(m) and pi ∈Fr for i∈[k], such that the map f 7→(f(p0), . . . ,f(pk))induces a well-defined map fromΣ[det,m]toPk. Such a setSis called a (strict) e.s.o.p.(explicit system of parameters). In addition, an e.s.o.p.Sisseparatingif for any distinct f,g

satisfying[f],[g]∈∆[det,m], we have f(pi)6=g(pi)for some pi∈S.

As shown in [45], the problem of constructing a separating e.s.o.p. reduces to that of constructing hit-ting sets forVP[45, Theorem 4.5]. Combining this reduction with ourPSPACEalgorithm of constructing hitting sets forVP, we obtain the following result.

Theorem 1.5(NNL for∆[det,m]inPSPACE). The problem of constructing a separating e.s.o.p. for

∆[det,m]belongs toPSPACE.

The results for∆[det,m]also hold for other varieties, as long as such varieties satisfy some explicitness condition. This is captured by the notion of an explicit family {Wn} of varieties in [45]. See [45, Definition 5.1] for the formal definition. For such varietiesWn, constructing a separating e.s.o.p. also reduces to constructing hitting sets forVP[45, Theorem 5.11]. So we have the following result.

Theorem 1.6(NNL for explicit varieties inPSPACE). Let{Wn}be an explicit family of varieties as in [45, Definition 5.1]. Then the problem of constructing a separating e.s.o.p. for Wnbelongs toPSPACE.

More applications?The exact polynomials satisfiability question HN (overF) is highly expressive and, naturally, many computational problems can be expressed that way. We claim that in a similar spirit, the APS question expresses those computer science problems that involve “infinitesimal approximation.” Since finite fields do not have a topology allowing approximations, algebraic approximations over arbitrary fields are needed. The latter has been useful in fast matrix multiplication algorithms.

One prominent example of algebraic approximation is the concept ofborder rankof tensor polynomi-als (used in matrix multiplication algorithms and Geometric Complexity Theory (GCT), see [12,35,36]). Border rank computation of a given tensor (overF) can easily be reduced to an APS instance and, hence, now solved inPSPACE. This brings the complexity of border rank closer to the complexity of tensor rank [54]. From the point of view of Gröbner basis theory, APS is a problem that seems a priori much harder than HN. Now that both of them have aPSPACEalgorithm, one may wonder whether it can be brought all the way down toNPorAM? (In fact, HN_Cis known to be inAM, conditionally under GRH [32].)

equivalent formulation of APS is precisely the problem of deciding if0is in the Zariski closure of the image of a given morphism from an affine space.

Our methods in the proof ofTheorem 1.3also imply an interesting “degree bound” related to the (prime) idealI of annihilators of the setfof polynomials. Namely,I=√I≤d, whereI≤d refers to the subideal generated by degree≤dpolynomials inI,dis the Perron-like bound(maxi∈[m]deg(fi))k, and k:=trdeg(f). This is equivalent to the geometric fact, which we prove, that the varieties defined by the two idealsIandI≤d are equal (Theorem 4.6). This again is an exponential improvement over what one expects to get from the general Gröbner basis methods, because, the generators ofImay well have doubly exponential degree.

The result on hitting set (Theorem 1.4) can be applied to compute, inPSPACE, the explicit system of parameters (e.s.o.p.) of theinvariant ringof the variety∆[det,s], overFq, with a given group action [45, Theorem 4.9]. Also, we can now construct, inPSPACE, polynomials inFq[x1, . . . ,xn]that cannot even be approximated by “small” algebraic circuits. Such results were previously known only for fields of characteristic zero, see [21, Theorems 1.1–1.4]. Bringing this complexity down toPis the longstanding problem of blackbox PIT (and lower bounds), see [52,58,53]. Mulmuley [44] pointed out that small hitting sets forVPcan be designed inEXPSPACEwhich is a far worse complexity than that forVP. He called it the GCT Chasm. We bridge it somewhat, as the proof ofTheorem 1.4shows that small hitting sets forVP

F can be designed inPSPACE(like those forVP) foranyfieldF.

In another application, the null-cone problem defined in [13] can be seen as a special case of APS and using our algorithm, it can be solved inPSPACE. Bürgisser et al. [13] gave an exponential-time algorithm for the above problem (bringing it down fromEXPSPACE).

Finally, another motivation for AnnAtZero or APS comes from the study of thegeometric ideal proof systemin algebraic proof complexity, introduced in [23], where they essentially discuss AnnAtZero for systems of polynomial equations corresponding to Boolean tautologies. See [23, Appendix B] for more details.

Remark 1.7. Given a system of polynomial equations f1(x1, . . . ,xn) =0, . . . ,fm(x1, . . . ,xn) =0, we can always homogenize the fi by introducing a new variablez. The question ofprojective polynomials satisfiabilityfor a given system of equations is whether there is a nonzero solution (called aprojective solution) to the homogenized system. We have seen that APS does not directly reduce to (affine) polynomials satisfiability, as there are unsatisfiable systems, e. g.,{X=0,XY=1}, that have approximate solutions. We note that APS does not directly reduce to projective polynomials satisfiability either. Consider, for example, the system{X+Y=0,X+Y =1}, corresponding to two parallel affine lines. It has no approximate solution, but the homogenized system{X+Y =0,X+Y =Z}has a projective solution(1,−1,0).

Nonetheless, it is true that if the equations f1=0, . . . ,fm=0 have an approximate solution, then the homogenized equations ˆf1=0, . . . ,fˆm=0 have a projective solution (inPn). We sketch a proof of this fact. Supposea₁, . . . ,an∈F(ε)⊆F((ε))form an approximate solution of the original system. Let an+1=1. Then fi(a1, . . . ,an) = fˆi(a1, . . . ,an,an+1)fori∈[m]. Choose the smallestk∈Zsuch thatεkai is in the ring of formal power seriesF[[ε]]for alli∈[n+1]. We havek≥0 asan+1=1. Fori∈[n+1], let

¯

ai∈Fbe the constant term ofεkai. Minimality ofkguarantees that ¯ai6=0 for somei∈[n+1]. Assigning ¯

1.2 Proof ideas

Proof idea of Theorem 1.1. Suppose we are given polynomials (represented as algebraic circuits) f:={f1, . . . ,fm}computing inFq[x1, . . . ,xn]. For theAMandcoAMprotocols, we consider the following system of equations over a “small” extensionFq0.

Forb= (b1, . . . ,bm)∈Fmq0, define the system of equations fi(x1, . . . ,xn) =bi, fori∈[m]. We denote the number of solutions of the above system inFnq0 asNb. Let f :Fnq0 →Fm_q0 be the polynomial map a7→(f₁(a), . . . ,fm(a)).

AMgap.[Theorem 3.5] We establish bounds for the numberNf(a), whereais a random point inFnq0.

If f₁, . . . ,fmare independent, we show thatNf(a)is relatively small. On the other hand, if the polynomials are algebraically dependent thenNf(a)is much greater.

Assumefis algebraically independent. Without loss of generality [47, Section 2] we can assume that

m=nand for alli∈[n],xi,f1, . . . ,fnare algebraically dependent. The first step is to show that with high probability, the zero set defined by the system of equations, for randomf(a), is finite (i. e., it has dimension ≤0 as a variety). This is proved using the Perron degree bound on the annihilator of{xi,f1, . . . ,fn}. Next, one can apply an affine version of Bézout’s theorem to obtain an upper bound onNf(a). On the other hand, supposefis algebraically dependent, say with annihilatorQ. Let Im(f):= f(Fnq0)be the image of f. SinceQvanishes on Im(f), we know that Im(f)is relatively small, whence we deduce thatN_f(a)is large for “most” values ofa.

coAMgap.[Theorem 3.8] We pick a random pointb= (b₁, . . . ,bm)∈Fmq0 and boundNb, which is the number of solutions of the system defined above. In the dependent case, we show thatNb=0 for “most” values ofb. But in the independent case, we show thatNb≥1 for “many” (maybe not “most”!) values of b. The ideas are based on those sketched above.

The two kinds of gaps shown above are based on the sets f−1(f(x))and Im(f), resp.

Note that membership in either of these sets is testable inNP(the latter requires nondeterminism). Based on this and the gaps between the respective cardinalities, we can invokeLemma 2.1and devise the

AMandcoAMprotocols for AD(Fq0), which also apply to AD(_Fq).

Remark 1.8. Our proof ofTheorem 1.1employs the fact that we could efficiently sample a random point in the set Im(f). In contrast, it is not clear how to efficiently sample a random point in the zero set Zer(f):={x∈_Fn

q0 | f(x) =0}. Thus, we manage to side-step theNP-hardness associated with most zero set properties. E. g., computing the dimension of Zer(f)isNP-hard.

Proof idea ofTheorem 1.3. Let polynomialsf:={f1, . . . ,fm}inF[x1, . . . ,xn]be given over a fieldF, represented by algebraic circuits. We want to determine if the constant term of every annihilator forfis zero. Redefine the polynomial map f :Fn→Fm;a7→(f1(a), . . . ,fm(a)). For a subsetSof an affine or projective space, writeSfor itsZariski closurein that space, i. e., for the smallest subset that containsS

and equals the zero set Zer(I)of some polynomial idealI (homogeneous ideal in the projective case). APSvs.AnnAtZero. [Theorem 4.2] Now, we interpret the problem AnnAtZero in a geometric way throughLemma 4.1:

The constant term of every annihilator offis zero iff the origin point0∈Im(f).

i=1, . . . ,m. The latter problem (call it HN for Hilbert’s Nullstellensatz) is known to be inAMifF=Q and GRH is assumed [32]. However, Im(f)is not necessarily Zariski closed; equivalently, it may be strictly smaller than Im(f). So, we need new ideas to test0∈Im(f).

Next, we observe that although0∈Im(f)is not equivalent to the existence of a solutionx∈_Fnto

f(x) =0, itisequivalent to the existence of an “approximate solution”x∈_F(ε)n, which is ann-tuple of rational functions in a formal variableε. The proof idea of this uses a degree bound onε due to [37].

We called this problem APS. As AnnAtZero problem is already known to beNP-hard [30], APS is also

NP-hard.

Upper bound onAPS.We now know that solving APS forfis equivalent to solving AnnAtZero forf. AnnAtZero was previously known to be inPSPACEin the special case when the transcendence degreek

ofF(f)/Fequalsmorm−1, but the general case remained open (best beingEXPSPACE).

In this article we prove that AnnAtZero is inPSPACEeven whenk<m−1. Our simple idea is to reduce the input to a smallerm=k+1 instance, by choosing new polynomialsg1, . . . ,gk+1 that

are random linear combinations of the polynomials fi. We show that with high probability, replacing

{f₁, . . . ,fm}by{g1, . . . ,gk+1}preserves YES/NO instances as well as the transcendence degree. This

gives a randomized poly-time reduction from the casek<m−1 tok=m−1 (Theorem 4.6). The latter has a standardPSPACEalgorithm.

For notational convenience viewFas theaffine lineA. DefineV :=Im(f)⊆Am. Proving that the above reduction (ofm) does preserve YES/NO instances amounts to proving the following geometric statement: IfV does not contain the originO∈_Am_{, then with high probability, the varietyV}0_:=

π(V)

does not contain the originO0∈_Ak+1_{either, where}

π:Am→Ak+1is a random linear map.

Asπ is picked at random, the kernelWofπis a random linear subspace ofAm. We haveO06∈π(V)

wheneverV∩W=/0, but this is not sufficient for provingO06∈π(V), sinceV may “get arbitrarily close

toW” inAmand meetW “at infinity.”

Example 1. The hyperbolaV:=V(X₁X₂−1)and the lineW:=V(X₁)do not meet in the affine spaceA2, even though they get closer and closer as we increase the absolute value of the coordinateX2. However,

note that the projective closures of these two varietiesVc:=V(X1X2−X₀2)andWc:=V(X1)do meet at

the point(0,0,1)of the projective spaceP2, which is thought of as one of the “points at infinity.”

Inspired by the above observation, we consider projective geometry instead of affine geometry, and prove thatO06∈V0holds as long as the projective closure ofV and that ofW are disjoint. The proof uses a construction of a projective subvariety—thejoin—to characterizeπ−1(V0), and eventually rules out

W⊆π−1(V0)(Lemma 4.8).

Moreover, we show that this holds with high probability ifO6∈V, by (repeatedly) using the fact that a general (=random) hyperplane section reduces the dimension of a variety by one.

Proof idea ofTheorem 1.4. DefineA:=Fqand assume w.l.o.g. q≥Ω(sr2)[1]. [27, Theorem 4.4] showed that a hitting set, of sizeh:=O(s2n2) inFnq, existsfor the class of degree-r polynomials, in A[x1, . . . ,xn], that can be infinitesimally approximated by size-salgebraic circuits. So, we can search over all possible subsets of sizehfromFnqand “most” of them are hitting sets.

How do we certify that a candidate set H is a hitting set? The idea is to use universal circuits. Auniversal circuithasnessential variablesx={x1, . . . ,xn}ands0:=O(sr4) auxiliary variablesy=

homogeneous circuit of size-s, approximating a degree-rpolynomial inVP_A. Given a universal circuitΨ,

certification of a hitting setHis based on the following observation, that follows from the definitions: A candidate setH=:{v1, . . . ,vh}is a hitting set iff

∀y∈_A(ε)s 0

,Ψ(y,x)∈/εA[ε][x]⇒ ∃i∈[h],Ψ(y,vi)∈/εA[ε].

Equivalently, a candidate setH={v₁, . . . ,vh}isnota hitting set iff

∃y∈_A(ε)s 0

,Ψ(y,x)∈/εA[ε][x] and ∀i∈[h],Ψ(y,vi)∈εA[ε].

Note that certification of hitting set is more challenging than the one against polynomials inVP, because the degree bounds forε are exponentially high and moreover, we do not know how to frame the

first “non-containment” condition as an APS instance. To translate it to an APS instance, our key idea is the following.

Pickq≥Ω(s0r2)so that a hitting set exists, inFnq, that works against polynomials infinitesimally approximated by the specializations ofΨ. SupposeΨ(α,x)is not inεA[ε][x], for someα ∈A(ε)s0.

This means that we can write it as∑−m≤i≤m0 _εig_i(x) withg−_m6=0 andm≥0. Clearly, _εm·Ψ(_α,x)

infinitesimally approximates the nonzero polynomialg−m∈A[x]. By the conditions onΨ, we know that

g−mis a homogeneous polynomial of degreer(and approximative complexitys0). Thus, byLemma 2.2, there exists aβ ∈Fnqsuch thatg−m(β) =:ais a nonzero element inA. We can normalize by this and considera−1εm·Ψ(y,x), which evaluates to 1+εA[ε]at(α,β). Since this normalization factor only

affects the auxiliary variablesy, we get another equivalent criterion:

A candidate setH={v₁, . . . ,vh}isnota hitting set iff∃y∈A(ε)s0 and∃x∈_Fn

qsuch that,

Ψ(y,x)−1∈εA[ε] and ∀i∈[h],Ψ(y,vi)∈εA[ε].

We reach closer to APS, but how do we implement∃?x∈_Fn

q(it takes exponential space)?

The idea is to rewrite it, instead using the(r+1)-th roots of unityZr+1⊆A, as∃x∈A(ε)n,∀i∈[n],

xr_i+1−1∈εA[ε]. This gives us a criterion that is an instance of APS withn+h+1 input polynomials

(Theorem 5.2). ByTheorem 1.3it can be done inPSPACE, finishing the proof. Moreover, thisPSPACE

algorithm is independent of the characteristic of the underlying field. (E. g., it can be seen as an alternative to [21] over the complex field.)

2

Preliminaries

Jacobian.Although this work would not need it, we define the classical Jacobian: Given polynomialsf=

{f1,· · ·,fm}inF[x1,· · ·,xn], theJacobianoffis the matrixJx(f):= (∂xjfi)m×n, where∂xjfi:=∂fi/∂xj. The Jacobian criterion [29,7] states that for degree≤dand trdeg≤rpolynomialsf, if char(F) =0 or char(F)>dr, then trdeg(f) =rankF(x)Jx(f). This yields a randomized poly-time algorithm byLemma 2.2. For other fields, the Jacobian criterion fails due to inseparability and AD(F)is open.

in which the randomized poly-time verifier (Arthur) and the all-powerful prover (Merlin) have only constantly many rounds of exchange.AMcontains interesting problems like verifying that two graphs are non-isomorphic.AM∩coAMis the class of decision problems for which both YES and NO answers can be verified by anAMprotocol. It can be thought of as the randomized version ofNP∩coNP. See [31] for a few natural algebraic problems inAM∩coAM. If such a problem isNP-hard (even under random reductions) then the polynomial hierarchy collapses to the second level, i. e.,PH=Σ2.

In this article,AMprotocols will only be used to distinguish whether a setSis “small” or “large.” Formally, we refer to the Goldwasser-Sipser [22] Set Lowerbound method.

Lemma 2.1(Goldwasser-Sipser [22]). Let m∈_Nbe given in binary. Suppose S is a set whose membership can be tested in nondeterministic polynomial time and its size is promised to be either≤m or≥2m. Then, the problem of deciding whether|S| ≥2m is inAM.

See, e. g., [4, Chapter 8] for a proof. We also use the Polynomial Identity Lemma several times. The version we use is stated below.

Lemma 2.2(Polynomial Identity Lemma [57,60,16]). Let f(x1, . . . ,xn)be a nonzero polynomial of total degree d over a fieldF. Let S be a finite subset ofF. Then the total number of roots of f(x1, . . . ,xn) in Snis bounded by d|S|n−1_.

For a proof, see [4, Lemma A.36].

Geometry.Our proof requires some basic facts and definitions from classical algebraic geometry, which are listed below. One can also refer to a standard text, e. g., [25,26].

LetA:=Fbe the algebraic closure of a fieldF. Ford∈N+, writeAd for thed-dimensional affine spaceoverA. It is defined to be the setAd, equipped with theZariski topology, defined as

Definition 2.3(Zariski topology). A subsetSofAdisclosedif it is the set of common zeros of some set of polynomials inA[X1, . . . ,Xd]. For a setSof polynomials, theclosure Sis defined as the smallest closed set containingS. A setSisdenseinAdifS=Ad. The complement of a closed set is called anopenset.

A closed set is called ahypersurfaceif it is definable by a single polynomial. A hypersurface is a

hyperplaneif its defining polynomial is linear.

DefineA×:=A\ {0}. WritePd for thed-dimensional projective space overA, defined to be the quotient set(Ad+1\ {(0, . . . ,0)})/∼where(x0, . . . ,xd)∼(y0, . . . ,yd)iff there existsc∈A×such that yi=cxi for 0≤i≤d. We use(d+1)-tuples(x0, . . . ,xd)to represent points inPd. Such a(d+1)-tuple is called a list ofhomogeneous coordinatesof the point it represents.

The setPdis again equipped with theZariski topology.

Definition 2.4(Zariski topology on a projective space). A subsetS ofPd is closed if it is the set of common zeros of some set ofhomogeneouspolynomials inA[X0, . . . ,Xd].

(In some references, varieties are not required to be irreducible, but in this paper we always assume they are.) An algebraic setV can be uniquely represented as the union of finitely many varieties, and these varieties are called theirreducible componentsofV.

Affine zero sets are in 1-1 correspondence withradicalideals; varieties correspond toprimeideals. Irreducible decomposition of an affine variety mirrors the factoring of an ideal into primary ideals. Finally, note that the affine points are in 1-1 correspondence with maximal ideals; this is a simple reformulation of Hilbert’s Nullstellensatz.

The affine spaceAd may be regarded as a subset ofPd via the map (x1, . . . ,xd)7→(1,x1, . . . ,xd). Then the subspace topology ofAd induced from the Zariski topology ofPd is just the Zariski topology of Ad. The setPd\Ad is the projective subspace ofPddefined byX0=0, called thehyperplane at infinity.

For an algebraic subsetV ofAd⊆Pd, the smallest algebraic subsetV0ofPd containingV (i. e., the intersection of all algebraic subsets containingV) is theprojective closureofV, and we haveV0∩_Ad_=V. To see this, note that forP= (x1, . . . ,xd)∈Ad\V, there exists a polynomialQ∈A[X1, . . . ,Xd]of degree D∈Nnot vanishing onP(but vanishing onV). Then its homogenizationQ0∈A[X0, . . . ,Xd], defined by replacing each monomial

M=

d

∏

Xdi

i by X

D−deg(M)

0

d

∏

Xdi i ,

does not vanish on(1,x1, . . . ,xd). So,(1,x)∈/V0.

For distinct pointsP= (x0, . . . ,xd),Q= (y0, . . . ,yd)∈Pd, writePQfor theprojective linepassing through them, i. e.,PQconsists of the points(ux0+vy0, . . . ,uxd+vyd), where(u,v)∈A2\ {(0,0)}.

Definition 2.6(Dimension and degree). Thedimensionof a varietyV is defined to be the largest integer

msuch that there exists a chain of varieties /0(V0(V1(· · ·(Vm=V. More generally, the dimension of an algebraic setV, denoted by dimV, is the maximal dimension of its irreducible components. E. g., we have dimAd=dimPd=d. The dimension of the empty set is−1 by convention. One-dimensional varieties are calledcurves.

Thedegreeof a varietyV inAd(or inPd) is the number of intersections ofV with a general3affine subspace (projective subspace, resp.) of dimensiond−dimV. More generally, we define the degree of an algebraic setV, denoted by deg(V), to be the sum of the degrees of its irreducible components. The degree of an algebraic subset ofAdcoincides with the degree of its projective closure inPd.

SupposeV ⊆_Ad _{is an algebraic set, defined by polynomials f}

1, . . . ,fk. Let(a1, . . . ,ad)∈Ad. Then the set{(x1+a1, . . . ,xd+ad):(x1, . . . ,xd)∈V} is called atranslateofV. It is also an algebraic set, defined by fi(X1−a1, . . . ,Xd−ad),i=1, . . . ,k.

Definition 2.7(Morphism). LetV ⊆_An_{,W ⊆}

Ambe affine varieties. A morphismfromV toW is a function f:V→W that is a restriction of a polynomial mapAn→Am.

A morphism f:V →W is calleddominantif Im(f) =W. The preimage of a closed subset under a morphism is closed (i. e., morphisms arecontinuousin the Zariski topology).

For a polynomial map f :An→Amand an affine varietyV⊆An,W := f(V)is also an affine variety (i. e., it is irreducible). To see this, assume to the contrary thatWis the union of two proper closed subsets

W1 andW2. By the definition of closure, f(V) is not contained in eitherW1 orW2, i. e., it intersects

both. Then f−1(W1)∩V and f−1(W2)∩V are two proper closed subsets ofV, and their union isV. This

contradicts the irreducibility ofV.

ThegraphΓf of a morphism f is the set{(x,f(x)):x∈V} ⊆V×W ⊆An×Am. HereV×W =

{(x,y):x∈V,y∈W}denotes theproductofV andW, which is a subvariety of the(n+m)-dimensional affine spaceAn×Am∼=An+m. Note the graph Γf is closed inAn×Am: Suppose f sends x∈V to

(f1(x), . . . ,fm(x))∈Am, where fi ∈A[X1, . . . ,Xn]for i∈[m]. And supposeV is defined by an ideal I⊆A[X1, . . . ,Xn]. Then Γf is defined by the ideal ofA[X1, . . . ,Xn,Y1, . . . ,Ym]generated byI and the polynomialsYi−fi(X1, . . . ,Xn),i=1, . . . ,m.

3

Algebraic dependence testing: Proof of

Theorem 1.1

Given f1, . . . ,fm∈Fq[x1, . . . ,xn], we want to decide if they are algebraically dependent. For this problem, AD(Fq), we could assume, with some preprocessing, thatm=n. For,m>nmeans that it is a YES instance. Ifm<nthen we could apply a “random” linear map on the variables to reducentom, preserving the YES/NO instances. Also, the transcendence degree does not change when we move to the algebraic closureFq. The details can be found in [47, Lemmas 2.7–2.9]. So, we assume the input instance to be

f:={f1, . . . ,fn}with nonconstant polynomials.

In the following, letD:=∏i∈[n]deg(fi) >0 andD0:=maxi∈[n]deg(fi) >0. Letd∈N+ andq0=qd. The value ofdwill be determined later. Let f :Fnq0→F

n

q0 be the polynomial mapa7→(f1(a), . . . ,fn(a)). Forb= (b1, . . . ,bn)∈Fnq0, denote byNbthe size of the preimage

f−1(b) =

n

x∈Fnq0

f(x) =b o

.

Define A:=Fq and Nb:=#{x∈An | fi(x) =bi, for alli∈[n]} which might be ∞. Let Q∈ Fq[y1, . . . ,yn]be a nonzero annihilator, of minimal degree, of f1, . . . ,fn. If it exists then deg(Q)≤Dby Perron’s bound.

3.1 AMprotocol

First, we study the independent case.

Lemma 3.1(Dimension-zero preimage). Supposefis independent. Then Nf(a)isfinitefor all but at

most(nDD0/q0)-fraction of a∈Fnq0.

Proof. Fori∈[n], letGi∈Fq[z,y1, . . . ,yn]be the annihilator of{xi,f1, . . . ,fn}. We have deg(Gi)≤D by Perron’s bound. Consider a∈_Fn

q0 such that G0_i(z):=Gi(z,f1(a), . . . ,fn(a))∈Fq[z]is a nonzero polynomial for everyi∈[n]. We claim thatN_f(a)is finite for sucha.

To see this, note that for anyb= (b1, . . . ,bn)∈Ansatisfying the equations fi(b) = fi(a),i∈[n], we have

Hence, eachbiis a root ofG0i. It follows thatNf(a)≤∏i∈[n]deg(G0i)<∞, as claimed. It remains to prove that the number ofa∈_Fn

q0 satisfyingG 0

i=0, for some indexi∈[n], is bounded by nDD0q0−1·q0n. Fixi∈[n]. SupposeGi=∑dj=i 0Gi,jzj, wheredi:=degz(Gi)andGi,j∈Fq[y1, . . . ,yn], for 0≤ j≤di. The leading coefficientGi,di is a nonzero polynomial. As f1, . . . ,fnare algebraically inde-pendent, the polynomialGi,di(f1, . . . ,fn)∈Fq[x1, . . . ,xn]is also nonzero. Its degree is≤D

0_deg(G i,di)≤

D0deg(Gi)≤DD0. ByLemma 2.2, for all but at most(DD0/q0)-fraction ofa∈Fnq0, we have

Gi,di(f1(a), . . . ,fn(a))6=0,

which implies

G0_i(z) =Gi(z,f1(a), . . . ,fn(a)) = di

∑

Gi,j(f1(a), . . . ,fn(a))zj 6=0.

The claim now follows from the union bound.

We need the following affine version of Bézout’s Theorem. Its proof can be found in [55, Theo-rem 3.1].

Theorem 3.2(Bézout’s). Let g1, . . . ,gn∈A[x1, . . . ,xn]. Then the number of common zeros of g1, . . . ,gn in_Anis either infinite, or at most∏i∈[n]deg(gi).

CombiningLemma 3.1with Bézout’s Theorem, we obtain

Lemma 3.3(Small preimage). Supposefis independent. Then Nf(a)≤D for all but at most(nDD0/q0)

-fraction of a∈Fnq0.

Next, we study the dependent case (with an annihilatorQ).

Lemma 3.4(Large preimage). Supposefis dependent. Then for k>0, we have Nf(a)>k for all but at

most(kD/q0)-fraction of a∈Fnq0.

Proof. Let Im(f):= f(Fnq0)be the image of the map. Note thatQvanishes on all the points in Im(f). So,

|Im(f)| ≤Dq0n−1byLemma 2.2.

LetB:={b∈Im(f):Nb≤k}be the “bad” images. We can estimate the bad domain points as,

#{a∈_Fn_q0:N_f(a)≤k} =#{a∈_Fn_q0: f(a)∈B} ≤ k|B| ≤k|Im(f)| ≤kDq0n−1.

which proves the lemma.

Theorem 3.5(AM). Testing algebraic dependence offis inAM.

Proof. Fixq0=qd>4nDD0+4kDandk:=2D. Note thatdwill be polynomial in the input size. For an

a∈_Fn

q0, consider the set f−1(f(a)):={x∈Fn_q0 | f(x) = f(a)}.

ByLemma 3.3andLemma 3.4, when Arthur picksarandomly, with high probability,|f−1(f(a))|=

3.2 coAMprotocol

We first study the independent case.

Lemma 3.6(Large image). Supposefis independent. Then Nb>0for at least(D−1−nD0q0−1)-fraction of b∈_Fn

q0.

Proof. LetS:={a∈Fnq0 :Nf(a)≤D}. Then|S| ≥(1−nDD0q0−1)·q0nbyLemma 3.3. As everyb∈ f(S) has at mostDpreimages inSunder f, we have|f(S)| ≥ |S|/D≥(D−1−nD0q0−1)·q0n. This proves the lemma sinceNb>0 for allb∈ f(S).

Next, we study the dependent case.

Lemma 3.7(Small image). Supposefis dependent. Then Nb=0for all but at most(D/q0)-fraction of b∈Fnq0.

Proof. By definition, Nb >0 iffb∈Im(f):= f(Fnq0). It was shown in the proof ofLemma 3.4that

|Im(f)| ≤Dq0n−1. The lemma follows.

Theorem 3.8(coAM). Testing algebraic dependence offis incoAM.

Proof. Fixq0=qd >D(2D+nD0). Note thatdwill be polynomial in the input size. Forb∈Fnq0, consider

the set f−1(b):={x∈_Fn

q0 | f(x) =b}of sizeNb. Define S:=Im(f). Note that b∈_Fn

q0 satisfies Nb >0 iff b∈S. Thus, by Lemma 3.6, |S| ≥

(D−1−nD0q0−1)q0n >2Dq0n−1 when f is independent, and by Lemma 3.7, |S| ≤Dq0n−1 when f is dependent.

Note that an upper bound on∏i∈[n]deg(fi) can be deduced from the size of the input circuits for the polynomials fi; thus, we knowDq0n−1. Moreover, containment inScan be tested inNP. Thus, by Lemma 2.1, AD(Fq)is incoAM.

Proof ofTheorem 1.1. The statement immediately follows fromTheorem 3.5andTheorem 3.8.

4

Approximate polynomials satisfiability: Proof of

Theorem 1.3

Theorem 1.3is proved in two parts. First, we show that APS is equivalent to AnnAtZero problem, which means that it isNP-hard [30]. Next, we utilize the beautiful underlying geometry to devise aPSPACE

algorithm.

4.1 APSis equivalent toAnnAtZero

LetAbe the algebraic closure ofF. Note that for the given polynomialsf:={f1, . . . ,fm}inF[x], there is an annihilator overFwith a nonzero constant term iff there is an annihilator overAwith a nonzero constant term. This is because ifQis an annihilator overAwith a nonzero constant term, w.l.o.g. 1, then by basic linear algebra, the linear system defined by the equationQ(f) =0, in terms of the unknown coefficients ofQ, would also have a solution inF. Thus, there is an annihilator overFwith constant term 1. This proves that it suffices to solve AnnAtZero over the algebraically closed fieldA. This provides us with a better geometry.

Write f:An→Amfor the polynomial map sending a point

x= (x₁, . . . ,xn)∈An to (f1(x), . . . ,fm(x))∈Am.

For a subsetSof an affine or projective space, writeSfor its Zariski closure in that space. We will useO

to denote the origin0of an affine space.

The following lemma reinterprets AnnAtZero in a geometric way.

Lemma 4.1(classical). The constant term of every annihilator forfis zero iff O∈Im(f).

Proof. Note thatQ∈_A[Y₁, . . . ,Ym]vanishes on Im(f)iffQ(f)vanishes onAn, which holds iffQ(f) =0, i. e., Q is an annihilator for f. So Im(f) =V(I), where the ideal I ⊆_A[Y1, . . . ,Ym] consists of the annihilators forf. Also note that{O}=V(m), wheremis the maximal idealhY1, . . . ,Ymi.

Let us study the conditionO∈Im(f). By the ideal-variety correspondence,{O}=V(m)⊆Im(f) =

V(I)is equivalent toI⊆m, i. e.,Qmodm=0 forQ∈I. ButQmodmis just the constant term of the annihilatorQ. Hence, we have the equivalence.

As a special case, the above lemma proves that wheneverfis algebraicallyindependent, we have Am=Im(f). E. g., f1=X1and f2=X1X2−1.

In general, Im(f)is not necessarily closed in the Zariski topology.

Example 2. Letn=2,m=3. Consider f₁= f₂=X₁and f₃=X₁X₂−1. The annihilators are multiples of(Y1−Y2), which means byLemma 4.1thatO∈Im(f). But there is no solution to f1= f2=f3=0,

i. e.,O∈/Im(f).

Approximation.AlthoughO∈Im(f)is not equivalent to the existence of a solutionx∈Anto fi=0, i∈[m], it is equivalent to the existence of an “approximate solution”x∈_A[ε,ε−1]n, which is a tuple of

Laurent polynomials in a formal variableε. The formal statement is as follows. W.l.o.g. we assumefto

bemnonconstant polynomials.

Theorem 4.2(classical). O∈Im(f)iff there exists x= (x1, . . . ,xn)∈A(ε)nsuch that fi(x)∈εA[ε], for all i∈[m]. Moreover, when such x exists, it may be chosen such that

xi ∈ ε−∆A[ε]∩ε∆ 0

A[ε−1] =

(

∆0

∑

cjεj:cj ∈A

)

, i∈[n],

For example, choosingx= (x1,x2):= (ε,1/ε)inExample 2, we havef1(x) =f2(x) =εand f3(x) =0.

ThusTheorem 4.2gives another way of seeingO∈Im(f)inExample 2.

As mentioned previously,Theorem 4.2was essentially proved in [37]. We include a proof here for the sake of completeness.

First, we recall a tool to reduce the domain from a variety to a curve, proven in [37].

Lemma 4.3([37, Proposition 1]). Let V⊆_An_{, W ⊆}

Ambe affine varieties,ϕ:V →W dominant, and t∈W\ϕ(V). Then there exists a curve C⊆_An_{such that t∈}

ϕ(C)anddeg(C)≤deg(Γϕ), whereΓϕ

denotes the graph ofϕ embedded inAn×Am.

Next, [37] essentially shows that in the case of a curve one can approximate the preimage of f by using asingleformal variableε and working inA(ε). HereA[[ε]]denotes a formal power series inε

with coefficients fromA.

Lemma 4.4([37, Corollary of Proposition 3]). Let C⊆Anbe an affine curve. Let f :C→Ambe a mor-phism sending x∈C to(f1(x), . . . ,fm(x))∈Am, where f1, . . . ,fm∈A[X1, . . . ,Xn]. Let t= (t1, . . . ,tm)∈

f(C). Then there exists p1, . . . ,pn∈ε−deg(C)A[[ε]]such that fi(p1, . . . ,pn)−ti ∈εA[[ε]], for all i∈[m].

Finally, we use the above two lemmas to prove the connection of APS withO∈Im(f), and hence with AnnAtZero (byLemma 4.1).

Proof ofTheorem 4.2. First assume that anx, satisfying the conditions inTheorem 4.2, exists. Pick such anx. Iffis algebraically independent then byLemma 4.1we have thatAm=Im(f)and we are done. So, assume that there is a nonzero annihilatorQfor f. We haveQ(f1(x), . . . ,fm(x)) =0∈εA[ε]. On

the other hand, as fi(x)∈εA[ε], for alli∈[m]; we deduce thatQ(f1(x), . . . ,fm(x))modεA[ε]isQ(0),

which is the constant term ofQ. So it equals zero. ByLemma 4.1, we haveO∈Im(f)and again we are done.

Conversely, assumeO∈Im(f)and we will prove thatxexists. IfO∈Im(f), then we can choose

x∈_An_{and we are done. So assumeO∈Im(f)\Im(f). Regard f as a dominant morphism from} Anto Im(f). Its graphΓf is cut out inAn×AmbyYi−fi(X1, . . . ,Xn),i∈[m]. So deg(Γf)≤∏mi=1deg(fi) =∆ by Bézout’s Theorem.

ByLemma 4.3, there exists a curveC⊆_An_{such thatO∈ f(C)and deg(C)≤deg(Γ}

f)≤∆. Pick such a curveC. ApplyLemma 4.4toC, f|CandO, and let p1, . . . ,pn∈ε−deg(C)A[[ε]]⊆ε−∆A[[ε]]be as

given by the lemma. Then fi(p1, . . . ,pn)∈εA[[ε]], for alli∈[m].

Fori∈[n], letxibe the Laurent polynomial obtained frompiby truncating the terms of degree greater than∆0. When evaluating f1, . . . ,fm, at(p1, . . . ,pn), such truncation does not affect the coefficient ofεk fork≤0 by the choice of∆0. So fi(x1, . . . ,xn)∈εA[ε], for alli∈[m].

Remark 4.5. The lower bound−∆=−∏mi=1deg(fi)for the least degree ofxiinε can be achieved up to

a factor of 1+o(1). Consider the polynomials f₁= f₂=X₁, f₃=X₁d−1X₂−1, and fi=Xid−2−Xi−1for i=4, . . . ,m, wherem=n+1. Then we are forced to choosex1∈εA[ε]andxi ∈ε−(d−1)di

−2

·A[ε−1],

4.2 PuttingAPSinPSPACE

Owing to the exponential upper bound on the precision (= degree w.r.t.ε) shown inTheorem 4.2, one

expects to solve APS inEXPSPACEonly. Surprisingly, in this section, we give aPSPACEalgorithm. This we do by reducing the general AnnAtZero instance to a very special instance, that is easy to solve.

LetAbe the algebraic closure of the field F. Let f1, . . . ,fm∈F[X1, . . . ,Xn]be given. Denote by k the transcendence degree of F(f1, . . . ,fm)/F. Computing k can be done in PSPACE using linear algebra [49,15,9,43]. We assumek<m−1, since the casesk=m−1 andk=mare again easy. In the casek=m, the input instances are always in APS since Im(f) =Am. And in the casek=m−1, the ideal of the annihilators is a principal ideal, and hence has a unique generator (up to scaling). The degree of this generator is at most∏mi=1deg(fi). Thus checking whether it has a nonzero constant term can be solved inPSPACEby solving an exponential sized linear system of equations using [15,9,43].

We reduce the number of polynomials frommtok+1 as follows: Fix a finite setS⊆_F, and choose

ci,j ∈Sat random fori∈[k+1]and j∈[m]. For this to work, we need a large enoughSandF. For i∈[k+1], letgi:=∑mj=1ci,jfj.

Letδ := (k+1)(maxi∈[m]deg(fi))k/|S|. Our algorithm is immediate once we prove the following claim.

Theorem 4.6(Random reduction). It holds, with probability≥(1−δ), that

(1) the transcendence degree ofF(g1, . . . ,gk+1)/Fequals k, and

(2) the constant term of every annihilator for g1, . . . ,gk+1is zero iff the constant term of every annihilator for f1, . . . ,fmis zero.

First, we reformulate the two items ofTheorem 4.6in a geometric way, and later we will analyze the error probability.

Ford∈_N, denote byAd (and byPd) thed-dimensional affine space (projective space, resp.) over A:=F. Let f :An→Am(resp.g:An→Ak+1) be the polynomial map sendingxto(f1(x), . . . ,fm(x)) (resp.(g₁(x), . . . ,gk+1(x))). LetOandO0be the origin ofAmand that ofAk+1respectively. Define the affine varietiesV:=Im(f)⊆_Am_andV0_:=Im(g)⊆

Ak+1. Then dimV =trdeg(f) =k.

Letπ :Am→Ak+1be the linear map sending(x1, . . . ,xm) to(y1, . . . ,yk+1)whereyi=∑mj=1ci,jxj. Theng=π◦f andV0=π(V).4 Now (1) ofTheorem 4.6is equivalent to dimV0=k, and (2) is equivalent

toO0∈V0iffO∈V.

An V =Im(f) Am

V0=Im(g) Ak+1 f

g

⊆

π|V π

⊆

We will give sufficient conditions of (1) and (2) in terms of incidence properties. Note thatO∈V implies

O0∈V0, sinceπ(O) =O0. Now supposeO6∈V. LetW :=π−1(O0), which is a linear subspace ofAm. ThenO06∈π(V)iffV∩W=/0. However,V∩W=/0 does not implyO06∈V0, asV may “get infinitesimally

close toW” without actually meetingW, so thatO0∈π(V) =V0. This is shown by the following example.

4_{To seeV}0_⊇

Example 3. Let m=4 and(f1,f2,f3,f4) = (X1,X2,X1X2−1,X1+X2). Then k:=trdeg(f) =2. Let

(g1,g2,g3) = (f1,f3,f1+f2−f4) = (X1,X1X2−1,0). SupposeAmhas coordinatesY1, . . . ,Y4andAk+1

has coordinatesZ1, . . . ,Z3.

ThenV⊆_Am_{is defined byY}

1Y2−Y3−1=0 andY1+Y2−Y4=0, andWis defined byY1=0,Y3=0,

andY2−Y4=0. SoV∩W =/0. ButV0⊆Ak+1is the planeZ3=0, which contains the origin.

To overcome the above problem, we consider projective geometry instead of affine geometry. Suppose Amhave coordinatesX1, . . . ,XmandPmhave homogeneous coordinatesX0, . . . ,Xm. RegardAmas a dense open subset ofPmvia(x1, . . . ,xm)7→(1,x1, . . . ,xm). ThenH:=Pm\Am ∼=Pm−1is thehyperplane at infinity, defined byX₀=0. Denote byVc(andWc) theprojective closureofV (W, resp.) inPm. Then V=Vc∩Am. LetWH:=Wc∩H, which is a projective subspace ofH.

For distinct pointsP,Q∈_Pm_{, writePQfor the projective line passing through them. The following} two lemmas provide sufficient conditions for dimV0=kandO06∈V0, respectively.

Lemma 4.7. If Vc∩WH=/0, thendimV0=k.

Proof. Assume dimV0<k. ChooseP∈π(V). The dimension ofπ−1(P)∩V is at least dimV−dimV0 ≥

1 [25, Theorem 11.12]. Denote byY andZ the projective closure ofπ−1(P) and that ofπ−1(P)∩V

inPm, respectively. Then Z⊆Y∩Vc. As dimZ=dimπ−1(P)∩V ≥1 and dimH=m−1, we have Z∩H6= /0 [25, Proposition 11.4].

Asπis a linear map,π−1(P) =Y∩Amis a translate ofπ−1(O0) =W=Wc∩Am. It is well known that two projective subspacesW₁,W26⊆Hhave the same intersection withHiffW1∩AmandW2∩Amare translates of each other.5 So,Y∩H=Wc∩H=WH. Therefore,Vc∩WH =Vc∩Y∩H ⊇Z∩H 6=/0.

Lemma 4.8. If Vc∩Wc=/0, then O06∈V0.

Proof. Assume to the contrary thatVc∩Wc=/0 butO0∈V0. We will derive a contradiction. AsWH⊆Wc, we haveVc∩WH=/0 and hence dimV0=kbyLemma 4.7.

Denote byJ(Vc,WH)thejoinofVcandWH, which is defined to be the union of the projective linesPQ, whereP∈VcandQ∈WH. It is known thatJ(Vc,WH), as the join of twodisjointprojective subvarieties, is again a projective subvariety ofPm[25, Example 6.17].

ForP∈V, denote byWPthe unique translate ofW containingP. We need the following two claims.

Claim 4.9. J(Vc,WH)∩Am =SP∈VWP.

Proof ofClaim 4.9. ConsiderP∈Vc andQ∈WH. IfP∈H, the linePQlies inHand does not meet

Am. Now supposeP∈Vc\H=V. ThenPQmeetsOQat the pointQ. So PQ∩Am is a translate of OQ∩_Am _⊆W

c∩Am=W. It follows thatPQ∩Am⊆WP. This provesJ(Vc,WH)∩Am⊆SP∈VWP. Conversely, letP∈V. Let`Pbe an affine line contained inWPand passing throughP(note thatWP is the union of such lines). Then`P is a translate of an affine line`⊆W. As`P and`are translates of each other, their projective closures intersectHat the same pointQ. We haveQ∈`∩H⊆WH. So `P=PQ∩Am⊆J(Vc,WH)∩Am.

5_Indeed,Wi∩

Claim 4.10. J(Vc,WH)∩Am=π−1(V0).

Proof ofClaim 4.10. Asπ is a linear map,Claim 4.9impliesJ(Vc,WH)∩Am⊆π−1(V0). We prove the other direction by comparing dimensions. It is known that for twodisjointprojective subvarietiesV1and V₂, dimJ(V₁,V2) =dimV1+dimV2+1 [25, Proposition 11.37 and Exercise 11.38]. Therefore,

dimJ(Vc,WH) =dimVc+dimWH+1=dimV+dimW=k+dimW.

So, dimJ(Vc,WH)∩Am =k+dimW. On the other hand, we haveπ−1(V0)∼=V0×W. So dimπ−1(V0) =

dimV0+dimW=k+dimW. NowJ(Vc,WH)∩Amandπ−1(V0)are (irreducible) affine varieties of the

same dimension, and one is contained in the other. So they must be equal. This proves the claim.

Soπ−1(V0) =S

P∈VWPbyClaim 4.9andClaim 4.10. AsO0∈V0, we haveW=π−1(O0)⊆π−1(V0) =

S

P∈VWP. SoWP=Wfor someP∈V, sinceW is a linear space. But thenP∈V∩WP=V∩W⊆Vc∩Wc, contradicting the assumptionVc∩Wc=/0.

Remark 4.11. The converse ofLemma 4.8is false, as shown by the following example.

Example 4. ConsiderExample 3but choose f4 to beX1+X2+1 instead ofX1+X2. Now we have g3=1,V is defined byY1Y2−Y3−1=0 andY1+Y2−Y4+1=0, andV0is the planeZ3=1. SoO06∈V0.

On the other hand, supposePmhas coordinatesY0, . . . ,Y4. ThenVc∩His defined byY0=Y1Y2= Y1+Y2−Y4=0, andWHis defined byY0=Y1=Y2−Y4=Y3=0. So(0,0,1,0,1)∈Vc∩WH⊆Vc∩Wc.

Error probability.It remains to bound the probability of failure of the conditionsVc∩WH= /0 and (in the caseO6∈V)Vc∩Wc=/0. We need the following lemma.

Lemma 4.12(Cut by hyperplanes). Let V⊆_Pm_{be a projective variety of dimension r and degree d. Let} r0≥r+1. Choose ci,j∈S at random, for i∈[r0]and0≤ j≤m. Let W ⊆Pmbe the projective subspace cut out by the equations∑mj=0ci,jXj=0, i=1, . . . ,r0, where X0, . . . ,Xmare homogeneous coordinates of

Pm. Then V∩W=/0holds with probability at least1−(r+1)d/|S|.

Proof. For i∈[r0], let Hi ⊆Pm be the hyperplane defined by ∑mj=0ci,jXj =0. By ignoring Hi for i>r+1, we may assumer0=r+1. LetV₀:=V andVi:=Vi−1∩Hifori∈[r0]. It suffices to show that dimVi=dimVi−1−1 holds with probability at least 1−d/|S|, for eachi∈[r0](the dimension of the

empty set is−1 by convention).

Fixi∈[r0]andci0,j, fori0∈[i−1]and 0≤ j≤m. SoVi−1 is also fixed. Note thatVi−16= /0 since

taking a hyperplane section reduces the dimension by at most one. If dimVi 6=dimVi−1−1, then

dimVi=dimVi−1, andHi contains some irreducible component ofVi−1[25, Exercise 11.6]. LetY be

an irreducible component ofVi−1, and fix a pointP∈Y. ThenY ⊆Hionly ifP∈Hi, which holds only ifci,0, . . . ,ci,msatisfy a nonzero linear equation determined byP. This occurs with probability at most 1/|S|(e. g., by fixing all but oneci,j). We also have deg(Vi−1)≤deg(V)≤d, and hence the number

Proof ofTheorem 4.6. As mentioned above,Theorem 4.6is equivalent to showing that, with probability at least 1−δ: (1) dimV0=k, and (2)O0∈V0iffO∈V. Note thatWcis cut out inPmby the linear equations

∑mj=1ci,jXj=0,i=1, . . . ,k+1. SoWHis cut out inH∼=Pm−1(corresponding toX0=0) by the linear

equations∑mj=1ci,jXj=0,i=1, . . . ,k+1. We also have deg(Vc∩H)≤deg(Vc)≤(maxi∈[m]deg(fi))k (see, e. g., [12, Theorem 8.48]).

AssumeO∈V. ThenO0∈V0 sinceπ(O) =O0. ApplyingLemma 4.12to each of the irreducible

components ofVc∩H andWH, as subvarieties of H∼=Pm−1, we seeVc∩WH = (Vc∩H)∩WH = /0 holds with probability at least 1−kdeg(Vc∩H)/|S| ≥1−δ. So byLemma 4.7, dimV0=kholds with probability at least 1−δ.

Now assumeO6∈V. LetπO,H:Vc→H be theprojection of Vcfrom O to H, defined byP7→OP∩H forP∈Vc. It is well defined since O6∈Vc. The imageπO,H(Vc)is a projective subvariety ofH [25, Theorem 3.5]. IfVc∩Wc contains a point P, then πO,H(Vc)∩WH containsπO,H(P). Conversely, if πO,H(Vc)∩WH contains a pointQ, then there existsP∈Vc such thatQ=πO,H(P), and we haveP∈ OQ⊆Wc. We conclude thatπO,H(Vc)∩WH=/0 iffVc∩Wc= /0, which impliesVc∩WH=/0.

Note that dimπO,H(Vc) =dimVc=k, since

πO,H(Vc) =J({O},Vc)∩H.

We also have deg(πO,H(Vc))≤deg(Vc)[25, Example 18.16]. ApplyingLemma 4.12toπO,H(Vc)andWH, as subvarieties ofH∼=Pm−1, we seeπO,H(Vc)∩WH=/0 holds with probability at least

1−(k+1)deg(πO,H(Vc))

|S| ≥1−δ.

We have shown above thatπO,H(Vc)∩WH= /0 implyVc∩WH=/0 andVc∩Wc=/0. ByLemma 4.7 andLemma 4.8,Vc∩WH=/0 andVc∩Wc=/0 imply dimV0=kandO06∈V0, respectively. Therefore, it holds with probability at least 1−δ that dimV0=kandO06∈V0.

Proof ofTheorem 1.3. AnnAtZero is known to beNP-hard [30]. TheNP-hardness of APS follows from

Lemma 4.1andTheorem 4.2.

Given an instance f of APS, we can first find k=trdeg(f). Fix a set S ⊆_A to be larger than 2(k+1)(maxi∈[m]deg(fi))k (which can be scanned using only polynomial space). Consider the points

((ci,j |i∈[k+1],j∈[m]))∈S(k+1)×m; for each such point defineg:=

gi:=∑mj=1ci,jfj |i∈[k+1] . Compute the transcendence degree ofg, and if it iskthen solve AnnAtZero for the instanceg. Output NO iff somegfailed the AnnAtZero test.

All these steps can be achieved in space polynomial in the input size, using the uniqueness of the annihilator forg[39, Theorem 47], Perron’s degree bound [49] and linear algebra [15,9,43].

5

Hitting set for

VP

: Proof of

Theorem 1.4

Approximative closure ofVP.[10] A family(fn|n) of polynomials fromA[x]is in theclass VPAif there are polynomials fn,iand a functiont:N7→Nsuch thatgnhas a algebraic circuit of size poly(n)and degree poly(n), over the fieldA(ε), computinggn(x) = fn(x) +εfn,1(x) +ε2fn,2(x) +· · ·+εt(n)fn,t(n)(x). That is,gn≡ fn modεA[ε][x].

The smallest possible circuit size ofgnis called theapproximative complexityof fn, namely size(fn). It may happen thatgnis much easier than fnin terms of traditional circuit complexity. That possibility makes the definition interesting and opens up a long line of research.

Hitting set forVP_A Given functions s=s(n) and r=r(n), a finite set H ⊆_An _{is called a hitting} setfor degree-r polynomials of approximative complexitys, if for every such nonzero polynomial f: ∃v∈H,f(v)6=0.

Explicitness.Heintz and Schnorr [27] proved that poly(n,s)-sized hitting sets exist aplenty (for degree-r

size-spolynomials) as follows.

Lemma 5.1([27, Theorem 4.4]). There exists a hitting setH⊆_Fn

qof size O(s2n2)(assuming q≥Ω(sr2)) that hits all nonzero degree-r n-variate polynomials inA[x]that can be infinitesimally approximated by size-s algebraic circuits.

The ultimate goal is computing such a hitting set in time poly(n,s,logqr). Before our work, the best result known wasEXPSPACE[44,45].

Note that for the hitting-set design problem it suffices to focus only on homogeneous polynomials. They are known to be computable by homogeneous circuits, where each gate computes a homogeneous polynomial [58].

Universal circuit.It can simulate any circuit of sizescomputing a degree-rhomogeneous polynomial inA(ε)[x1, . . . ,xn]. We define theuniversal circuitΨ(y,x)as a circuit innessential variablesxand s0:=O(sr4)auxiliary variablesy. The variablesyare the ones that one can specialize inA(ε), to compute

a specific polynomial inA(ε)[x1, . . . ,xn]. Every specialization gives a homogeneous degree-r size-s0 polynomial. Moreover, the set of these polynomials is closed under constant multiples [21, Theorem 2.2].

Note that by [27] there is a hitting set, withm:=O(s02n2)points inFnq(∵q≥Ω(s0r2)), for the setP of the polynomials infinitesimally approximated by the specializations ofΨ(y,x). A universal circuit construction can be found in [51,58]. Using the above notation, we give a criterion to decide whether a candidate set is a hitting set.

Theorem 5.2(hitting-set criterion). SetH=:{v₁, . . . ,vm} ⊆Fnqisnota hitting set for the familyPof the polynomials infinitesimally approximated by the specializations ofΨ(y,x)iff there is a satisfying assignment(α,β)∈A(ε)s

0

×_A(ε)nsuch that:

(1)∀i∈[n],βir+1−1 ∈εA[ε], where r is the degree of the specializations ofΨ(y,x), (2)Ψ(α,β)−1∈εA[ε], and

(3)∀i∈[m],Ψ(α,vi) ∈εA[ε].

Remark 5.3. The above criterion holds for algebraically closed fieldsAofanycharacteristic. Thus, it reduces the hitting set verification problems for the familyPover these fields to APS as well.

Proof. First we show that∃x∈A(ε),xr+1−1 ∈εA[ε]impliesx∈A[[ε]]∩A(ε)(= rational functions

Claim 5.4. ∃x∈_A(ε),xr+1−1 ∈εA[ε]implies x∈Zr+1+εA[[ε]], where Zr+1is the set of(r+1)-th roots of unity inA.

Proof. Recall the formal power seriesA[[ε]]and its group of unitsA[[ε]]∗. Note that for any polynomial

a=

_∑

i0≤i≤d aiεi

!

withai06=0, the inverse

a−1=ε−i0·

_∑

aiεi−i0

!−1

is inε−i0_·A[[ε]]∗_{. This is just a consequence of the identity(1−ε)}−1₌

∑i≥0εi. In other words, any

rational functiona∈_A(ε)can be written as an element inε−iA[[ε]]∗, for somei≥0. Thus, writexas ε−i·(b0+b1ε+· · ·)fori≥0 andb0∈A∗. This gives

xr+1−1 =ε−i(r+1)(b0+b1ε+b2ε2+· · ·)r+1 − 1.

For this to be inεA[ε], clearlyihas to be 0 (otherwise,ε−i(r+1)remains uncancelled), implying that x∈_A[[ε]].

Moreover, we deduce thatbr₀+1−1=0. Thus, Condition (1) implies thatb0is one of the(r+1)-th

roots of unityZr+1⊆A(recall that, sincep-(r+1),|Zr+1|=r+1). Thus,x∈Zr+1+εA[[ε]].

[⇒]: Suppose H is not a hitting set for P. Then, there is a specialization α ∈A(ε)s0 of the universal circuit such thatΨ(α,x)computes a polynomial inA[ε][x]\εA[ε][x], but still “fools”H, i. e.,

∀i∈[m],Ψ(α,vi) ∈εA[ε]. What remains to be shown is that Conditions (1) and (2) can be satisfied too.

Consider the polynomialg(x):=Ψ(α,x)

ε=0. It is a nonzero polynomial, inA[x]of degreer, that

“fools”H. ByLemma 2.2, there is aβ ∈Z_rn₊₁such thata:=g(β)is inA∗. Clearly,β_ir+1−1=0, for

alli. Considerψ0:=a−1·Ψ(α,x). Note thatψ0(β)−1∈εA[ε], andψ0(vi) ∈εA[ε]for alli. Moreover, the normalized polynomialψ0(x)can easily be obtained from the universal circuitΨby changing one

of the coordinates ofα (e. g., the incoming wires of the root of the circuit). This means that the three

conditions (1) – (3) can be simultaneously satisfied by (some)(α0,β)∈A(ε)s 0

×Z_rn₊₁.

[⇐]: Suppose the satisfying assignment is (α,β0) ∈A(ε)s 0

×A(ε)n. As shown in Claim 5.4,

Condition (1) impliesβ_i0∈ Zr+1+εA[[ε]]for alli∈[n]. Let us defineβi:=βi0

ε=0, for alli∈[n]; they

are inZr+1⊆A. By Condition (3),∀i∈[m],Ψ(α,vi) ∈εA[ε].

Since any rational functiona∈A(ε)can be written as an element inε−iA[[ε]]∗, for somei≥0, we

get thatΨ(α,x)is inε−jA[[ε]][x], for some j≥0. Expand the polynomialΨ(α,x), w.r.t.ε, as

g−j(x)ε−j+· · ·+ε−2g−2(x) +g−1(x)ε−1+g0(x) +εg1(x) +ε2g2(x) +. . . .

Let us study Condition (2). If for each 0≤`≤ j, polynomialg−`(x)is zero, then Ψ(α,β0)

ε=0=0

contradicting the condition. Thus, we can pick the largest 0≤`≤ jsuch that the polynomialg−`(x)6=0.

Note that the normalized circuitε`·Ψ(α,x)equalsg−`atε=0. This means thatg−`∈P, and it is a

Proof ofTheorem 1.4. Given a prime pand parametersn,r,sin unary (w.l.o.g.p_-(r+1)), fix a fieldFq withq≥Ω(sr6). Fix the universal circuitΨ(y,x)withnessential variablesxands0:=Ω(sr4)auxiliary variablesy. Fixm:=Ω(s02n2).

For every subsetH=:{v₁, . . . ,vm} ⊆Fnqsolve the APS instance described by Conditions (1) – (3) in Theorem 5.2. These are(n+m+1)algebraic circuits of degree poly(srn,logp)and a similar bit size. Using the algorithm fromTheorem 1.3it can be solved in space poly(srn,logp).

The number of subsetsHis at mostqnm. So we can go over all of them in space poly(nmlogq). If APS fails on one of them (sayH) then we know thatHis a hitting set forP. SinceΨis universal, for

homogeneous degree-rsize-spolynomials inA[x], we outputHas the desired hitting set.

Remark 5.5. One advantage of our method compared to the one in [21] is that in our construction, the bit complexity of the coordinates of the points in the hitting set isO(logrs), whereas the bit complexity of those in [21] is poly(n,s,r).

6

Conclusion

Our result that algebraic dependence testing is in AM∩coAM gives some hope that a randomized polynomial-time algorithm for the problem might exist. Studying the following special case might be helpful to get an idea for designing better algorithms.

Given quadratic polynomials f₁, . . . ,fn∈F2[x1, . . . ,xn], test if they are algebraically dependent in randomized polynomial time [47].

In addition, Ilya Volkovich [59] asked whether AD(F)∈SBP∩coSBPis true, and whether our method can be used to prove this stronger result. Here SBP, introduced in [8], stands for “small bounded-error probability” and is a subclass ofAM.

As indicated in this paper, approximate polynomials satisfiability, or equivalently testing zero-membership in the Zariski closure of the image, may have further applications to problems in computa-tional algebraic geometry and algebraic complexity.

We know that HN is inAMover fields of characteristic zero, assuming GRH [32]. Can we prove that AnnAtZero (or APS) is also inAMover fields of characteristic zero assuming GRH [30]? This would also imply a better hitting set construction forVP.

Acknowledgements. We thank Anurag Pandey and Sumanta Ghosh for insightful discussions on the approximate polynomials satisfiability and explicit construction of hitting set for the closure ofVP. We are also grateful to the anonymous reviewers for their careful reading of the manuscript and for their many helpful comments. We thank the editor-in-chief Laci Babai for his meticulous suggestions regarding grammar and style. Finally, N.S. thanks the funding support from DST (DST/SJF/MSA-01/2013-14), and Z.G. thanks the funding support from DST and Research I Foundation of CSE, IITK.

References

[2] MANINDRAAGRAWAL, SUMANTAGHOSH, ANDNITIN SAXENA: Bootstrapping variables in algebraic circuits. Proc. Nat. Acad. of Sciences (USA), 116(17):8107–8118, 2019. Preliminary version inSTOC’18. [doi:10.1073/pnas.1901272116] 4

[3] MANINDRAAGRAWAL, CHANDANSAHA, RAMPRASADSAPTHARISHI,ANDNITINSAXENA: Jacobian hits circuits: Hitting sets, lower bounds for depth-Doccur-kformulas and depth-3 tran-scendence degree-kcircuits. SIAM J. Comput., 45(4):1533–1562, 2016. Preliminary version in

STOC’12. [doi:10.1137/130910725,arXiv:1111.0582] 4

[4] SANJEEVARORA ANDBOAZBARAK:Computational Complexity: A Modern Approach. Cam-bridge Univ. Press, 2009. [doi:10.1017/CBO9780511804090] 3,5,10,11

[5] VIKRAMAN ARVIND, PUSHKAR S. JOGLEKAR, PARTHA MUKHOPADHYAY, AND S. RAJA: Randomized polynomial-time identity testing for noncommutative circuits. Theory of Computing, 15(7):1–36, 2019. Preliminary version inSTOC’17. [doi:10.4086/toc.2019.v015a007] 3

[6] LÁSZLÓBABAI: Trading group theory for randomness. InProc. 17th STOC, pp. 421–429. ACM Press, 1985. [doi:10.1145/22145.22192] 10

[7] MALTEBEECKEN, JOHANNESMITTMANN,ANDNITINSAXENA: Algebraic independence and blackbox identity testing.Inform. and Comput., 222:2–19, 2013. Preliminary version inICALP’11. [doi:10.1016/j.ic.2012.10.004,arXiv:1102.2789] 2,4,10

[8] ELMARBÖHLER, CHRISTIANGLASSER,AND DANIELMEISTER: Error-bounded probabilistic computations between MA and AM. J. Comput. System Sci., 72(6):1043–1076, 2006. Preliminary version inMFCS’03. [doi:10.1016/j.jcss.2006.05.001] 24

[9] ALLANBORODIN, JOACHIM VON ZURGATHEM,ANDJOHNHOPCROFT: Fast parallel matrix and GCD computations. Inf. Control, 52(3):241–256, 1982. Preliminary version inFOCS’82. [doi:10.1016/S0019-9958(82)90766-5] 2,18,21

[10] KARLBRINGMANN, CHRISTIANIKENMEYER,ANDJEROENZUIDDAM: On algebraic branching programs of small width. J. ACM, 65(5):32:1–32:29, 2018. Preliminary version in CCC’17. [doi:10.1145/3209663,arXiv:1702.05328] 22

[11] PETERBÜRGISSER: The complexity of factors of multivariate polynomials. Foundations of Compu-tational Mathematics, 4(4):369–396, 2004. Preliminary version inFOCS’01. [ doi:10.1007/s10208-002-0059-5] 5

[12] PETER BÜRGISSER, MICHAEL CLAUSEN, ANDAMINSHOKROLLAHI: Algebraic Complexity Theory. Springer, 1997. [doi:10.1007/978-3-662-03338-8] 6,15,21

[13] PETERBÜRGISSER, ANKITGARG, RAFAELOLIVEIRA, MICHAELWALTER,ANDAVI WIGDER -SON: Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory. InProc. 9th Innovations in Theoretical Computer Science Conf. (ITCS’18), pp. 24:1–24:20. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2018. [doi:10.4230/LIPIcs.ITCS.2018.24,