Special-case Algorithms for Blackbox Radical Membership, Nullstellensatz and Transcendence Degree
aa r X i v : . [ c s . CC ] J un Special-case Algorithms for Blackbox RadicalMembership, Nullstellensatz andTranscendence Degree
Abhibhav GargCSE, Indian Institute of Technology Kanpur [email protected]
Nitin SaxenaCSE, Indian Institute of Technology Kanpur [email protected]
Abstract
Radical membership testing, resp. its special case of Hilbert’s Nullstel-lensatz (HN), is a fundamental computational algebra problem. It is NP-hard; and has a famous PSPACE algorithm due to effective
Nullstellensatzbounds. We identify a useful case of these problems where practical algo-rithms, & improved bounds, could be given— When transcendence degree r of the input polynomials is smaller than the number of variables n . If d is the degree bound on the input polynomials, then we solve radical mem-bership (even if input polynomials are blackboxes ) in around d r time. Theprior best was > d n time (always, d n > d r ). Also, we significantly improveeffective Nullstellensatz degree-bound, when r ≪ n .Structurally, our proof shows that these problems reduce to the case of r + > r . This input instance (cor-responding to none or a unique annihilator ) is at the core of HN’s hardness.Our proof methods invoke basic algebraic-geometry. Given a set of polynomials f , . . . , f n , there is a natural certificate for the ex-istence of a common root, namely the root itself. Hilbert’s Nullstellensatz[Rab30, Zar47, Kru50] states that there is also a natural certificate for the nonex-istence of a common root, when the underlying field is algebraically closed . For-mally, the theorem states that the polynomials have no common root if andonly if there exist polynomials g , . . . , g n such that 1 = P f i g i . We refer tothe latter type of certificate as a Nullstellensatz certificate . These certificates arenot polynomial sized: every common root can have exponential bit complex-1ty, and every set of witness polynomials g i can have exponential degrees. Thisproblem is naturally of computational interest, since the generality of the state-ment affords reductions from many problems of interest. Effective versionsof the Nullstellensatz have been extensively studied [Jel05, KPS +
01, KPS99,Som99, Som97, BS91, Kol88, Bro87], and they allow the decision problem of ex-istence of common roots (called HN) to be solved in polynomial space. Koiran[Koi96] proved that under generalized Riemann Hypothesis, HN can be solvedin AM [AB09, Ch.8], for fields of characteristic zero.In this work, we relate the complexity of HN to the transcendence degreeof the input polynomials. The transcendence degree of polynomials f , . . . , f m is defined as the size of any maximal subset of the polynomials that are al-gebraically independent. This notion is well defined since algebraic indepen-dence satisfies matroid properties [Oxl06]. We show that HN can be solved intime single-exponential in transcendence degree This can be seen as a gener-alization of the fact that HN can be solved in time exponential in the numberof polynomials (or variables) in the system. We state our result in terms of thequestion of radical membership : f ∈ ? p h f , . . . , f m i . Note that the standard al-gorithms for both ideal membership [Her26] and radical computation [Lap06]are far slower than ours. Given a set of polynomials f , . . . , f m with transcendence degree at most r , asblackboxes, we can perform radical membership tests for the ideal generated by f , . . . , f m in time polynomial in d r , m , n , where d is the degree-bound on the polynomials and n is the number of variables. We also relate the transcendence degree of the input polynomials to thedegrees of the Nullstellensatz certificates, that is the degrees of g i in P f i g i =
1; improving the best bounds by [Jel05].
Given a set of polynomials f , . . . , f m with transcendence degree r and without anycommon roots, there exist polynomials g i of degree at most d r + such that P f i g i = . We also give an output-sensitive algorithm to compute the transcendencedegree of polynomials. Slightly more formally, we show:
Given a set of polynomials f , . . . , f m , we can compute their transcendence degreein time polynomial in d r and m , n . All three of the problems stated above have been extensively studied. We there-fore only list some of the previously known results, and direct readers to thesurveys [May97, BS91].
Nullstellensatz.
The decidability of the ideal membership problem was es-tablished by Hermann [Her26] when she proved a doubly-exponential boundon witnesses to ideal membership. A lower bound of the same complexityby Mayr and Meyer [May89, MM82] showed that this problem is EXPSPACEcomplete. A number of different algorithms were developed for operations onideals, most prominently the method of Gröbner basis [Buc65]. The proof ofsingle-exponential bounds for the Nullstellensatz (discussed below) allowed2pecial cases of the ideal membership problem, such as the case of unmixedand zero dimensional ideals to be solved in single-exponential time [DFGS91].It also allowed the general Nullstellensatz problem to be solved in PSPACE.Giusti and Heintz [GH94] proved that the dimension of a variety can be com-puted by a randomized algorithm in single-exponential time, with the expo-nent being linear in n , which gives an algorithm of the same complexity forHN (by testing if the dimension is − C and the polynomials have integer coefficients. His methodis completely different from the previous methods (of using the effective Null-stellensatz to reduce the system to a linear one). The positive characteristic caseis an open problem, and the best known complexity remains PSPACE. Effective Nullstellensatz.
The projective version of the effective Nullstel-lensatz follows from the fundamental theorem of elimination theory [Laz77].An affine version was first proved by Brownawell [Bro87] in characteristic 0 us-ing analytic methods. It was later improved by Kollár [Kol88] who used localcohomology to improve the bounds and remove the condition on the character-istic. A more elementary proof that used bounds on the Hilbert function wasgiven by Sombra [Som97], who also gave improved bounds based on somegeometric properties of related varieties [Som99]. An even more elementaryand significantly shorter proof was given by Jelonek [Jel05], who obtained im-proved bounds when the number of polynomials is lesser than the number ofvariables.
Transcendence degree.
Algebraic independence was studied in computerscience by [DGW07] in their study of explicit extractors. They proved that therank of the Jacobian matrix is the same as the transcendence degree for fieldsof characteristic zero (or large enough) which gives an efficient randomizedmethod for computing the transcendence degree. The problem was studiedfurther in [Kay09], where the condition on the characteristic for the above al-gorithm was relaxed, and some hardness results were established. [GSS18]showed that the problem is in coAM ∩ AM, making it unlikely to be NP-hard,and conjecturing that the problem is in coRP for all characteristics. Algorithmi-cally, the best known method for computing the transcendence degree in fieldsof positive characteristic still has PSPACE complexity, by using the bounds ofPerron [Per51, Pł05] to reduce the problem to solving an exponential sized lin-ear system. This method takes time polynomial in d r using the methods of[Csa75]. We refer the reader to the thesis [Sin19] for an exhaustive survey ofrelated results; and applications in [ASSS12, PSS16].Certain radical membership methods were developed by Gupta [Gup14] inhis work on deterministic polynomial identity testing algorithms for heavilyrestricted depth-four circuits. The focus there however was on a deterministicalgorithm for the above problem. Further, he restricts his attention to systemswhere the underlying field is C . 3 .2 Our results Our algorithms will be Monte Carlo algorithms. We assume that our base field k is algebraically closed, but our algorithms only use operations in the field inwhich the coefficients of the inputs lie, which we denote by k i . For example, k i might be F p , and k would then be F p . By time complexity we mean opera-tions in k i , where operations include arithmetic operations, finding roots, andcomputing GCD of polynomials. Our results are valid for any field where theabove procedures are efficient, for example finite fields.We relate the complexity of radical membership, and the degree bounds ineffective Nullstellensatz, to the transcendence degree of the input set of poly-nomials. We do this by showing that given a system of polynomials, we canreduce both the number of variables and the number of polynomials to onemore than the transcendence degree, while preserving the existence (resp. non-existence) of common roots. In particular, when the transcendence degree ofthe input polynomials is constant, we get efficient algorithms for these prob-lems. Theorem 1.1 (Radical membership) . Suppose f , . . . , f m and g are polynomials, invariables x , . . . , x n , of degrees d , . . . , d m and d g respectively, given as blackboxes.Suppose that trdeg ( f , . . . , f m ) r . Define d := max ( max i d i , d g ) .Then, testing if g belongs to the radical of the ideal generated by f , . . . , f m can bedone in time polynomial in n , m and d r , with randomness. Remarks:(1)
The tr.deg r can be much smaller than n , and this improves the complex-ity significantly to d r from the prior d n [LL91]. On the other hand, the usualreduction from SAT to HN results in a set of polynomials with transcendencedegree n , due to the presence of polynomials x i − x i (that enforce the binary0/1 values). (2) We also show that the tr.deg itself can be computed in time d r , indepen-dent of the characteristic (Theorem 1.3). In the above statement therefore, wecan always pick r = trdeg ( f ) , and we can assume that r is not part of the input. (3) The transcendence degree is upper bounded by the number of polyno-mials, and therefore we generalize the case of few polynomials. It is surprisingif one contrasts this case with that of ideal membership— where the instancewith three polynomials (i.e. transcendence degree = 3) is as hard as the generalinstance making it EXPSPACE-complete. Next, we show that taking constant-free random linear combinations pre-serves the zeroset of the polynomials, if the number of linear combinations isat least one more than the transcendence degree. This allows us to get boundson the Nullstellensatz certificates that depend on the transcendence degree. Suppose g ∈ h f , . . . , f m i is an instance of ideal membership. This is equivalent to z m z m g ∈ D z m + , z m + , P i f i z ii z m − i E . Here, z , z are fresh variables. This reduces the general instance ofideal membership to an instance where the ideal is generated by 3 elements. This transformationis from [Sap19]. heorem 1.2 (Effective Nullstellensatz) . Suppose f , . . . , f m are polynomials in x , . . . , x n , of degrees d > · · · > d m respectively, with an empty zeroset. Supposefurther that trdeg ( f , . . . , f m ) = r .Then, there exist polynomials h i such that deg f i h i Q r + i = d i that satisfy P f i h i = . Remark:
The prior best degree-bound for the case of ‘small’ transcendencedegree is Q mi = d i [Jel05]. Our bound is significantly better when the transcen-dence degree r is ‘smaller’ than the number of polynomials m .Finally, as stated before, we show that the transcendence degree of a givensystem of polynomials can be computed in time polynomial in d r (and m , n ),where d is the maximum degree of the input polynomials, and r is their tran-scendence degree. The algorithm is output-sensitive in the sense that the time-complexity depends on the output number r . Theorem 1.3 (Transcendence degree) . Given as input polynomials f , . . . , f m , invariables x , . . . , x n , of degrees at most d , we can compute the transcendence degree r of the polynomials in time polynomial in d r , n , m . Remark:
In the case when the characteristic of the field is greater than d r ,there is a much more efficient (namely, randomized polynomial time) algo-rithm using the Jacobian criterion [BMS13]. The algorithm presented here isuseful when the characteristic is ‘small’; whereas the previous best knowntime-complexity was > d r if one directly implements the PSPACE algorithm.Eg. for d = O ( ) and r = O ( log n ) our complexity is polynomial-time unlikethe prior known algorithms. A motivating example where our results are better than the known results iswhen the input blackboxes are implicitly of the form f i ( h , . . . , h r ) , i ∈ [ m ] , for r ≪ n , where each h i is an n -variate polynomial, and m = n +
1. Here, f i ’shave transcendence degree r . Thus, our algorithms take time d r ; significantlyless than d n . Pf. idea Theorem 1.1:
We first use the Rabinowitsch trick to reduce to HN: thecase g =
1. Next, we perform a random linear variable-reduction. We showthat replacing each x i with a linear combination of r new variables z j preservesthe existence of roots. This is done by using the fact that a general linear hy-perplane intersects a variety properly (Lemma 3.1). Once we are able to re-duce the variables, we can interpolate to get dense representation of our poly-nomials, and invoke existing results about testing nonemptiness of varieties(Theorem 2.6). Pf. idea Theorem 1.2:
For the second theorem, we show that random linearcombinations of the input polynomials, as long as we take at least r + Pf. idea Theorem 1.3:
The image of the polynomial map defined by the poly-nomials is such that the general fibre has codimension equal to the transcen-dence degree. We first show that a random point, with coordinates from asubset which is not ‘too large’, satisfies this property. In order to efficientlycompute the dimension of this fibre, we take intersections with hyperplanes;and apply Lemma 3.1 and Theorem 2.6.
We reserve n for the number of variables ( x , . . . , x n ), m for the number ofpolynomials ( f , . . . , f m ) in our inputs. The polynomials have total degrees d , . . . , d m . We assume that the polynomials are labeled such that d > d > · · · > d m .We use boldface to denote sequence of objects, when the indexing set isclear; for example, x denotes x , . . . , x n and f denotes f , . . . , f m . The point (
0, . . . , 0 ) will be represented by . We use k to denote the underlying fieldwhich we assume is algebraically closed, and k i to denote the field in whichthe coefficients of the inputs lie. We use A n to denote the n dimensional affinespace over k . Given a variety X , we use k [ X ] to denote its coordinate ring, andwhen X is irreducible we use k ( X ) to denote its function field. We use A n and P n to denote the n dimensional affine and projective spaces respectively, and P n ∞ to denote the hyperplane at infinity. We use elementary facts from algebraic-geometry, for which [CLO07, SR13]are good references. We do not assume that our varieties (or zerosets) are irre-ducible. We will use the
Noether normalization lemma . The following statementis useful, as it characterizes the linear maps which are Noether normalizing.
Theorem 2.1. [SR13, Thm.1.15] If X ⊆ P N is a closed subvariety disjoint from an ℓ -dimensional linear subspace E ⊆ P N then the projection π : X → P N − ℓ − withcentre E defines a finite map X → π ( X ) . Here, by projection with center E we mean that the coordinate functions of themap are the same as a set of defining linear equations for E . By the above theo-6em, proving that a given map is Noether normalizing for a particular varietyreduces to proving that the variety is disjoint from a linear subspace.We will also use the following two statements from dimension theory, namelythe theorem on the dimension of intersections with hypersurfaces, and the the-orem on the dimension of fibres. Theorem 2.2. [SR13, Thm.1.22]
If a form F is not zero on an irreducible projectivevariety X then dim ( X ∩ V ( F )) = dim X − . Theorem 2.3 (Fibre dimension) . [SR13, Thm.1.25] Let f : X → Y be a surjective regular map between irreducible varieties. Then dim Y dim X , and for every y ∈ Y ,the fibre f − ( y ) satisfies dim f − ( y ) > dim X − dim Y (equiv. codim f − ( y ) dim Y ).Further, there is a nonempty open subset U ⊂ Y : for every y ∈ U , dim f − ( y )= dim X − dim Y (equiv. codim f − ( y ) = dim Y ). The above theorem also holds if we replace surjective by dominant . Everyfibre either is empty (if the point is not in the image) or has the above bound onthe dimension. We sketch a proof of a special case of the above in appendix Asince we require an intermediate statement in the proof of Theorem 1.3.We will also require the Bézout inequality. The definition of degree we useis the version more common in computational complexity. The degree of a va-riety is the sum of the degrees of all its irreducible components, as opposedto just the components of highest dimension. For irreducible varieties, the de-gree is the number of points when intersected with a general linear subspaceof complementary dimension. This definition affords the following version ofthe
Bézout inequality [Hei83], which holds without any conditions on the typeof intersection.
Theorem 2.4 (Bézout [Hei83]) . Let X , Y be subvarieties of A n . Then deg ( X ∩ Y ) deg X · deg Y . Following is a recent version of effective Nullstellensatz [Jel05].
Theorem 2.5. [Jel05, Thm.1.1]
Let f , . . . , f m be nonconstant polynomials, from thering k [ x , . . . , x n ] with k algebraically closed, that have no common zeros. Assume deg f i = d i with d > · · · > d m , and also m n . Then, there exist polynomials h i such that deg f i h i Q mi = d i satisfying P f i h i = . We will need the following algorithm for checking if a variety has dimen-sion 0 (dim is an integer in the range [− n ] ). The statement assumes that thepolynomials are given in the monomial (also called dense ) representation. Weonly state the part of the theorem that we require. A discussion is providedin Appendix B. We note that the below theorem itself invokes results from[Laz81], section 8 of which proves that the operations occur in a field exten-sion of degree at most d n of the field k i . Theorem 2.6. [LL91, Part of Thm.1]
Let f , . . . , f m be polynomials of degree at most d in n variables. There exists a randomized algorithm that checks if the dimension of he zeroset of f , . . . , f m is or not, in time polynomial in d n , m . The error-probabilityis − d n . We will also require a bound on the degrees of annihilators of algebraicallydependent polynomials. We refer to this bound as the Perron bound. It alsoplays a crucial role in the new proofs of effective Nullstellensatz (Theorem 2.5).
Theorem 2.7 (Perron bound) . [BMS13, Cor.5] Let f , . . . , f m be algebraically de-pendent polynomials of degrees d , . . . , d m . Then there exists a nonzero polynomial A ( y , . . . , y m ) of degree at most Q mi = d i such that A ( f , . . . , f m ) is identically zero. We note that the theorem statement in [BMS13] has the bound as ( max d i ) m ,however their method of constructing a linear faithful homomorphism andthen applying the bound from [Pł05] actually gives the above mentioned bound(even for the weighted -degree of A ).In the course of our proof, we will study the image of the polynomial mapwhose coordinate functions are f , . . . , f m . We list some properties of this im-age. Lemma 2.8 (Polynomial map) . Let f , . . . , f m be polynomials of degrees at most d ,in variables x , . . . , x n . Set r := trdeg ( f , . . . , f m ) . Let F : A n → A m be a polynomialmap defined as F ( a , . . . , a n ) = ( f ( a , . . . , a n ) , . . . , f m ( a , . . . , a n )) .Let Y be the (Zariski) closure of the image of A n under F , that is Y := F ( A n ) . Then,1. Y is irreducible.2. dim Y = r .3. deg Y d r .Proof of Lemma 2.8. The first statement is a consequence of the fact that Y isthe image of an irreducible set (namely A n ) under a continuous map. Since k [ Y ] = k [ f , . . . , f m ] , we have trdeg ( k ( Y )) = r , whence dim Y = r by defini-tion. Here we used the fact that the dimension of an irreducible variety is thetranscendence degree of its function field over the ground field. A proof of thethird part can be found in [BCS97, 8.48]. We require a bound on the probability that a random linear hyperplane inter-sects a variety of a given dimension properly, that is such that the dimensionof the variety decreases by exactly one. It is well known that the set of suchhyperplanes form a Zariski open set in the space of all hyperplanes. We use anexplicit bound on the probability of such an intersection based on the degree ofthe variety, both for the projective and the affine case. We will require that ourintersecting hyperplanes have some structure: that their defining equationsdepend only on a few variables, depending on the dimension of the variety to8e intersected. We establish all these facts in the next subsection. In the threesubsections following that, we use this lemma to prove our three main results–Theorem 1.1, Theorem 1.2, and Theorem 1.3.
Lemma 3.1.
Let V ⊆ P n be a projective variety of dimension r and degree D . Let S be a finite subset, of the underlying field k , not containing . Let ℓ be a linear form in x , x , . . . , x n − r with each coefficient picked uniformly and independently from S . Let H be the hyperplane defined by ℓ . Then, with probability at least − D/ | S | we have dim V ∩ H = dim V − .Analogously, if V ⊆ A n is affine, ℓ is a linear polynomial in x , . . . , x n − r + and H its hyperplane; then dim V ∩ H = dim V − with probability at least − D/ | S | .Proof of Lemma 3.1. First we prove the projective case. Let ℓ := c x + · · · + c n − r x n − r , where the c i are the coefficients picked uniformly at random from S . Let ∪ dj = V j be the decomposition of the dimension- r part of V into irreduciblecomponents. Then by definition, deg V > P deg V j , and hence d D . Picka point p j in V j , for each j . We can always pick p j so that not all of its first n − r + V j would have tobe contained in the variety defined by x = x = · · · = x n − r =
0, which hasdimension only r −
1. By Theorem 2.2, dim H ∩ dim V j = dim V j if and only if V j ⊆ H (since V j and H are irreducible), and otherwise dim H ∩ V j = dim V j − p j ∈ H . For a fixed j , this is equivalent to ℓ ( p j ) =
0. Since not all of thefirst n − r + p j are zero, the above is bounded by 1 / | S | , byfixing all but one of the coordinates. By a union bound, with probability atmost d/ | S | , there exists some j where dim H ∩ V j = dim V j . Therefore, withprobability at least 1 − D/ | S | , we get dim V j ∩ H = dim V j − j ,whence dim V ∩ H = dim V − V is affine. The difference from the projective case is thatthe intersection V ∩ H might be empty, and we need to bound the probabil-ity of this event. Let V p be its projective closure. Then dim V p = dim V anddeg V p = deg V . By the previous part, we have dim V p ∩ H p = dim V p − − D/ | S | . Then, the case V ∩ H = ∅ only happens if dim V p ∩ H p ∩ P n ∞ = dim V p −
1, where P n ∞ is the hyperplane x = P n . The irreduciblecomponents of V are in bijection with those of V p , and hence V p has no irre-ducible component contained in P n ∞ . Therefore, dim V p ∩ P n ∞ = dim V p − V p ∩ P n ∞ deg V p .Now H p ∩ P n ∞ is a hyperplane in P n ∞ defined by the nonconstant part of ℓ . Inparticular, it is a hyperplane whose defining equation has coefficients pickeduniformly and independently and we can apply the projective version of thislemma on P n ∞ . Therefore the probability that its intersection with V p ∩ P n ∞ doesnot result in a reduction in the dimension is at most D/ | S | . By a union bound,with probability at least 1 − D/ | S | it holds that dim V p ∩ H p = dim V − V p ∩ H p ∩ P n ∞ = dim V −
2, whence dim V ∩ H = dim V − n − r + n − r + Using the above lemma, we complete the proof of the main theorem:
Proof of Theorem 1.1.
We first assume g =
1, which is the Nullstellensatz prob-lem HN. Define D := Q mi = d i , and V := V ( h f i ) . The set of common zeroesof these polynomials is the fibre of the point under the map F defined inLemma 2.8. The problem HN is thus equivalent to testing if a particular fibre ofa polynomial map is nonempty. By the fibre dimension theorem (Theorem 2.3),the codimension of the zeroset—if it is nonempty—is bounded above by thedimension of the image of the map, which by Lemma 2.8 is r . The zeroset V is therefore either empty, or has dimension at least n − r . Assume that V is nonempty. By repeated applications of Bézout’s theorem (Theorem 2.4),deg V D . Let S be a subset of the underlying field k i (or an extension) of sizeat least 6 ( n − r ) D that does not contain 0. We can sample from S in time poly-nomial in d , n , m , since S has size exponential in these parameters. Further,if we were required to go to an extension to form S , the degree of the exten-sion would be polynomial in d , n , m . Pick n − r random linear polynomials ℓ , . . . , ℓ n − r with coefficients from S , and call their zero sets H , . . . , H n − r re-spectively. By Lemma 3.1, the intersection V ∩ H has dimension r − − / ( ( n − r )) . Further, by Bézout’s theorem we getdeg V ∩ H deg V D , since each H i has degree one. Again by Lemma3.1, the intersection ( V ∩ H ) ∩ H has dimension r − − / ( ( n − r )) , and deg V ∩ H ∩ H D . Repeating this for all H i and usingthe union bound, we get dim V ∩ H ∩ · · · ∩ H n − r > / f have nonempty zeroset and are restrictedto the r dimensional affine subspace ∩ H i , the new zeroset has dimension atleast 0, and in particular is nonempty. If the zeroset of the polynomials wasempty to begin with, then the restriction to the linear subspace also results inan empty zeroset.This restriction can be performed by a variable reduction, as follows. Treat-ing A n as a vector space of dimension n over k , let H be the linear subspacecorresponding to the affine subspace H := ∩ H i . H has dimension r , and hencehas basis a , . . . , a r . Further, let vector b be such that H = H + b . Define linearforms c , . . . , c n in new variables z , . . . , z r as c i := P rj = a ji z j + b i , where a ji is the i th component of a j . Define f ′ i := f i ( c , . . . , c n ) . Then by construction,the zeroset of f ′ , . . . , f ′ m is equal to V ∩ ( ∩ H i ) . Further, deg f ′ i = deg f i , and10hese polynomials are in r variables. Also, the construction of these f ′ i can bedone in a blackbox manner, given blackboxes for f i . This construction takestime polynomial in m , r , n .We now repeatedly invoke Theorem 2.6 to check if f ′ i s have a common root.First we must convert them to a sparse representation. The polynomial f ′ i hasat most (cid:0) r + d i r (cid:1) many monomials, and therefore we can find every coefficientin time polynomial in (cid:0) r + d i r (cid:1) by simply solving a linear system. ApplyingTheorem 2.6, we can test whether the dimension of the zeroset of f ′ , . . . , f ′ m is 0 or not. However, we want to check if the dimension is at least 0. For this,we randomly sample r more hyperplanes H ′ , . . . , H ′ r as in the previous partof the proof, this time in the new variables z , . . . , z r . Let V ′ be the zerosetof f ′ , . . . , f ′ m . We first use Theorem 2.6 to check if V ′ has dimension 0. If not,then we check if V ′ ∩ H ′ has dimension 0. If not, then we check V ′ ∩ H ′ ∩ H ′ ,and so on. We return success if any one of the above iterations returns success(implying that the corresponding variety has dimension 0). Performing calcu-lations similar to the ones earlier in the proof, we see that with high probabilityeach intersection reduces the dimension by 1. If V ′ originally had dimension r ′ , then after intersecting with r ′ hyperplanes, the algorithm of Theorem 2.6returns success. If V ′ was empty, then the algorithm does not return successin any of the above iterations. This allows us to decide if V ′ has dimension atleast 0. Finally, using the fact that the dimension of the zeroset of f ′ , . . . , f ′ m isat least 0 if and only if dim V >
0, we get the required algorithm for HN.We now estimate the time taken. Computing the dense representation takestime polynomial in d r and m . Each of the at most r applications of Theorem 2.6also take the same amount of time. The sampling steps take time polynomial inlog nD (in turn polynomial in d , m ) and only requires an extension of degreepolynomial in n and log d . The total time taken is therefore polynomial in m , d r .Now assume that g is an arbitrary polynomial. We reduce the problem tothe case of g = g belongsto the radical of the ideal h f i if and only if the polynomials f , 1 − yg have nocommon root (here y is a new variable). Further, if f have transcendence degree r , then the set f , 1 − yg has transcendence degree r +
1. We therefore reduce theradical membership problem to HN problem, with a constant increase in thetranscendence degree, number of polynomials and the number of variables. Bythe result in the previous paragraph, we can solve this in time polynomial in n , m and d r . We now prove that by taking random linear combinations of the input poly-nomials, we can reduce the number of polynomials to be one more than thetranscendence degree while preserving the existence of roots. This reductiongives degree bounds for the Nullstellensatz certificates. Note that this reduc-tion does not help in Section 3.2’s root-testing procedure, since we will only be11aving a factor in m if we reduce the number of polynomials. Theorem 3.2 (Generator reduction) . Let f , . . . , f m be polynomials, in x , . . . , x n ,of degrees atmost d and of transcendence degree r . Let g , . . . , g r + be polynomialsdefined as g i := P mj = i c ij f j , where each c ij is randomly picked from a finite subset S of k . Then with probability at least − d ( r + ) m / | S | , we have V ( h f i ) = V ( h g i ) . That we pick the linear combinations so that the first involves all polyno-mials, the second involves all except f , the third involves all except f , f andso on is crucial for the improvement in the degree bounds. Proof of Theorem 3.2.
We prove this by studying the set Y defined in Lemma 2.8.Let F : A n → A m be the map with coordinate functions f i . Let Y := F ( A n ) , theclosure of the image of F in A m . We use y , . . . , y m to denote the coordinatefunctions of A m . By Lemma 2.8, Y has dimension r , and degree at most D := d r .Let Y p be the projective closure of Y . Then Y p also has dimension r and degreeat most D . Let ℓ , . . . , ℓ r + be the linear polynomials ℓ i := P i j m c ij y j .Consider the subspace defined by y , ℓ , . . . , ℓ r in P m . The variety Y p ∩ P m ∞ ,which is the intersection of Y p with the hyperplane at infinity defined by y =
0, has dimension r −
1. Since ℓ , . . . , ℓ r are random linear polynomials and Y p ∩ P m ∞ is a variety of, degree at most D and, dimension r −
1, we can repeat-edly apply Lemma 3.1 to get a bound on the probability of proper intersections.Let H i be the hyperplane defined by ℓ i . We apply Lemma 3.1 starting from H r .The equation ℓ r has m − r + D , and dimension decreased by one. We then apply the theoremwith H r − and so on, as in the proof of Theorem 1.1. In each iteration the va-riety considered has one less dimension than the previous iteration, but ourlinear polynomial has one more variable, and therefore we will always satisfythe conditions of Lemma 3.1.We can now invoke Theorem 2.1 to say that the map P m → P r with coor-dinate functions ( y , ℓ , . . . , ℓ r ) is Noether normalizing for Y p . We call this map L ′ . We use z , . . . , z r to denote the coordinate functions of P r . The map L ′ sendsthe affine chart y = z =
0. Let L be the restriction of L ′ tothis affine chart. Then L defines a map from A m to A r , which is Noether nor-malizing for the variety Y ; we also call this restricted map L . More explicitly,the map L has coordinate functions ( ℓ , . . . , ℓ r ) . Also, let the map A m → A r + with coordinate functions ( ℓ , . . . , ℓ r + ) be labelled M .Since the map L is Noether normalizing, it has finite fibres. Let Q be the fi-bre of in Y . We bound the size of this set. The map L is Noether normalizing,and hence it is surjective. The image A r is normal, and hence the cardinal-ity | Q | of the fibre is bounded by the degree of the map [SR13, Theorem 2.28].Here, by the degree of the map we mean the degree of k ( Y ) over the pullback L ∗ ( k ( A r )) . Note that k ( Y ) = k ( f , . . . , f m ) , and L ∗ ( k ( A r )) = k ( ℓ ( f ) , . . . , ℓ r ( f )) after applying the same isomorphism. By Perron’s bound, for each i there ex-ists an annihilator of f i , l ( f ) , . . . , l r ( f ) of degree at most d r + . The degree of theextension, and hence | Q | , is bounded by d m ( r + ) .12urther, no point of Q , other than , has all of the last m − r coordinatesas zero. This follows from the fact that L − ( ) is a linear space of dimension m − r , and its intersection with y r + = y r + = · · · = y m = ℓ r + . For every = q ∈ Q , the probability that ℓ r + ( q ) = / | S | . Therefore, with probability at least 1 − d m ( r + ) / | S | ,the polynomial ℓ r + is nonzero on every nonzero point of Q .Consider the polynomials g , . . . , g r + , and let G be the polynomial map A n → A r + with coordinate functions g i . By the choice of ℓ i in the previousparagraph, the map G is exactly the composition of the map F : A n → A m with M : A m → A r + . Let Q be as defined earlier, the fibre of under L .By construction, the set M − ( ) is a subset of Q . But since the polynomial ℓ r + is nonzero on every nonzero point of Q , the set M − ( ) consists only of .Therefore, F − ( M − ( )) = F − ( ) . Since G = M ◦ F we get G − ( ) = F − ( ) ;which is the same as V ( h f i ) = V ( h g i ) .We use the above to prove our 2nd main result: Proof of Theorem 1.2.
Using Theorem 3.2, there exists polynomials g , . . . , g r + of degrees d , . . . , d r + that do not have a common root. By Theorem 2.5, thereexist h ′ , . . . , h ′ r + such that deg g i h ′ i Q r + i = d i such that P g i h ′ i =
1. Inthis equation, substitute back the linear-combination of f , . . . , f m for each g i ;whence we get the required h i ’s. We give a method of ‘efficiently’ computing the transcendence degree of inputpolynomials f , . . . , f m . By Lemma 2.8 and the second part of Theorem 2.3, thetranscendence degree can be computed if we know the dimension of a gen-eral fibre. We need to get a bound on the points that violate the equality inTheorem 2.3. For this we follow the classical proof of the theorem and give ef-fective bounds wherever required. For convenience we have provided a proofsketch in appendix A, for the special case we need. Lemma 3.3.
Let h , . . . , h m be polynomials of degree at most d in n variables, and let W be the Zariski closure of the image of the map h with coordinates h i . Let S ⊂ k beof size nd n . If a , . . . , a n are randomly picked from S , then with probability at least / , the fibre of ( h ( a ) , · · · , h m ( a )) has codimension exactly dim W .Proof. First assume that the h i are algebraically independent . Then W = A m .Let the input variables be labelled such that x , . . . , x n − m , h , . . . , h m are al-gebraically independent, and let A j ( z , z , . . . , z n − m , w , . . . , w m ) be the (mini-mal) annihilator of x j over this set of variables, that is A j ( x j , x , . . . , x n − m , h , . . . , h m ) =
0. By the proof of Theorem 2.3 (appendix A), a sufficient condition for point a , . . . , a n to be such that h ( a ) has fibre of dimension exactly n − m is that A j ( x j , x , . . . , x n − m , h ( a ) , . . . , h m ( a )) is a nonzero polynomial. The polynomial A j , when treated as polynomials in variables z , . . . , z n − m with coefficients in k [ w , . . . , w m ] are such that the leading monomial has coefficient a polynomial13n w , . . . , w m of weighted-degree at most Q mi = d i (by Perron bound). By thepolynomial identity lemma [Ore22, DL78, Sch80, Zip79], if we pick each a i ran-domly from a set of size 6 Q mi = d i then, with probability at least 5 /
6, none ofthe polynomials A j ( x j , x , . . . , x n − m , h ( a ) , . . . , h m ( a )) is zero. In this case, thecodimension of the fibre of h ( a ) is exactly m as claimed.In the general case, the h i may be algebraically dependent, and W is a sub-variety of A m . Suppose dim W = trdeg ( h ) =: s . Then we take s many randomlinear combinations g i of the h i , as in the proof of Theorem 1.2. The map de-fined by the g i is dense in A s and therefore the g i ( i ∈ [ s ] ) are algebraicallyindependent. By the previous paragraph, point a picked coordinatewise from S is such that the fibre of g ( a ) has codimension s . The fibre of h ( a ) is a sub-set of the fibre of g ( a ) , and therefore it has codimension at least s . Finally, byTheorem 2.3, the fibre has codimension at most s , whence the fibre of h ( a ) hascodim = s as required. Proof of Theorem 1.3.
For each i , upwards from 1 to n , we do the followingsteps. We iterate till i reaches transcendence degree r of the m polynomials. Inthe i -th iteration, we intersect A n with n − i random hyperplanes ℓ , . . . , ℓ n − i ,as in the proof of Theorem 1.1 (Sec.3.2). Here, the coefficients are picked froma set S of size at least n · Q mi = d i . We therefore reduce the problem to i variables.Randomly pick point a where each coordinate (of the n many) is pickedrandomly from S . By Lemma 3.3 (& 3.1), with error-probability / n , thepoint f ( a ) has intersected fibre of dimension ( n − r ) − ( n − i ) = ( i − r ) . Weneed to check this algorithmically; which is done by interpolating the polyno-mials f after hyperplane intersections, and then using Theorem 2.6 (as detailedin Sec.3.2). If the intersected fibre dimension is zero, we have certified tran-scendence degree = i = r ; so we halt and return i as output. Else, we move tothe next i i +
1. The interpolation step above is performed by solving a linearsystem which has size polynomial in d i which is the count of the monomialsof degree at most d in i variables.Note that for i < r , with error-probability / n , the fibre of f ( a ) has anempty intersection with ℓ , . . . , ℓ n − i ; which is dim = − /
6, the above algo-rithm gives the correct answer. For each i , the time complexity of the abovesteps is polynomial in d i , m , which is the time taken for the interpolation stepand to verify zero-dimension of the fibre. Therefore the algorithm as a wholetakes time polynomial in d r , n , m as claimed. We give algorithms for radical membership and transcendence degree of sys-tems of polynomials, in time that depends on the transcendence degree. Inboth cases, our algorithms generalize the cases of ‘few’ input polynomials. We14urther give bounds on the degree of the Nullstellensatz certificates that de-pend on the transcendence degree of the input polynomials. In all three cases,our bounds are significantly better than the previously known results in theregime when the transcendence degree is much smaller than the number ofvariables and the number of polynomials.Our work leaves the natural open problem of designing efficient algorithmswhen the transcendence degree is ‘larger’. • For the blackbox radical membership problem, given the NP-hardness ofHN, it is unlikely that a significantly better algorithm exists (unless otherrestrictions are put on the input polynomials). • Could our methods, and the core hard instance thus identified, help inproving that HN is in AM? Currently, this is known only partially [Koi96]. • For the transcendence degree problem however, we know that the prob-lem is in coAM ∩ AM, making it unlikely to be NP hard. It is thereforelikely that there is an efficient randomized algorithm whose time com-plexity is polynomial in n and m . This is already known in the casewhen the field has large/zero characteristic, and it is an open problemto extend this to other fields. A first step might be to give a subexponen-tial time algorithm for the problem that works without any assumptions. Acknowledgements.
We thank Ramprasad Saptharishi for introducing us tothe universality of 3-generators ideal membership problem. Nitin Saxena thanksthe funding support from DST (DST/SJF/MSA-01/2013-14).
References [AB09] Sanjeev Arora and Boaz Barak.
Computational complexity: a modernapproach . Cambridge University Press, 2009. 2[ASSS12] M. Agrawal, C. Saha, R. Saptharishi, and N. Saxena. Jacobian hitscircuits: Hitting-sets, lower bounds for depth-D occur-k formulas &depth-3 transcendence degree-k circuits. In
Proceedings of the 44thACM Symposium on Theory of Computing (STOC) , pages 599–614, 2012.(SICOMP spl.issue, 45(4), 1533–1562, 2016). 3[BCS97] Peter Bürgisser, Michael Clausen, and Mohammad Amin Shokrollahi.
Algebraic complexity theory , volume 315 of
Grundlehren der mathematis-chen Wissenschaften . Springer, 1997. 8[BMS13] M. Beecken, J. Mittmann, and N. Saxena. Algebraic independenceand blackbox identity testing.
Information and Computation , 222:2 – 19,2013. (Also, 38th International Colloquium on Automata, Languagesand Programming, ICALP 2011). 5, 815Bro87] W. Dale Brownawell. Bounds for the degrees in the Nullstellensatz.
Annals of Mathematics , 126(3):577–591, 1987. 2, 3[BS91] Carlos A. Berenstein and Daniele C. Struppa. Recent improvementsin the complexity of the effective Nullstellensatz.
Linear Algebra andits Applications , 157:203 – 215, 1991. 2[Buc65] B. Buchberger.
Ein Algorithmus zum Auffinden der Basiselemente desRestklassenringes nach einem nulldimensionalen Polynomideal . PhD the-sis, University of Innsbruck, 1965. 2[Can90] John Canny. Generalised characteristic polynomials.
Journal of Sym-bolic Computation , 9(3):241 – 250, 1990. 20[CLO07] David A. Cox, John Little, and Donal O’Shea.
Ideals, Varieties, and Al-gorithms: An Introduction to Computational Algebraic Geometry and Com-mutative Algebra, 3/e (Undergraduate Texts in Mathematics) . Springer-Verlag, Berlin, Heidelberg, 2007. 6[Csa75] L. Csanky. Fast parallel matrix inversion algorithms. , pages 11–12, 1975. 3[DFGS91] Alicia Dickenstein, Noaï Fitchas, Marc Giusti, and Carmen Sessa.The membership problem for unmixed polynomial ideals is solvablein single exponential time.
Discrete Applied Mathematics , 33(1-3):73–94,1991. 3[DGW07] Z. Dvir, A. Gabizon, and A. Wigderson. Extractors and rank ex-tractors for polynomial sources. In , pages 52–62, Oct 2007. 3[DL78] Richard A. Demillo and Richard J. Lipton. A probabilistic remark onalgebraic program testing.
Information Processing Letters , 7(4):193 – 195,1978. 14[GH94] Marc Giusti and Joos Heintz. La détermination des points isolés et dela dimension d’une variété algébrique peut se faire en temps polyno-mial.
Computational Algebraic Geometry and Commutative Algebra , 34, 021994. 3[GSS18] Zeyu Guo, Nitin Saxena, and Amit Sinhababu. Algebraic dependen-cies and pspace algorithms in approximative complexity. In
Proceed-ings of the 33rd Computational Complexity Conference , CCC ’18, 2018. 3[Gup14] Ankit Gupta. Algebraic geometric techniques for depth-4 PIT &Sylvester-Gallai conjectures for varieties.
Electronic Colloquium onComputational Complexity (ECCC) , 21:130, 2014. 316Hei83] Joos Heintz. Definability and fast quantifier elimination in alge-braically closed fields.
Theoretical Computer Science , 24(3):239 – 277,1983. 7[Her26] Grete Hermann. Die Frage der endlich vielen Schritte in der Theorieder Polynomideale.
Mathematische Annalen , 95(1):736–788, Dec 1926.2[Jel05] Zbigniew Jelonek. On the effective Nullstellensatz.
Inventiones mathe-maticae , 162(1):1–17, Oct 2005. 2, 3, 5, 7[Kay09] N. Kayal. The complexity of the annihilating polynomial. In , pages 184–193,July 2009. 3[Koi96] Pascal Koiran. Hilbert’s Nullstellensatz is in the polynomial hierarchy.
J. Complexity , 12(4):273–286, 1996. 2, 3, 15[Kol88] János Kollár. Sharp effective Nullstellensatz.
Journal of the AmericanMathematical Society , 1(4):963–975, 1988. 2, 3[KPS99] Teresa Krick, Luis Miguel Pardo, and Martín Sombra. ArithmeticNullstellensätze.
ACM SIGSAM Bulletin , 33(3):17, 1999. 2[KPS +
01] Teresa Krick, Luis Miguel Pardo, Martín Sombra, et al. Sharp es-timates for the arithmetic Nullstellensatz.
Duke Mathematical Journal ,109(3):521–598, 2001. 2[Kru50] Wolfgang Krull. Jacobsonsches Radikal und Hilbertscher Nullstellen-satz. In
Proceedings of the International Congress of Mathematicians, Cam-bridge, Mass , volume 2, pages 56–64, 1950. 1[Lap06] Santiago Laplagne. An algorithm for the computation of the radicalof an ideal. In
Proceedings of the International Symposium on Symbolicand Algebraic Computation , pages 191–195, 2006. 2[Laz77] Daniel Lazard. Algèbre linéaire sur k [ x , . . . , x n ] et élimination. Bul-letin de la Société Mathématique de France , 105:165–190, 1977. 3[Laz81] Daniel Lazard. Résolution des Systèmes d’Équations algébriques.
Theoretical Computer Science , 15(1):77 – 110, 1981. 7, 19, 20[LL91] Y. N. Lakshman and Daniel Lazard.
On the Complexity of Zero-dimensional Algebraic Systems , pages 217–225. Birkhäuser Boston, 1991.4, 7[Mac02] Francis Sowerby Macaulay. Some formulae in elimination.
Proceed-ings of the London Mathematical Society , 1(1):3–27, 1902. 1917May89] Ernst Mayr. Membership in polynomial ideals over Q is exponentialspace complete. In B. Monien and R. Cori, editors,
STACS 89 , pages400–406, Berlin, Heidelberg, 1989. Springer Berlin Heidelberg. 2[May97] Ernst W. Mayr. Some complexity results for polynomial ideals.
J.Complexity , 13(3):303–325, 1997. 2[MM82] Ernst W Mayr and Albert R Meyer. The complexity of the word prob-lems for commutative semigroups and polynomial ideals.
Advances inMathematics , 46(3):305 – 329, 1982. 2[Ore22] Øystein Ore. Über höhere kongruenzen.
Norsk Mat. Forenings Skrifter ,1(7):15, 1922. 14[Oxl06] James G. Oxley.
Matroid Theory (Oxford Graduate Texts in Mathematics) .Oxford University Press, Inc., USA, 2006. 2[Per51] O. Perron.
Algebra: Die Grundlagen . Number v. 1 in GöschensLehrbücherei : 1. Gruppe, Reine u. angewandte Mathematik. Walterde Gruyter & Company, 1951. 3[PSS16] Anurag Pandey, Nitin Saxena, and Amit Sinhababu. Algebraic inde-pendence over positive characteristic: New criterion and applicationsto locally low algebraic rank circuits. In , pages 74:1–74:15, 2016. (Comput.Compl.,27(4), 617–670, 2018). 3[Pł05] Arkadiusz Płoski. Algebraic dependence of polynomials afterO.Perron and some applications.
Computational Commutative and Non-Commutative Algebraic Geometry , pages 167–173, 2005. 3, 8[Rab30] JL Rabinowitsch. Zum Hilbertschen Nullstellensatz.
MathematischeAnnalen , 102(1):520–520, 1930. 1, 11[Sap19] Ramprasad Saptharishi. Private Communication, 2019. 4[Sch80] J. T. Schwartz. Fast probabilistic algorithms for verification of polyno-mial identities.
J. ACM , 27(4):701–717, October 1980. 14[Sin19] Amit Kumar Sinhababu.
Power series in complexity: Algebraic Depen-dence, Factor Conjecture and Hitting Set for Closure of VP . PhD thesis,Indian Institute of Technology Kanpur, 2019. 3[Som97] Martín Sombra. Bounds for the Hilbert function of polynomial idealsand for the degrees in the Nullstellensatz.
Journal of Pure and AppliedAlgebra , 117-118:565 – 599, 1997. 2, 3[Som99] Martín Sombra. A sparse effective Nullstellensatz.
Advances in Ap-plied Mathematics , 22(2):271 – 295, 1999. 2, 318SR13] I.R. Shafarevich and M. Reid.
Basic Algebraic Geometry 1: Varieties inProjective Space . SpringerLink : Bücher. Springer Berlin Heidelberg,2013. 6, 7, 12, 19[Zar47] Oscar Zariski. A new proof of Hilbert’s Nullstellensatz.
Bulletin of theAmerican Mathematical Society , 53(4):362–368, 1947. 1[Zip79] Richard Zippel. Probabilistic algorithms for sparse polynomials. In
Proceedings of the International Symposiumon on Symbolic and AlgebraicComputation , EUROSAM ’79, page 216–226, Berlin, Heidelberg, 1979.Springer-Verlag. 14
A Effective proof of fibre dimension theorem for adominant map
In this section, we sketch a proof of Theorem 2.3 and show that the conditionon A j in the proof of Theorem 1.3 is indeed sufficient. We follow the proofin [SR13]. We only sketch the proof in the case when Y = A M and X = A N ;however we do not need ‘surjectivity’ and only assume a dominant f .Let ( b , . . . , b M ) ∈ A M be a point in the image of f . This point is defined bythe equations y − b , . . . , y M − b M . If f , . . . , f M are the coordinate functions of f , then the fibre f − ( b ) is defined by the equations f i − b in A N . The nonemptyfibre is defined therefore by M equations; thus by Theorem 2.2, every compo-nent has dimension at least N − M .Now we prove the second part. Since the map f is dominant, the inducedmap f ∗ : k [ Y ] → k [ X ] is injective, and we identify k ( Y ) with its image in k ( X ) via f ∗ . Then by dimension definition, k ( X ) has transcendence degree N − M over k ( Y ) . Let y , . . . , y M ∈ k ( Y ) be the generators of k ( Y ) over k and let x , . . . , x N be the generators of k ( X ) over k . Let them be numbered so that x , . . . , x N − M , y , . . . , y M are algebraically independent, and let A j be the an-nihilator of x j over this set. The fibre f − ( b ) of a point has coordinate ringgenerated by the restrictions of x , . . . , x N to the fibre. Suppose the point b is such that the annihilators A j are nonzero polynomials when we substitute b for y . Then A j continue to be annihilators of the restrictions of x j over therestrictions of x , . . . , x N − M . This shows that the fibre has dimension at most N − M , and combining this with the previous part, the fibres have dimensionexactly N − M .Therefore, the sufficient condition for the fibres to have this dimension isthat all the A j remain nonzero polynomials when evaluated at the given point b . This completes our proof sketch. B Tools for zero-dimension testing
We briefly discuss Theorem 2.6. The two main tools used here are Lazard’salgorithm [Laz81] for computing a multiple of the U-resultant [Mac02], and19anny’s [Can90] study of this U-resultant.Given input polynomials f , . . . , f m , first the polynomials are reduced to n + n = m . The polynomials are then deformed and homogenized: set h i to be the homogenization of the polynomial f i + sx deg f i i . Here s is a new variable. This is due to [Can90], and is usefulbecause it gives control over the newly introduced roots at the hyperplane atinfinity due to homogenization.The next step is to use the algorithm of [Laz81] to compute matrices M , . . . , M n (with each entry a polynomial in s ) that are such that the determinant of P u i M i is the U-resultant up to a multiple. To compute these matrices, first the matricesof multiplication by x i considered as a map between the homogeneous degree D − D := P d i − n components of the coordinate ring of the ze-roset is considered. A simultaneous base change is performed on these matri-ces, and then a subset of the columns of each matrix is returned as M , . . . , M n .The computation of these matrices is the most expensive step in the algorithm.Since the number of n -variate monomials of degree D is d O ( n ) , the algorithmtakes time just polynomial in d n , m .The coefficient of the lowest degree term (in s ) in the above determinant isa product of linear forms (in u ), whose coefficient correspond to the isolatedzeros (that is, zeroes not part of a higher dimension component) of the originalpolynomial [Can90]. However, computing this coefficient is prohibitively ex-pensive, and therefore some specializations of the U-resultant are computed.It is at this specialization stage that the algorithm decides the zero dimension-ality. First, a change of basis is done to ensure that all the roots of the homoge-nized polynomials have distinct first coordinates. The variables U , . . . , U n arespecialized to ( y , 1, 0, . . . , 0 ) where y is a new variable. The determinant of thematrix P U i M i is now a polynomial in y and s , and the coefficient of the low-est degree term is nonzero in y if and only if the original system of polynomialshas zeroset of dimension exactly 0. This last test can be done just by studyingthe characteristic polynomial of the matrix M M − , and this can also be donein time polynomial in d n , mm