We give an improved polynomial bound on the complexity of the equation solvability problem, or more generally, of finding the value sets of polynomials over finite nilpotent rings. Our proof depends on a result in additive combinatorics, which may be of independent interest

Károlyi, Gyula

Szabó, Csaba

English

Diposit Digital de Documents de la UAB

Publ. Mat. 66 (2022), 197–205DOI: 10.5565/PUBLMAT6612208EVALUATION OF POLYNOMIALS OVER FINITERINGS VIA ADDITIVE COMBINATORICSGyula Károlyi and Csaba SzabóAbstract: We give an improved polynomial bound on the complexity of the equationsolvability problem, or more generally, of finding the value sets of polynomials overfinite nilpotent rings. Our proof depends on a result in additive combinatorics, whichmay be of independent interest.2010 Mathematics Subject Classification: Primary: 16Z05; Secondary: 11B75,13M10, 16P10.Key words: additive combinatorics, Chevalley’s theorem, dichotomy, nilpotent rings,Olson’s theorem, polynomial method, equation solvability problem.1. IntroductionSeveral ‘classical’ algebraic problems are investigated from the com-putational perspective. The equivalence problem and the equation solv-ability problem for various algebraic structures have received increasingattention recently. Both problems originate from the theory of rings andfields. The equivalence problem over a finite ring asks whether or not twopolynomials define the same function over the given ring. The equationsolvability problem is basically looking for the existence of a root or forthe solution of an equation. It asks whether or not two polynomials overa ring have at least one substitution, where they attain the same value.Both of these problems are decidable for finite rings: it is enough tosubstitute every element of the ring into the variables.Early investigations into the computational complexity of the equiva-lence problem were carried out by computer scientists at Syracuse Uni-versity. In the early 1990s it was shown by Hunt and Stearns [11] that theequivalence problem of a finite commutative ring either can be decidedin polynomial time or has co-NP-complete complexity. Later Burris andLawrence proved in [3] that the same holds for rings in general: theproblem can be decided in polynomial time if the ring is nilpotent (thispart is already found in [11]), and it is co-NP-complete otherwise.198 G. Károlyi, C. SzabóThe solvability problem has a somewhat shorter history. Althoughthe borderline is again nilpotency, the proofs are more complicated. Fol-lowing the idea of Burris and Lawrence it is not hard to verify that thesolvability problem for non-nilpotent rings is NP-complete. Horváth ([7])has proved that for nilpotent rings the problem can be decided in polyno-mial time. More precisely, let R be a nilpotent ring of size m and nilpo-tency class `, and let f be a polynomial over R in the (non-commuting)variables x1, . . . , xn. In [7] it is shown that O(||f ||t) substitutions sufficeto decide, whether f has a root or not. Here ||f || represents the numberof operations used to present f . The exponent t = t(m, `), obtained interms of the Ramsey number for a particular coloured hypergraph, israther enormous, though.In the present paper, by applying a recent result in additive combi-natorics the exponent t is reduced to m logm; see Theorem 4.3 for amore precise formulation. In particular, it is proved that the range of apolynomial can be found using O(nm(`−1)) many substitutions.Note that the equivalence and solvability problems for finite groupshave proved to be far more challenging. For results and detailed refer-ences see for example [9, 10] and the most recent paper of Horváth [8],where most of the existing results for equivalence and equation solvabil-ity are brought under a unified theory. For results about these problemson finite monoids and semigroups we refer to [1, 12, 15, 16]; concerninggeneral algebras see e.g. [6].2. TerminologyPolynomials. Having a diverse readership in mind we introduce ournotions in a somewhat informal, albeit precise manner. Let R be anarbitrary ring, and x1, . . . , xn symbols we think of as non-commutingvariables. By a polynomial in these variables over R we mean an expres-sion that can be constructed in a finite number of steps according to thefollowing rules: (i) every element of {x1, . . . , xn} ∪ R is a polynomial;(ii) if f is a polynomial, then −f is also a polynomial; (iii) if f , g arepolynomials, then f + g is also a polynomial; and (iv) if f , g are poly-nomials, then fg is also a polynomial. The set of all such polynomialswe denote by R[x1, . . . , xn].Evaluating a polynomial f ∈R[x1, . . ., xn] at the point c=(c1, . . ., cn)∈Rn, or substituting c in f , means replacing each occurrence of thevariable xi in f by the corresponding ring element ci, for every 1 ≤i ≤ n; the result is an element of R denoted by f(c). This way f de-fines a function from Rn to R; two polynomials are equivalent if theyEvaluation of Polynomials over Finite Rings 199define the same function. For example, for a, b ∈ R the polynomi-als (ax)b, a(xb),−(ax)(−b) ∈ R[x] are equivalent, but may not be equiv-alent to (ab)x. One may even identify the first three, but this is irrelevantto our purposes. The polynomials x1(x2x1) and x1x2 are also equivalentover Z/2Z.Each polynomial is equivalent to a ‘standard’ polynomial, that is, toone that can be written as a sum of monomials. More precisely, a mono-mial is a polynomial that can be constructed using only the rules (i),(ii), and (iv), where (ii) is only applied in the last step, if at all. Sup-pressing the brackets, each monomial can be represented in a canonicalform as z1 · · · zt or −z1 · · · zt, where each zi ∈ {x1, . . . , xn} ∪ R. If twomonomials have the same canonical form, then they are clearly equiva-lent. A standard polynomial is one obtained from monomials using onlythe rule (iii). Once again, up to equivalence it does not matter in whichorder the additions are executed.The computational model. In our model R denotes a fixed finitering, presented by its addition and multiplication tables. Thus, its car-dinality is |R| = O(1). We may assume that the table of additive in-verses and the zero element are also given, for they can be computedfrom the addition table in O(1) time. In accordance with rules (i)–(iv),the length ||f || of a polynomial f ∈ R[x1, . . . , xn] is defined as follows:||f || = 0 if f ∈ {x1, . . . , xn} ∪ R, and if ||f ||, ||g|| are defined, then set|| − f || = ||f || + 1 and ||f + g|| = ||fg|| = ||f || + ||g|| + 1. Thus, ||f || isthe number of operations used to define f , which may be regarded asthe complexity of evaluating f at any single point of Rn. Our goal is tomeasure the complexity of the problems mentioned in the introductionin terms of ||f ||.Nilpotency. In what follows, R(i) will denote the ideal consisting ofall finite sums of terms of the form c1 · · · ci, where c1, . . . , ci ∈ R. Thering R is nilpotent if there is a positive integer i such that R(i) = 0; thesmallest such i is called the nilpotency class of R. The standard examplefor a ring of nilpotency class ` is the ring of strictly upper triangular`× ` matrices over an arbitrary field.3. A result in additive combinatoricsA standard application of the pigeonhole principle gives that forany sequence g1, g2, . . . , gn of a finite group G there is a subsequencegi1 , gi2 , . . . , git with t < |G| such that gi1gi2 · · · git = g1g2 · · · gn. Herewe need a more general result that we can establish for finite abelianp-groups.200 G. Károlyi, C. SzabóLet p denote a prime and let G be an arbitrary finite abelian p-group,written additively. Then there exist unique positive integers r and α1 ≤· · · ≤ αr such that G ∼= Z/pα1Z⊕ · · · ⊕ Z/pαrZ. For a finite set H anda non-negative integer k we denote by Pk(H) the set of all subsets Xof H with |X| ≤ k. Given a function ϕ : Pk(H)→ G, associate with eachsubset U of H the valueϕ(U) =∑X⊆U, |X|≤kϕ(X).Thus, ϕ(H) is simply the total sum of the values of ϕ over Pk(H).Theorem 3.1. Let H be a finite set, k a non-negative integer. For anyfunction ϕ : Pk(H)→ Z/pα1Z⊕· · ·⊕Z/pαrZ, there is a set U ⊆ H suchthat|U | ≤ kr∑j=1(pαj − 1)and ϕ(U) = ϕ(H).Remark 3.2. It is easily seen that the case k = 1 of the above theoremis equivalent to Olson’s classical result [13] on the Davenport constantof finite abelian p-groups.Remark 3.3. The bound on the cardinality of U cannot be improvedupon for any k. Assume that |H| ≥ k∑rj=1(pαj − 1). Select pairwisedisjoint k-element subsets Hj,l of H for 1 ≤ j ≤ r, 1 ≤ l < pαj . Denoteby ej any generating element of the jth direct summand Z/pαjZ, andconsider the function ϕ that assigns ej to each set Hj,l and 0 to any otherelement of Pk(H). Then ϕ(H) =∑rj=1(pαj −1)ej , and it is obvious thatϕ(U) = ϕ(H) implies U ⊇ ∪Hj,l.Our proof depends on the following result of Brink [2], which can beviewed as a generalization of Chevalley’s well-known theorem [4] and itssomewhat forgotten extension by Schanuel [14].Theorem 3.4. Let p denote a prime and let A1, . . . , An be non-emptysubsets of Z such that the natural homomorphism from Z to Z/pZ re-stricted to Ai is injective for each 1 ≤ i ≤ n. Assume that the polynomi-als f1, . . . , fr ∈ Z[x1, . . . , xn] satisfyn∑i=1(|Ai| − 1) >r∑j=1(pαj − 1) deg fj .If the set {a ∈ A1 × · · · × An | fj(a) ≡ 0 (mod pαj ), 1 ≤ j ≤ r} is notempty, then it has at least two different elements.Evaluation of Polynomials over Finite Rings 201Proof of Theorem 3.1: Putting |H| = n, enumerate the elements of Has h1, . . . , hn. We may assume that n ≥ k∑rj=1(pαj − 1) + 1, for thestatement is valid with U = H otherwise. For 1 ≤ j ≤ r denote byϕj : Pk(H)→ Z/pαjZ the jth coordinate function of ϕ, and consider thepolynomial fj ∈ (Z/pαjZ)[x1, . . . , xn] defined byfj(x) =k∑ν=0( ∑1≤i1<···<iν≤nϕj({hi1 , . . . , hiν})ν∏α=1xiα).Thus, letting f = (f1, . . . , fr) we have f(0) = ϕ(∅) = ϕ(∅), f(1) = ϕ(H),andf(x) = ϕ(U) with U = U(x) = {hi | xi = 1}for x ∈ {0, 1}n in general.Define sequences Is ⊆ {1, . . . , n} and a(s) ∈ {0, 1}n recursively asfollows. Put I0 = {1, . . . , n} and a(0) = 1, thenI0 = {i | a(0)i = 1} and f(a(0)) = ϕ(H).Assume that Is and a(s) are already defined such thatIs = {i | a(s)i = 1} and f(a(s)) = ϕ(H).If |Is| > k∑rj=1(pαj − 1), then put Ai = {0, 1} for i ∈ Is and Ai = {0}for i ∈ I0 \ Is. Note that a(s) ∈ A1 × · · · × An. For each 1 ≤ j ≤ r thepolynomial gj = fj−ϕj(H) ∈ (Z/pαjZ)[x1, . . . , xn] vanishes at a(s) andn∑i=1(|Ai| − 1) = |Is| > kr∑j=1(pαj − 1) ≥r∑j=1(pαj − 1) deg gj .It follows from Theorem 3.4 that there exists an elementa ∈ (A1 × · · · ×An) \ {a(s)}such that gi(a) = 0 in Z/pαiZ for every i = 1, 2, . . . , r, and accordinglyf(a) = ϕ(H). Leta(s+ 1) = a, Is+1 = {i | a(s+ 1)i = 1};then Is+1 is a proper subset of Is.This process must terminate at some t ≥ 1 with |It| ≤ k∑rj=1(pαj −1), meaning that the statement of the theorem is valid withU = {hi | i ∈ It} = U(a(t)).202 G. Károlyi, C. Szabó4. Nilpotent ringsAll the results claimed in Section 1 can be easily reduced to the fol-lowing general statement.Proposition 4.1. Let R be an arbitrary finite ring and g ∈ R[x1, . . . , xn]a standard polynomial, written as a sum of monomials. If each suchmonomial contains at most k different variables, then O(nk|R|) evalua-tions suffice to determine the range of g.Proof: Assume first that the cardinality of R is a power of a prime p.That is, the additive group of R is a finite abelian p-group:R+ ∼= Z/pα1Z⊕ · · · ⊕ Z/pαrZ.Suppose that g admits the value v: there are elements c1, . . . , cn ∈ R suchthat g(c1, . . . , cn) = v. For each subset X of H = {1, 2, . . . , n} with |X| ≤k, denote by gX the sum of those monomial terms of g which containexactly the variables xi with i ∈ X. Writing ϕ(X) = gX(c1, . . . , cn)we have v =∑X∈Pk(H) ϕ(X) = ϕ(H). Theorem 3.1 guarantees theexistence of a subset U of {1, 2, . . . , n} with|U | ≤ kr∑j=1(pαj − 1) ≤ k|R|and ϕ(U) = ϕ(H) = v. Putting c′i = ci for i ∈ U and c′i = 0 for i 6∈ Uwe havegX(c′1, . . . , c′n) ={gX(c1, . . . , cn) if X ⊆ U ,0 otherwise.It follows thatg(c′1, . . . , c′n) =∑X∈Pk(H)gX(c′1, . . . , c′n) =∑X⊆UgX(c1, . . . , cn)=∑X⊆Uϕ(X) = ϕ(U) = ϕ(H) = v.Consequently, to determine the range of g it suffices to substitute into gonly those n-tuples (c1, . . . , cn) in which all but at most k|R| elementsare equal to 0. The number of such n-tuples isk|R|∑i=0(ni)|R|i ≤ (k|R|+ 1)|R|k|R|nk|R| = O(nk|R|),where the implied constant only depends on k and R.Evaluation of Polynomials over Finite Rings 203Turning to the general case, assume that R+ = G1 ⊕ · · · ⊕Gs, whereGi is an abelian group of order pβii , the primes p1, . . . , ps being pairwisedifferent. Each set defined byRi = {r ∈ R | pβii r = 0}is an ideal in R whose additive group must be identical to Gi. It followsthat R = R1⊕ · · · ⊕Rs, and there are natural homomorphisms πi : R→Ri such that π1 + · · · + πs is the identical map from R to R. Thus, ifthe n-tuple c ∈ Rn is written as c = c1 + · · · + cs with ci ∈ (Ri)n,then g(c) = g1(c1) + · · · + gs(cs), where gi is the polynomial over Riobtained from g by replacing each coefficient in g by its image under πi.Accordingly, the value set of g is the Minkowski sum of the value sets ofthe polynomials gi, and in view of the first part of the proof it can befound usingO(s∏i=1nk|Ri|)= O(nk∑si=1 |Ri|)= O(nk|R|)substitutions.Remark 4.2. As part of the preprocessing, one can compute in O(1) timethe factorization |R| = pβ11 · · · pβss , and then the ideals Ri in the directdecomposition of R. To compute the range of g it is then enough toevaluate g on the elements of the small setS = {c ∈ Rn : |{j : (ci)j 6= 0}| ≤ kpβii for every 1 ≤ i ≤ s}.According to the above proof, |S| = O(nk|R|).Theorem 4.3. Let R be a nilpotent ring of size m. For an n-variablepolynomial f ∈ R[x1, x2, . . . , xn], it can be decided in O(||f ||nm logm)time whether or not f has a root in R.Proof: Denote by ` the nilpotency class of R. Thus, R(`) = 0. Thereforef is equivalent to a standard polynomial f ∈ R[x1, x2, . . . , xn] in whichevery monomial term contains at most `−1 different variables. By Propo-sition 4.1, the range of f , which is the same as the range of f , can befound by evaluating f on a subset S of Rn, of size O(n(`−1)m); see Re-mark 4.2 for an explicit description. In particular, one can check whetherf admits the value 0, and it can be also decided if f is identically zeroor not. Note that the chain of ideals R . R(2) . · · · . R(`) = 0 is strictlydecreasing. As |R(i)/R(i+1)| ≥ 2, it follows that `− 1 ≤ log2m.Remark 4.4. During the preparation of this article Földvári ([5])proved a similar result with an entirely different method. His bound204 G. Károlyi, C. Szabóis O(||f ||m2 logm log5m). Note that one may assume that ||f || ≥ n − 1,otherwise the set of variables that occur in f can be restricted to a propersubset of {x1, . . . , xn}. If this is the case, then Theorem 4.3 implies thebound O(||f ||m logm).Acknowledgements. The first author was supported by the BolyaiResearch Fellowship and by National Research, Development, and Inno-vation Office NKFIH Grant K 120154. The research of the second authorwas supported by NKFIH Grant K 109185.References[1] Zh. Almĕıda, M. V. Volkov, and S. V. Gol’dberg, Complexity of theidentity checking problem for finite semigroups (Russian), Zap. Nauchn. Sem.S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 358 (2008), 5–22, 301;translation in J. Math. Sci. (N.Y.) 158(5) (2009), 605–614. DOI: 10.1007/s10958-009-9397-z.[2] D. Brink, Chevalley’s theorem with restricted variables, Combinatorica 31(1)(2011), 127–130. DOI: 10.1007/s00493-011-2504-z.[3] S. Burris and J. Lawrence, The equivalence problem for finite rings, J. Sym-bolic Comput. 15(1) (1993), 67–71. DOI: 10.1006/jsco.1993.1004.[4] C. Chevalley, Démonstration d’une hypothèse de M. Artin, Abh. Math. Sem.Univ. Hamburg 11(1) (1935), 73–75. DOI: 10.1007/BF02940714.[5] A. Földvári, The complexity of the equation solvability problem over semipat-tern groups, Internat. J. Algebra Comput. 27(2) (2017), 259–272. DOI: 10.1142/S0218196717500126.[6] T. A. Gorazd and J. Krzaczkowski, Term equation satisfiability over finitealgebras, Internat. J. Algebra Comput. 20(8) (2010), 1001–1020. DOI: 10.1142/S021819671000600X.[7] G. Horváth, The complexity of the equivalence and equation solvability prob-lems over nilpotent rings and groups, Algebra Universalis 66(4) (2011), 391–403.DOI: 10.1007/s00012-011-0163-y.[8] G. Horváth, The complexity of the equivalence and equation solvability prob-lems over meta-Abelian groups, J. Algebra 433 (2015), 208–230. DOI: 10.1016/j.jalgebra.2015.03.015.[9] G. Horváth and C. Szabó, The extended equivalence and equation solvabilityproblems for groups, Discrete Math. Theor. Comput. Sci. 13(4) (2011), 23–32.DOI: 10.46298/dmtcs.536.[10] G. Horváth and C. Szabó, Equivalence and equation solvability problems forthe alternating group A4, J. Pure Appl. Algebra 216(10) (2012), 2170–2176.DOI: 10.1016/j.jpaa.2012.02.007.[11] H. B. Hunt, III and R. E. Stearns, The complexity of equivalence for com-mutative rings, J. Symbolic Comput. 10(5) (1990), 411–436. DOI: 10.1016/S0747-7171(08)80053-3.[12] O. Kĺıma, Complexity issues of checking identities in finite monoids, SemigroupForum 79(3) (2009), 435–444. DOI: 10.1007/s00233-009-9180-y.[13] J. E. Olson, A combinatorial problem on finite Abelian groups, I, J. NumberTheory 1 (1969), 8–10. DOI: 10.1016/0022-314X(69)90021-3.Evaluation of Polynomials over Finite Rings 205[14] S. H. Schanuel, An extension of Chevalley’s theorem to congruences moduloprime powers, J. Number Theory 6(4) (1974), 284–290. DOI: 10.1016/0022-314X(74)90025-0.[15] S. Seif and C. Szabó, Computational complexity of checking identities in 0-sim-ple semigroups and matrix semigroups over finite fields, Semigroup Forum 72(2)(2006), 207–222. DOI: 10.1007/s00233-005-0510-4.[16] P. Tesson and D. Thérien, Monoids and computations, International Confer-ence on Semigroups and Groups in honor of the 65th birthday of Prof. JohnRhodes, Internat. J. Algebra Comput. 14(5–6) (2004), 801–816. DOI: 10.1142/S0218196704001979.Gyula KárolyiAlfréd Rényi Institute of Mathematics, 13–15 Reáltanoda utca, 1053 Budapest, Hun-gary and Institute of Mathematics, Eötvös University, 1/C Pázmány Péter sétány,1117 Budapest, HungaryE-mail address: karolyi.gyula@renyi.huCsaba SzabóInstitute of Mathematics, Eötvös University, 1/C Pázmány Péter sétány, 1117 Bu-dapest, HungaryE-mail address: szabo.csaba.mathdid@ttk.elte.huReceived on April 14, 2020.Accepted on September 4, 2020.

Evaluation of polynomials over finite rings via additive combinatorics

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arXiv:1809.06543v1  [math.RA]  18 Sep 2018EVALUATION OF POLYNOMIALS OVER FINITERINGS VIA ADDITIVE COMBINATORICSGYULA KÁROLYI AND CSABA SZABÓAbstract. We give an improved polynomial bound on the com-plexity of the equation solvability problem, or more generally, offinding the value sets of polynomials over finite nilpotent rings.Our proof depends on a result in additive combinatorics, whichmay be of independent interest.1. IntroductionSeveral ‘classical’ algebraic problems are investigated from the compu-tational perspective. The equivalence problem and the equation solv-ability problem for various algebraic structures has received increasingattention recently. Both problems originate from the theory of ringsand fields. The equivalence problem over a finite ring asks whether ornot two polynomials define the same function over the given ring. Theequation solvability problem is basically looking for the existence of aroot or for the solution of an equation. It asks whether or not two poly-nomials over a ring have at least one substitution, where they attainthe same value. Both of these problems are decidable for finite rings:it is enough to substitute every element of the ring into the variables.Early investigations into the computational complexity of the equiv-alence problem were carried out by computer scientists at SyracuseUniversity. In the early 1990s it was shown by Hunt and Stearns [11]that the equivalence problem of a finite commutative ring either can bedecided in polynomial time or has co-NP-complete complexity. LaterBurris and Lawrence proved in [3] that the same holds for rings ingeneral: the problem can be decided in polynomial time if the ring isnilpotent (this part is already found in [11]), and it is co-NP-completeotherwise.The solvability problem has a somewhat shorter history. Althoughthe borderline is again nilpotency, the proofs are more complicated.Following the idea of Burris and Lawrence it is not hard to verifythat the solvability problem for non-nilpotent rings is NP-complete.Horváth [7] has proved that for nilpotent rings the problem can be12 GYULA KÁROLYI AND CSABA SZABÓdecided in polynomial time. More precisely, let R be a nilpotent ringof size m and nilpotency class ℓ, and let f be a polynomial over R in the(non-commuting) variables x1, . . . , xn. In [7] it is shown that O(||f ||t)substitutions suffice to decide, whether f has a root or not. Here ||f ||represent the number of operations used to present f . The exponentt = t(m, ℓ), obtained in terms of the Ramsey-number for a particularcoloured hypergraph, is rather enormous, though.In the present paper, applying a recent result in additive combina-torics the exponent t is reduced to m logm; see Theorem 4.2 for amore precise formulation. In particular, it is proved that the range ofa polynomial can be found using O(nm(ℓ−1))many substitutions.Note that the equivalence and solvability problems for finite groupshas proved to be far more challenging. For results and detailed ref-erences see for example [9, 10] and the most recent paper of Horváth[8], where most of the existing results for equivalence and equationsolvability are brought under a unified theory. For results about theseproblems on finite monoids and semigroups we refer to [1, 12, 15, 16];concerning general algebras see e.g. [6].2. TerminologyPolynomials. Having a diverse readership in mind we introduce ournotions in a somewhat informal, albeit precise manner. Let R be anarbitrary ring, and x1, . . . , xn symbols we think of as non-commutingvariables. By a polynomial in these variables over R we mean an ex-pression that can be constructed in a finite number of steps accordingto the following rules: (i) every element of {x1, . . . , xn} ∪ R is a poly-nomial; (ii) if f is a polynomial, then −f is also a polynomial; (iii)if f, g are polynomials, then f + g is also a polynomial; and (iv) iff, g are polynomials, then fg is also a polynomial. The set of all suchpolynomials we denote by R[x1, . . . , xn].Evaluating a polynomial f ∈ R[x1, . . . , xn] at the point c = (c1, . . . , cn) ∈Rn, or substituting c in f , means replacing each occurance of the vari-able xi in f by the corresponding ring element ci, for every 1 ≤ i ≤ n,the result is an element of R denoted by f(c). This way f definesa function from Rn to R; two polynomials are equivalent if they de-fine the same function. For example, for a, b ∈ R the polynomials(ax)b, a(xb),−(ax)(−b) ∈ R[x] are equivalent, but may not be equiv-alent to (ab)x. One may even identify the first three, but this is ir-relevant to our purposes. The polynomials x1(x2x1) and x1x2 are alsoequivalent over Z/2Z.EVALUATION OF POLYNOMIALS OVER FINITE RINGS 3Each polynomial is equivalent to a ‘standard’ polynomial, that is,to one that can be written as a sum of monomials. More precisely,a monomial is a polynomial that can be constructed using only therules (i), (ii) and (iv), where (ii) is only applied in the last step, if atall. Suppressing the brackets each monomial can be represented in acanonical form as z1 · · · zt or−z1 · · · zt, where each zi ∈ {x1, . . . , xn}∪R.If two monomials have the same canonical form, then they are clearlyequivalent. A standard polynomial is one obtained from monomialsusing merely the rule (iii). Once again, up to equivalence it does notmatter, in which order the additions are executed.The computational model. In our model R denotes a fixed finitering, presented by its addition and multiplication tables. Thus, itscardinality is |R| = O(1). We may assume that the table of additiveinverses and the zero element are also given, for they can be computedfrom the addition table in O(1) time. In accordance with rules (i)–(iv),the length ||f || of a polynomial f ∈ R[x1, . . . , xn] is defined as follows:||f || = 0 if f ∈ {x1, . . . , xn} ∪ R, and if ||f ||, ||g|| are defined, then set|| − f || = ||f ||+ 1 and ||f + g|| = ||fg|| = ||f ||+ ||g||+ 1. Thus, ||f ||is the number of operations used to define f , that may be regarded asthe complexity of evaluating f at any single point of Rn. Our goal is tomeasure the complexity of the problems mentioned in the introductionin terms of ||f ||.Nilpotency. In what follows, R(i) will denote the ideal consisting ofall finite sums of terms of the form c1 · · · ci, where c1, . . . , ci ∈ R. Thering R is nilpotent if there is a positive integer i such that R(i) = 0;the smallest such i is called the nilpotency class of R. The standardexample for a ring of nilpotency class ℓ is the ring of strictly uppertriangular ℓ× ℓ matrices over an arbitrary field.3. A result in additive combinatoricsA standard application of the pigeonhole principle gives that for any se-quence g1, g2, . . . , gn of a finite groupG there is a subsequence gi1, gi2 , . . . , gitwith t < |G| such that gi1gi2 . . . git = g1g2 . . . gn. Here we need a moregeneral result that we can establish for finite abelian p-groups.Let p denote a prime and let G be an arbitrary finite abelian p-group, written additively. Then there exist unique positive integersr and α1 ≤ . . . ≤ αr such that G ∼= Z/pα1Z ⊕ . . . ⊕ Z/pαrZ. For afinite set H and a nonnegative integer k we denote by Pk(H) the setof all subsets X of H with |X| ≤ k. Given a function ϕ : Pk(H) → G,4 GYULA KÁROLYI AND CSABA SZABÓassociate with each subset U of H the valueϕ(U) =∑X⊆U,|X|≤kϕ(X).Thus, ϕ(H) is simply the total sum of the values of ϕ over Pk(H).Theorem 3.1. Let H be a finite set, k a nonnegative integer. For anyfunction ϕ : Pk(H) → Z/pα1Z ⊕ . . . ⊕ Z/pαrZ, there is a set U ⊆ Hsuch that|U | ≤ kr∑j=1(pαj − 1)and ϕ(U) = ϕ(H).Remark 1. It is easily seen that the case k = 1 of the above theoremis equivalent to Olson’s classical result [13] on the Davenport constantof finite abelian p-groups.Remark 2. The bound on the cardinality of U cannot be improvedupon for any k. Assume that |H| ≥ k∑rj=1(pαj − 1). Select pairwisedisjoint k-element subsets Hj,l of H for 1 ≤ j ≤ r, 1 ≤ l < pαj . Denoteby ej any generating element of the jth direct summand Z/pαjZ, andconsider the function ϕ that assigns ej to each set Hj,l and 0 to anyother element of Pk(H). Then ϕ(H) =∑rj=1(pαj − 1)ej, and it isobvious that ϕ(U) = ϕ(H) implies U ⊇ ∪Hj,l.Our proof depends on the following result of Brink [2], which can beviewed as a generalization of Chevalley’s well-known theorem [4] andits somewhat forgotten extension by Schanuel [14].Theorem 3.2. Let p denote a prime and let A1, . . . , An be nonemptysubsets of Z such that the natural homomorphism from Z to Z/pZrestricted to Ai is injective for each 1 ≤ i ≤ n. Assume that thepolynomials f1, . . . , fr ∈ Z[x1, . . . , xn] satisfyn∑i=1(|Ai| − 1) >r∑j=1(pαj − 1) deg fj .If the set {a ∈ A1 × . . .×An | fj(a) ≡ 0 (mod pαj ), 1 ≤ j ≤ r} is notempty, then it has at least two different elements.Proof of Theorem 3.1. Putting |H| = n, enumerate the elements of Has h1, . . . , hn. We may assume that n ≥ k∑rj=1(pαj − 1) + 1, for thestatement is valid with U = H otherwise. For 1 ≤ j ≤ r denote byEVALUATION OF POLYNOMIALS OVER FINITE RINGS 5ϕj : Pk(H) → Z/pαjZ the jth coordinate function of ϕ, and considerthe polynomial fj ∈ (Z/pαjZ)[x1, . . . , xn] defined byfj(x) =k∑ν=0(∑1≤i1<...<iν≤nϕj({hi1 , . . . , hiν})ν∏α=1xiα).Thus, letting f = (f1, . . . , fr) we have f(0) = ϕ(∅) = ϕ(∅), f(1) =ϕ(H) andf(x) = ϕ(U) with U = U(x) = {hi | xi = 1}for x ∈ {0, 1}n in general.Define sequences Is ⊆ {1, . . . , n} and a(s) ∈ {0, 1}n recursively asfollows. Put I0 = {1, . . . , n} and a(0) = 1, thenI0 = {i | a(0)i = 1} and f(a(0)) = ϕ(H).Assume that Is and a(s) are already defined such thatIs = {i | a(s)i = 1} and f(a(s)) = ϕ(H).If |Is| > k∑rj=1(pαj − 1), then put Ai = {0, 1} for i ∈ Is and Ai = {0}for i ∈ I0 \ Is. Note that a(s) ∈ A1 × . . .×An. For each 1 ≤ j ≤ r thepolynomial gj = fj − ϕj(H) ∈ (Z/pαjZ)[x1, . . . , xn] vanishes at a(s)andn∑i=1(|Ai| − 1) = |Is| > kr∑j=1(pαj − 1) ≥r∑j=1(pαj − 1) deg gj.It follows from Theorem 3.2 that there exists an elementa ∈ (A1 × . . .×An) \ {a(s)}such that gi(a) = 0 in Z/pαiZ for every i = 1, 2, . . . , r, and accordinglyf(a) = ϕ(H). Leta(s + 1) = a, Is+1 = {i | a(s+ 1)i = 1},then Is+1 is a proper subset of Is.This process must terminate at some t ≥ 1 with |It| ≤ k∑rj=1(pαj −1) meaning that the statement of the theorem is valid withU = {hi | i ∈ It} = U(a(t)).4. Nilpotent ringsAll the results claimed in Section 1 can be easily reduced to the follow-ing general statement.6 GYULA KÁROLYI AND CSABA SZABÓProposition 4.1. Let R be an arbitrary finite ring, g ∈ R[x1, . . . , xn]a standard polynomial, written as a sum of monomials. If each suchmonomial contains at most k different variables, then O(nk|R|)evalu-ations suffice to determine the range of g.Proof. Assume first that the cardinality of R is a power of a prime p.That is, the additive group of R is a finite abelian p-group:R+ ∼= Z/pα1Z⊕ . . .⊕ Z/pαrZ.Suppose that g admits the value v: there are elements c1, . . . , cn ∈ Rsuch that g(c1, . . . , cn) = v. For each subset X of H = {1, 2, . . . , n}with |X| ≤ k, denote by gX the sum of those monomial terms of g,which contain exactly the variables xi with i ∈ X . Writing ϕ(X) =gX(c1, . . . , cn) we have v =∑X∈Pk(H)ϕ(X) = ϕ(H). Theorem 3.1guarantees the existence of a subset U of {1, 2, . . . , n} with|U | ≤ kr∑j=1(pαj − 1) ≤ k|R|and ϕ(U) = ϕ(H) = v. Putting c′i = ci for i ∈ U and c′i = 0 for i 6∈ Uwe havegX(c′1, . . . , c′n) ={gX(c1, . . . , cn) if X ⊆ U ,0 otherwise.It follows thatg(c′1, . . . , c′n) =∑X∈Pk(H)gX(c′1, . . . , c′n) =∑X⊆UgX(c1, . . . , cn)=∑X⊆Uϕ(X) = ϕ(U) = ϕ(H) = v.Consequently, to determine the range of g it suffices to substitute intog only those n-tuples (c1, . . . , cn) in which all but at most k|R| elementsare equal to 0. The number of such n-tuples isk|R|∑i=0(ni)|R|i ≤ (k|R|+ 1)|R|k|R|nk|R| = O(nk|R|),where the implied constant only depends on k and R.Turning to the general case, assume that R+ = G1⊕ . . .⊕Gs, whereGi is an abelian group of order pβii , the primes p1, . . . , ps being pairwisedifferent. Each set defined byRi = {r ∈ R | pβii r = 0}EVALUATION OF POLYNOMIALS OVER FINITE RINGS 7is an ideal in R whose additive group must be identical to Gi. Itfollows that R = R1 ⊕ . . .⊕Rs, and there are natural homomorphismsπi : R → Ri such that π1 + . . . + πs is the identical map form R toR. Thus, if the n-tuple c ∈ Rn is written as c = c1 + . . . + cs withci ∈ (Ri)n, then g(c) = g1(c1)+ . . .+gs(cs), where gi is the polynomialover Ri obtained from g by replacing each coefficient in g by its imageunder πi. Accordingly, the value set of g is the Minkowski-sum of thevalue sets of the polynomials gi, and in view of the first part of theproof it can be found usingO(s∏i=1nk|Ri|)= O(nk∑si=1 |Ri|)= O(nk|R|)substitutions. Remark 3. As part of the preprocessing, one can compute in O(1)time the factorization |R| = pβ11 · · · pβss , and then the ideals Ri in thedirect decomposition of R. To compute the range of g then it is enoughto evaluate g on the elements of the small setS ={c ∈ Rn :∣∣{j : (ci)j 6= 0}∣∣ ≤ kpβii for every 1 ≤ i ≤ s}.According to the above proof, |S| = O(nk|R|).Theorem 4.2. Let R be a nilpotent ring of sizem. Let f ∈ R[x1, x2, . . . , xn]be an n-variable polynomial. Then, it can be decided in O(||f ||nm logm)time whether or not f has a root in R.Proof. Denote by ℓ the nilpotency class of R. Thus, R(ℓ) = 0. There-fore f is equivalent to a standard polynomial f ∈ R[x1, x2, . . . , xn]in which every monomial term contains at most ℓ − 1 different vari-ables. By Proposition 4.1, the range of f , which is the same as therange of f , can be found by evaluating f on a subset S of Rn, of sizeO(n(ℓ−1)m); see Remark 3 for an explicit description. In particular,one can check whether f admits the value 0, and it can be also de-cided if f is identically zero, or not. Note that the chain of idealsR ⊲ R(2) ⊲ · · · ⊲ R(ℓ) = 0 is strictly decreasing. As |R(i)/R(i+1)| ≥ 2, itfollows that ℓ− 1 ≤ log2m. Remark 4. During the preparation of this article Földvári [5] proved asimilar result with an entirely different method. His bound is O(||f ||m2 logm log5 m).Note that one may assume that ||f || ≥ n− 1, otherwise the set of vari-ables that occur in f can be restricted to a proper subset of {x1, . . . , xn}.If this is the case, then Theorem 4.2 implies the bound O(||f ||m logm).8 GYULA KÁROLYI AND CSABA SZABÓThe first author was supported by Bolyai Research Fellowship and byNational Research, Development and Innovation Office NKFIH GrantK 120154. Research of the second author was supported by NKFIHGrant K 109185.References[1] J. Almeida, M. V. Volkov, and S. V. Goldberg, ‘Complexity of the identitychecking problem for finite semigroups’, J. Math. Sci. (N.Y.) 158 (2009) 605–614.[2] D. Brink, ‘Chevalley’s theorem with restricted variables’, Combinatorica 31(2011) 127–130.[3] S. Burris and J. Lawrence, ‘The equivalence problem for finite rings’, J. Sym-bolic Comput. 15 (1993) 67–71.[4] C. Chevalley, ‘Démonstration d’une hypothèse de M. Artin’, Abh. Math. Sem.Univ. Hamburg 11 (1935) 73–75.[5] A. Földvári, ‘The complexity of the equation solvability problem over semi-pattern groups’, Internat. J. Algebra Comput. 27 (2017) 259–272.[6] T. A. Gorazd and J. Krzaczkowski, ‘Term equation satisfiability over finitealgebras’, Internat. J. Algebra Comput. 20 (2010) 1001–1020.[7] G. Horváth, ‘The complexity of the equivalence and equation solvability prob-lems over nilpotent rings and groups’, Algebra Universalis 66 (2011) 391–403.[8] G. Horváth, ‘The complexity of the equivalence and equation solvability prob-lems over meta-Abelian groups’, J. Algebra 433 (2015) 208–230.[9] G. Horváth and Cs. Szabó, ‘The extended equivalence and equation solvabilityproblems for groups’, Discrete Math. Theor. Comput. Sci. 13 (2011) 23–32.[10] G. Horváth and Cs. Szabó, ‘Equivalence and equation solvability problems forthe group A4’, J. Pure Appl. Algebra 216 (2012) 2170–2176.[11] H. R. Hunt III and R. E. Stearns, ‘The complexity for equivalence for commu-tative rings’, J. Symbolic Comput. 10 (1990) 411–436.[12] O. Kĺıma, ‘Complexity issues of checking identities in finite monoids’, Semi-group Forum 79 (2009) 435–444.[13] J. E. Olson, ‘A combinatorial problem on finite Abelian groups. I’, J. NumberTheory 1 (1969) 8–10.[14] S. H. Schanuel, ‘An extension of Chevalley’s theorem to congruences moduloprime powers’, J. Number Theory 6 (1974) 284–290.[15] S. Seif and Cs. Szabó, ‘Computational complexity of checking identities in 0-simple semigroups and matrix semigroups over finite fields’, Semigroup Forum72 (2006) 207–222.[16] P. Tesson and D. Thérien, ‘Monoids and computations’, Internat. J. AlgebraComput. 14 (2004) 801–816.

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Evaluation of polynomials over finite rings via additive combinatorics

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