aa r X i v : . [ m a t h . L O ] J a n SOLVING SYSTEMS OF EQUATIONS IN SUPERNILPOTENTALGEBRAS
ERHARD AICHINGER
Abstract.
Recently, M. Kompatscher proved that for each finite supernilpo-tent algebra A in a congruence modular variety, there is a polynomial timealgorithm to solve polynomial equations over this algebra. Let µ be the maxi-mal arity of the fundamental operations of A , and let d := | A | log ( µ )+log ( | A | )+1 . Applying a method that G. K´arolyi and C. Szab´o had used to solve equationsover finite nilpotent rings, we show that for A , there is c ∈ N such that asolution of every system of s equations in n variables can be found by testingat most cn sd (instead of all | A | n possible) assignments to the variables. Thisalso yields new information on some circuit satisfiability problems. Introduction
We study systems of polynomial equations over a finite algebraic structure A .Such a system is given by equations of the form p ( x , . . . , x n ) ≈ q ( x , . . . , x n ),where p, q are polynomial terms of A ; a polynomial term of A is a term of thealgebra A ∗ which is obtained by expanding A with one nullary function symbolfor each a ∈ A . A solution to a system p i ( x , . . . , x n ) ≈ q i ( x , . . . , x n ) ( i =1 , . . . , s ) is an element a = ( a , . . . , a n ) ∈ A n such that p A i ( a ) = q A i ( a ) for all i ∈{ , . . . , s } . The problem to decide whether such a solution exists has been called PolSysSat ( A ), and PolSat ( A ) if the system consists of one single equation,and the terms of the input are encoded as strings over { x , . . . , x n } ∪ A ∪ F , where F is the set of function symbols of A . A survey of results on the computationalcomplexity of this problem is given, e.g., in [IK18, Kom18]. In algebras suchas groups, rings or Boolean algebras, one can reduce an equation p ( x ) ≈ q ( x )to an equation of the form f ( x ) ≈ y , where y ∈ A . A system of equations ofthis form then has the form f i ( x ) ≈ y i ( i = 1 , . . . , s ). For n ∈ N , let Pol n ( A )denote the n -ary polynomial functions on A [MMT87, Definition 4.4]. For a finitenilpotent ring or group A , [Hor11] establishes the existence of a natural number d A such that for every f ∈ Pol n ( A ) and for every a ∈ A n , there exists b suchthat f A ( a ) = f A ( b ) and b has at most d A components that are different from 0. Mathematics Subject Classification.
Key words and phrases.
Supernilpotent algebras, polynomial equations, polynomial map-pings, circuit satisfiability.Supported by the Austrian Science Fund (FWF):P29931.
Hence the equation f ( x ) ≈ y has a solution if and only if it has a solution withat most d A nonzero entries. Thus for the algebra A , testing only vectors with atmost d A nonzero entries is an algorithm, which, given an equation f ( x ) ≈ y oflength n , takes at most c ( A ) · n d A +1 many steps to find whether this equation issolvable: there are at most P d A i =0 (cid:0) ni (cid:1) ( | A |− i ≤ c ( A ) · n d A many evaluations to bedone, each of them taking at most c ( A ) · n many steps. The number d A in [Hor11]is obtained from Ramsey’s Theorem and therefore rather large. In [Kom18], itis proved that for every finite supernilpotent algebra in a congruence modularvariety, such a number d A exists, again using Ramsey’s Theorem. For rings,lower values of d A have been obtained in [KS18] (cf. [KS15]). In [F¨ol17, F¨ol18],A. F¨oldv´ari provides polynomial time algorithms for solving equations over finitenilpotent groups and rings relying on the structure theory of these algebras. Inthis paper, we extend the method developed in [KS18] from finite nilpotent ringsto arbitrary finite supernilpotent algebras in congruence modular varieties. Forsuch algebras, we compute d A as | A | log ( µ )+log ( | A | )+1 (Theorem 10). The techniquethat allows to generalize K´arolyi’s and Szab´o’s method is the coordinatizationof nilpotent algebras of prime power order by elementary abelian groups from[Aic18, Theorem 4.2]. The method generalizes to systems of equations: we showfor a given finite supernilpotent algebra A in a congruence modular variety, anda given s ∈ N , there is a polynomial time algorithm to test whether a system ofat most s polynomial equations over A has a solution. If s is not fixed in advance,then [LZ06, Corollary 3.13] implies that if A is not abelian, PolSysSat ( A ) isNP-complete.Let us finally explain to which class of algebras our results applies: A finitealgebra A from a congruence modular variety with finitely many fundamentaloperations is supernilpotent if and only if it is a direct product of nilpotent al-gebras of prime power order; modulo notational differences explained, e.g., in[Aic18, Lemma 2.4], this result has been proved in [Kea99, Theorem 3.14]. Suchan algebra is therefore always nilpotent, has a Mal’cev term (cf. [FM87, Theo-rem 6.2], [Kea99, Theorem 2.7]), and hence generates a congruence permutablevariety. For a more detailed introduction to supernilpotency and, for k ∈ N , to k -supernilpotency, we refer to [AM10, AMO18, Aic18].2. A theorem of K´arolyi and Szab´o
In this section, we state a special case of [KS18, Theorem 3.1]. Since theirresult is much more general than needed for our purpose, we also include a self-contained proof, which is a reduction K´arolyi’s and Szab´o’s proof to the case ofelementary abelian groups.For n ∈ N = { , , , . . . } , we denote the set { , , . . . , n } by n . Let A be a setwith an element 0 ∈ A , and let J ⊆ n . For a ∈ A n , a ( J ) is defined by a ( J ) ∈ A n , a ( J ) ( j ) = a ( j ) for j ∈ J and a ( J ) ( j ) = 0 for i ∈ n \ J . Suppose that 1 is anelement of A . Then by , we denote the vector (1 , , . . . ,
1) in A n , and for J ⊆ n , YSTEMS OF EQUATIONS IN SUPERNILPOTENT ALGEBRAS 3 ( J ) is the vector ( v , . . . , v n ) with v j = 1 if j ∈ J and v j = 0 if j J . For anysets C, D , we write C ⊂ D for ( C ⊆ D and C = D ).We first need the following variation of [Bri11, Theorem 1] and [KS18, The-orem 3.2], which is proved using several arguments from the proof of [Alo99,Theorem 3.1] and from [Bri11]. Lemma 1.
Let F be a finite field, let k, m, n ∈ N , let q := | F | , let p , . . . , p m ∈ F [ x , . . . , x n ] be polynomials such that for each i ∈ m , each monomial of p i contains at most k variables. Then there exists J ⊆ n such that | J | ≤ km ( q − and p i ( ( J ) ) = p i ( ) for all i ∈ m .Proof. We proceed by induction on n . If n ≤ km ( q − J := n .For the induction step, we assume that n > km ( q − J ⊂ n such that p i ( ( J ) ) = p i ( ) for all i ∈ m . Seekinga contradiction, we suppose that no such J exists. Following an idea from theproof of [Alo99, Theorem 3.1], we consider the polynomials q ( x , . . . , x n ) := Q mi =1 (1 − ( p i ( x ) − p i ( )) q − ) ,q ( x , . . . , x n ) := x x · · · x n − q ( x , . . . , x n ) . We first show that for all a ∈ { , } n , q ( a ) = 0. To this end, we first considerthe case a = . Then q ( a ) = 1 − Q mi =1 . If a ∈ { , } n \ { } , then by theassumptions, there is i ∈ m such that p i ( a ) = p i ( ). Then 1 − ( p i ( a ) − p i ( )) q − =0. Therefore q ( a ) = 0. Hence the polynomial q vanishes at { , } n . By theCombinatorial Nullstellensatz [Alo99, Theorem 1.1] applied to g j ( x j ) := x j − x j , q then lies in the ideal V of F [ x , . . . , x n ] generated by G = { x j − x j | j ∈ n } . Hence x x · · · x n − q ( x , . . . , x n ) ∈ V . Since the leading monomials of the polynomialsin G are coprime, G is a Gr¨obner basis of V (with respect to x > x > · · · > x n ,lexicographic order, cf. [Eis95, p.337]). Therefore, reducing q ( x , . . . , x n ) modulo G , we must obtain x x · · · x n as the remainder (as defined, e.g., in [Eis95, p.334]).Because of the form of all polynomials in G (all variables of g j occur in the leadingterm of g j ), none of the reduction steps increases the number of variables in anymonomial. Therefore, q ( x , . . . , x n ) must contain a monomial that contains all n variables. Computing the expansion of q by multiplying out all products from itsdefinition, we see that each monomial in q contains at most km ( q −
1) variables.Hence n ≤ km ( q − n > km ( q − J ⊂ n such that p i ( ( J ) ) = p i ( ) forall i ∈ m . Now we let n ′ := | J | , and we assume that J = { j , . . . , j n ′ } with j < · · · < j n ′ . For i ∈ m , we define p ′ i ∈ F [ y , . . . , y n ′ ] by p ′ i ( x j , . . . , x j n ′ ) = p i ( x ( J ) ) . By the induction hypothesis, there exists J ⊆ n ′ with | J | ≤ km ( q −
1) suchthat p ′ i ( ( J ) ) = p ′ i ( ) for all i ∈ m . Now we define J := { j t | t ∈ J } . Wehave J ⊆ J , and therefore ( J ) = ( ( J ) ) ( J ) . Then p i ( ( J ) ) = p i (( ( J ) ) ( J ) ) = E. AICHINGER p ′ i ( ( J ) ( j ) , . . . , ( J ) ( j n ′ )) = p ′ i ( ( J ) ) = p ′ i ( ) = p i ( ( J ) ) = p i ( ), which com-pletes the induction step. (cid:3) We will need the following special case of [KS18, Theorem 3.1]. Let P k ( n )denote the set { I ⊆ n : | I | ≤ k } of subsets of n with at most k elements. Theorem 2 (cf. [KS18, Theorem 3.1]) . Let n ∈ N , let k ∈ N , let p be a prime,and let m ∈ N . Let ϕ : P k ( n ) → Z mp . Then there is U ⊆ n with | U | ≤ km ( p − such that X J ∈P k ( n ) ϕ ( J ) = X J ∈P k ( U ) ϕ ( J ) . Proof.
We denote the vector ϕ ( J ) by (( ϕ ( J )) , . . . , ( ϕ ( J )) m ), and we define m polynomial functions f , . . . , f m ∈ Z p [ x , . . . , x n ] by f i ( x , . . . x n ) := X J ∈P k ( n ) (cid:0) ( ϕ ( J )) i · Y j ∈ J x j (cid:1) . for i ∈ m . By Lemma 1, there is a subset U of n with | U | ≤ km ( p −
1) suchthat for all i ∈ m , we have f i ( ) = f i ( ( U ) ). Hence P J ∈P k ( n ) ( ϕ ( J )) i = f i ( ) = f i ( ( U ) ) = P J ∈P k ( n ) ,J ⊆ U ( ϕ ( J )) i = P J ∈P k ( U ) ( ϕ ( J )) i . (cid:3) Absorbing components
Let A be a set, let 0 A be an element of A , let B = ( B, + , − ,
0) be an abeliangroup, let n ∈ N , let f : A n → B , and let I ⊆ n . By Dep( f ) we denote the set { i ∈ n | f depends on its i th argument } . We say that f is absorbing in its j thargument if for all a = ( a (1) , . . . , a ( n )) ∈ A n with a ( j ) = 0 A we have f ( a ) = 0.In the sequel, we will denote 0 A simply by 0. We say that f is absorbing in I ifDep( f ) ⊆ I and for every i ∈ I , f is absorbing in its i th argument. Lemma 3.
Let A be a set, let be an element of A , let B = ( B, + , − , be anabelian group, let n ∈ N , and let f : A n → B . Then there is exactly one sequence ( f I ) I ⊆ n of functions from A n to B such that for each I ⊆ n , f I is absorbing in I and f = P I ⊆ n f I . Furthermore, each function f I lies in the subgroup F of B A n that is generated by the functions x f ( x ( I ) ) , where I ⊆ n .Proof. We first prove the existence of such a sequence. To this end, we define f I by recursion on | I | . We define f ∅ ( a ) := f (0 , . . . ,
0) and for I = ∅ , we let f I ( a ) := f ( a ( I ) ) − X J ⊂ I f J ( a ) . By induction on | I | , we see that Dep( f I ) ⊆ I and that f I lies in the subgroup F . We will now show that each f I is absorbing in I , and we again proceedby induction on | I | . Let i ∈ I , and let a ∈ A n be such that a ( i ) = 0. Wehave to show f I ( a ) = 0. We compute f I ( a ) = f ( a ( I ) ) − P J ⊂ I f J ( a ). By theinduction hypothesis, we have f J ( a ) = 0 for those J with i ∈ J . Hence f ( a ( I ) ) − YSTEMS OF EQUATIONS IN SUPERNILPOTENT ALGEBRAS 5 P J ⊂ I f J ( a ) = f ( a ( I ) ) − P J ⊆ I \{ i } f J ( a ), and because of a ( I ) = a ( I \{ i } ) , this isequal to f ( a ( I \{ i } ) ) − P J ⊆ I \{ i } f J ( a ) = f ( a ( I \{ i } ) ) − P J ⊂ I \{ i } f J ( a ) − f I \{ i } ( a ).By the definition of f I \{ i } , the last expression is equal to f I \{ i } ( a ) − f I \{ i } ( a ) = 0.This completes the induction proof; hence each f I is absorbing in I . In orderto show f = P I ⊆ n f I , we choose a ∈ A n and compute P I ⊆ n f I ( a ) = f n ( a ) + P I ⊂ n f I ( a ) = f ( a ( n ) ) − P J ⊂ n f J ( a ) + P I ⊂ n f I ( a ) = f ( a ). This completes theproof of the existence of such a sequence.For the uniqueness, assume that f = P I ⊆ n f I = P I ⊆ n g I and that for all I , f I and g I are absorbing in I . We show by induction on | I | that f I = g I . Let I := ∅ .First we notice that f (0 , . . . ,
0) = P J ⊆ n f J (0 , . . . ,
0) = P J ⊆ n g J (0 , . . . , f J and g J are absorbing, the summands with J = ∅ are 0, and thus f ∅ (0 , . . . ,
0) = P J ⊆ n f J (0 , . . . ,
0) = f (0 , . . . ,
0) = P J ⊆ n g J (0 , . . . ,
0) = g ∅ (0 , . . . , f ∅ and g ∅ are constant functions, they are equal. For the induction step, we as-sume | I | ≥
1. Let a ∈ A n . Then P J ⊆ n f J ( a ( I ) ) = P J ⊆ n g J ( a ( I ) ). Only the sum-mands with J ⊆ I can be nonzero, and therefore P J ⊆ I f J ( a ( I ) ) = P J ⊆ I g J ( a ( I ) ).By the induction hypothesis, f J = g J for J ⊂ I . Therefore, f I ( a ( I ) ) = g I ( a ( I ) ).Since f I and g I depend only on the arguments at positions in I , we obtain f I ( a ) = f I ( a ( I ) ) = g I ( a ( I ) ) = g I ( a ). Thus f I = g I . (cid:3) Actually, the component f I can be computed by f I ( a ) = P J ⊆ I ( − | I | + | J | f ( a ( J ) ). Definition 4.
Let A be a set, let 0 be an element of A , let B = ( B, + , − ,
0) bean abelian group, let n ∈ N , let f : A n → B , and let J ⊆ n . Then we call thesequence ( f I ) I ⊆ n such that for each I ⊆ n , f I is absorbing in I , and f = P I ⊆ n f I the absorbing decomposition of f , and f J the J -absorbing component of f . Wedefine the absorbing degree of f by adeg( f ) := max ( {− }∪{| J | : J ⊆ n and f J =0 } ). Theorem 5.
Let A be a set, let be an element of A , let p be a prime, let k ∈ N ,let n ∈ N , and let f , . . . , f m : A n → Z p . We assume that each f i is of absorbingdegree at most k . Let a ∈ A n . Then there is U with | U | ≤ km ( p − such thatfor all i ∈ m , we have f i ( a ) = f i ( a ( U ) ) .Proof. We define a function ϕ : P k ( n ) → Z mp by ϕ ( J ) := (( f ) J ( a ) , . . . , ( f m ) J ( a )),where for i ∈ m , (cid:0) ( f i ) J (cid:1) J ⊆ n is the absorbing decomposition of f i . Then Theo-rem 2 yields a subset U of n with | U | ≤ km ( p −
1) such that P J ∈P k ( n ) ϕ ( J ) = P J ∈P k ( U ) ϕ ( J ). Since ( f i ) J = 0 for all J with | J | > k , we have P J ∈P k ( n ) ϕ ( J ) = P J ∈P k ( n ) (( f ) J ( a ) , . . . , ( f m ) J ( a )) E. AICHINGER = P J ⊆ n (( f ) J ( a ) , . . . , ( f m ) J ( a )) = ( f ( a ) , . . . , f m ( a )) and X J ∈P k ( U ) ϕ ( J ) = X J ∈P k ( U ) (( f ) J ( a ) , . . . , ( f m ) J ( a ))= X J ⊆ U (( f ) J ( a ) , . . . , ( f m ) J ( a )) = X J ⊆ U (( f ) J ( a ( U ) ) , . . . , ( f m ) J ( a ( U ) ))= X J ⊆ n (( f ) J ( a ( U ) ) , . . . , ( f m ) J ( a ( U ) )) = ( f ( a ( U ) ) , . . . , f m ( a ( U ) )) . (cid:3) Polynomial mappings
In this section, we develop a property of polynomial mappings of finite su-pernilpotent algebras in congruence modular varieties. We call an algebra A =( A, + , − , , ( f i ) i ∈ S ) an expanded group if its reduct A + = ( A, + , − ,
0) is a group,an expanded abelian group if A + is an abelian group, and an expanded elementaryabelian group if A + is elementary abelian, meaning that A + is abelian and allits nonzero elements have the same prime order. For an algebra A and n, s ∈ N ,we define the set Pol n,s ( A ) of polynomial maps from A n to A s as the set of allmappings a ( f ( a ) , . . . , f s ( a )) with f , . . . , f s ∈ Pol n ( A ). Lemma 6.
Let k, n ∈ N , let A be a k -supernilpotent expanded abelian group, andlet f ∈ Pol n ( A ) . Then f is of absorbing degree at most k .Proof. Let J ⊆ n with | J | > k , and let f J be the J -absorbing component of f . Let m := | J | and let J = { i , . . . , i m } . Using Lemma 3, we obtain that the function g : A m → A defined by g ( a i , . . . , a i m ) := f J ( a ) for a ∈ A n is an absorbingfunction in Pol m ( A ). Hence [Aic18, Lemma 2.3] and the remark immediatelypreceding that Lemma yield that g is the zero function. Thus f J = 0. Hence theabsorbing degree of f is at most k . (cid:3) We first consider polynomial mappings of supernilpotent expanded elementaryabelian groups of prime power order.
Theorem 7.
Let k, n, s, α ∈ N , let p ∈ P , and let A be a k -supernilpotentexpanded elementary abelian group of order p α . Let F = ( f , . . . , f s ) ∈ Pol n,s ( A ) ,and let a ∈ A n . Then there is U ⊆ n with | U | ≤ ksα ( p − such that F ( a ) = F ( a ( U ) ) .Proof. We let π be a group isomorphism from ( A, + , − ,
0) to Z αp , and for a ∈ A ,we denote π ( a ) by ( π ( a ) , . . . , π α ( a )). For each r ∈ s and each β ∈ α , let f r,β : A n → Z p be defined by f r,β ( a ) = π β ( f r ( a )); hence f r,β ( a ) is the β thcomponent of f r ( a ). Since f r ∈ Pol n ( A ) and A is k -supernilpotent, Lemma 6implies that each of these f r,β is of absorbing degree at most k . Setting m := sα , YSTEMS OF EQUATIONS IN SUPERNILPOTENT ALGEBRAS 7
Theorem 5 yields U with | U | ≤ ksα ( p −
1) such that f r,β ( a ) = f r,β ( a ( U ) ) for all r ∈ s and β ∈ α . Then clearly F ( a ) = F ( a ( U ) ). (cid:3) We apply this result to polynomial mappings of direct products of finite su-pernilpotent expanded elementary abelian groups. For a vector a ∈ A n , we callthe number of its nonzero entries the weight of a ; formally, wt( a ) := |{ j ∈ n : a ( j ) = 0 }| . Theorem 8.
Let n, s, t, k , . . . , k t ∈ N . For each i ∈ t , let B i a k i -supernilpotentexpanded elementary abelian group with | B i | = p α i i , where p i is a prime and α i ∈ N . Let A := Q ti =1 B i , let F ∈ Pol n,s ( A ) , and let a ∈ A n . Then there is y ∈ A n with wt( y ) ≤ P ti =1 k i sα i ( p i − such that F ( a ) = F ( y ) .Proof. For i ∈ t , let ν i be the i th projection kernel. Applying Theorem 7 to A /ν i , which is isomorphic to B i , and b := a /ν i , we obtain U i ⊆ n with | U i | ≤ k i sα i ( p i −
1) such that F A /ν i ( b ( U i ) ) = F A /ν i ( b ). Lifting b ( U i ) to A , we obtain( x i, , . . . , x i,n ) ∈ A n such that ( x i, , . . . , x i,n ) /ν i = b ( U i ) and x i,j = 0 for j ∈ n \ U i .Now for every j ∈ n , we define y j ∈ A by the equations y j ≡ ν i x i,j for all i ∈ t. For each i ∈ t , we have F ( y , . . . , y n ) /ν i = F A /ν i ( x i, /ν i , . . . , x i,n /ν i ) = F A /ν i ( b ( U i ) ) = F A /ν i ( b ) = F A /ν i ( a /ν i ) = F ( a ) /ν i . Hence F ( y ) = F ( a ). For j ∈ n \ ( U ∪ · · · ∪ U t ), and for all i ∈ t , we have x i,j = 0, and therefore y j = 0. Hence the numberof nonzero entries in y is at most P ti =1 | U i | = P ti =1 k i sα i ( p i − (cid:3) Now we consider arbitrary finite supernilpotent algebras in congruence mod-ular varieties. In these algebras, we can introduce group operations preservingnilpotency using [Aic18].
Lemma 9.
Let µ ∈ N , let A = ( A, ( f i ) i ∈ S ) be a finite supernilpotent algebra ina congruence modular variety all of whose fundamental operations have arity atmost µ , and let z ∈ A . Let t ∈ N , let p , . . . , p t be different primes, and let α , . . . , α t ∈ N such that | A | = Q ti =1 p α i i . For i ∈ t , let k i := ( µ ( p α i i − α i − . Then there are operations + (binary), − (unary), (nullary) on A such that A ′ = ( A, + , − , , ( f i ) i ∈ S ) is isomorphic to a direct product Q ti =1 B ′ i , where each B ′ i is a k i -supernilpotent expanded elementary abelian group, and A ′ = z .Proof. Since the result is true for | A | = 1, we henceforth assume | A | ≥
2. By[Kea99], A is isomorphic to a direct product Q ti =1 B i of nilpotent algebras ofprime power order. We let ( π ( a ) , . . . , π t ( a )) denote the image of a of the under-lying isomorphism. As a finite supernilpotent algebra in a congruence modularvariety, A is nilpotent (cf. [Aic18, Lemma 2.4]) and therefore has a Mal’cevterm [FM87, Theorem 6.2]. We use [Aic18, Theorem 4.2] to expand each B i with operations + i and − i such that the expansion B ′ i is a nilpotent expandedelementary abelian group with zero element π i ( z ). By [Aic18, Theorem 1.2], B ′ i is k i -supernilpotent. (cid:3) E. AICHINGER
We note that the supernilpotency degree of A ′ may be strictly larger than thesupernilpotency degree of A .Combining these results, we obtain the following result on polynomial map-pings on arbitrary finite supernilpotent algebras in congruence modular varieties. Theorem 10.
Let µ ∈ N , let A be a finite supernilpotent algebra in a congruencemodular variety all of whose fundamental operations have arity at most µ . Let p , . . . , p t be distinct primes, and let α , . . . , α t ∈ N such that | A | = Q ti =1 p α i i .Let F ∈ Pol n,s ( A ) be a polynomial map from A n to A s , and let z ∈ A . Thenfor every a ∈ A n there is y ∈ A n such that F ( y ) = F ( a ) and |{ j ∈ n : y ( j ) = z }| ≤ s P ti =1 ( µ ( p α i i − α i − α i ( p i − ≤ sµ − | A | log ( µ )+log ( | A | ) log ( | A | ) ≤ s | A | log ( µ )+log ( | A | )+1 . Proof.
Let A ′ = Q ti =1 B ′ i be the expansion of A produced by Lemma 9, and foreach i ∈ t , let p i ∈ P and α i ∈ N be such that | B i | = p α i i . Clearly, F is a also apolynomial map of A ′ . Let k i = ( µ ( p α i i − α i − . Then Theorem 8 yields y ∈ A n such that |{ j ∈ n : y ( j ) = z }| ≤ P ti =1 k i sα i ( p i − α i ≤ log ( | A | ), we obtain t X i =1 k i sα i ( p i −
1) = s t X i =1 ( µ ( p α i i − α i − log ( | A | )( p i − ≤ s log ( | A | ) t X i =1 µ α i − ( p α i i ) α i − p α i i ≤ s log ( | A | ) t X i =1 µ log ( | A | ) − ( p α i i ) α i ≤ sµ log ( | A | ) − log ( | A | ) t X i =1 ( p α i i ) log ( | A | ) ≤ sµ log ( | A | ) − log ( | A | )( t X i =1 p α i i ) log ( | A | ) ≤ sµ log ( | A | ) − log ( | A | )( t Y i =1 p α i i ) log ( | A | ) ≤ sµ log ( | A | ) − log ( | A | ) | A | log ( | A | ) = sµ − | A | log ( µ )+log ( | A | ) log ( | A | ) ≤ s | A | log ( µ )+log ( | A | )+1 . (cid:3) Systems of equations
We will now explain how these results give a polynomial time algorithm forsolving systems of a fixed number of equations over the finite supernilpotentalgebra A . The size m of a system of polynomial equations is measured as thelength of the polynomial terms used to represent the system. For measuring the“running time” of our algorithm, we count the number of A -operations: eachsuch A -operation, may, for example, be done by looking up one value in theoperation tables defining A . YSTEMS OF EQUATIONS IN SUPERNILPOTENT ALGEBRAS 9
Theorem 11.
Let A be a finite supernilpotent algebra in a congruence modularvariety all of whose fundamental operations are of arity at most µ , and let s ∈ N .We consider the following algorithmic problem s - PolSysSat ( A ) : Given: s polynomial terms f , g , . . . , f s , g s over A . Asked:
Does the system f ≈ g , . . . , f s ≈ g s have a solutionin A ?Let m be the length of the input of this system, and let e := s | A | log ( µ )+log ( | A | )+1 + 1 . Then we can decide s - PolSysSat ( A ) using at most O ( m e − ) evaluations of allterms occuring in the system. Therefore, we have an algorithm that determineswhether a system of s polynomial equations over A has a solution using O ( m e ) many A -operations.Proof. Let n be the number of different variables that occur in the given sys-tem. We may assume that these variables are x , . . . , x n , and that our system is V si =1 f i ( x , . . . , x n ) ≈ g i ( x , . . . , x n ). We choose an element z ∈ A , and we willshow: if this system has a solution in a ∈ A n , then it has a solution in C := { y ∈ A n : |{ j ∈ n : y ( j ) = z }| ≤ e − } . For proving this claim, we first observe that A is a finite nilpotent algebra in acongruence modular variety, and it therefore has a Mal’cev term d . We considerthe polynomial map H = ( h , . . . , h s ), where h i ( x ) := d ( f i ( x ) , g i ( x ) , z ) for i ∈ s and x ∈ A n . Since a is a solution of the system, H ( a ) = ( z, z, . . . , z ). ByTheorem 10, there is y ∈ C such that H ( y ) = H ( a ). Then for every i ∈ s ,we have d ( f i ( y ) , g i ( y ) , z ) = z . By [FM87, Corollary 7.4], the function x d ( x, g i ( y ) , z ) is injective. Since d ( f i ( y ) , g i ( y ) , z ) = z = d ( g i ( y ) , g i ( y ) , z ), thisinjectivity implies that f i ( y ) = g i ( y ). Hence y is a solution that lies in C .The algorithm for solving the system now simply evaluates the system at allplaces in C ; if a solution is found, the answer is “yes”. If we find no solutioninside C , we answer “no”, and by the argument above, we know that in thiscase, the system has no solution inside A n at all.We now estimate the complexity of this procedure: There is a c ∈ N suchthat for all n ∈ N , | C | ≤ c n e − , hence we have to do O ( n e − ) evaluations ofall the terms f i , g i in the system. Such an evaluation can be done using at most O ( m ) many A -operations. Since the length of the input m is at least the numberof variables n occuring in it, this solves s - PolSysSat ( A ) using at most O ( m e )many A -operations. (cid:3) Circuit satisfiability
With every finite algebra A , [IK18] associates a number of computationalproblems that involve circuits whose gates are taken from the fundamental op-erations of A . One of these problems is SCsat ( A ). It takes as an input 2 s circuits f , g , . . . , f s , g s over A with n input variables, and asks whether thereis a a ∈ A n such that the evaluations at a satisfy f i ( a ) = g i ( a ) for all i ∈ s .For finite algebras in congruence modular varieties, [LZ06, Corollary 3.13] impliesthat SCsat ( A ) is in P when A is abelian, and NP-complete otherwise. However,if we restrict the number s of circuits, we obtain a different problem, which wecall s - SCsat ( A ) in the sequel. Obviously, 1- SCsat ( A ) is the circuit satisfiabil-ity problem called Csat ( A ) in [IK18]. The method used to prove Theorem 11immediately yields: Theorem 12.
Let A be a finite supernilpotent algebra in a congruence modularvariety, and let s ∈ N . Then s - SCsat ( A ) is in P . Hence a supernilpotent, but not abelian algebra A has s - SCsat ( A ) in P,whereas SCsat is NP-complete. In the converse direction, Theorem 9.1 from[IK18] has the following corollary.
Corollary 13.
Let A be a finite algebra from a congruence modular variety. If A has no homomorphic image A ′ such that - SCsat ( A ′ ) is NP -complete, then A is nilpotent.Proof. Suppose that A has a homomorphic image A ′ for which Csat ( A ′ ) isNP-complete. Then also 2- SCsat ( A ′ ) is NP-complete because an algorithmsolving 2- SCsat can be used to solve an instance ( ∃ a )( f ( a ) = g ( a )) of Csat ( A ′ )by solving 2- SCsat on the input ( ∃ a )( f ( a ) = g ( a ) & f ( a ) = g ( a )). Thusthe assumptions imply that for no homomorphic image A ′ of A , the problem Csat ( A ′ ) is NP-complete. Now by [IK18, Theorem 9.1], A is isomorphic to N × D , where N is nilpotent and D is a subdirect product of 2-element algebraseach of which is polynomially equivalent to a two element lattice. If | D | >
1, thenthere is a homomorphic image A of A such that A is polynomially equivalentto a two element lattice. By [GK11], 2- SCsat ( A ) is NP-complete, contradictingthe assumptions. Hence | D | = 1, and therefore A is nilpotent. (cid:3) Acknowledgements
The author thanks M. Kompatscher for dicussions on solving equations overnilpotent algebras. These discussions took place during a workshop organized byP. Aglian`o at the University of Siena in June 2018. The author also thanks A.F¨oldv´ari, C. Szab´o, M. Kompatscher, and S. Kreinecker for their comments onpreliminary versions of the manuscript.
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Institute for Algebra, Johannes Kepler University Linz, Linz, Austria
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