Erdős-Burgess constant of the multiplicative semigroup of the quotient ring off q [x]

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1 Erdős-Burgess constant of the multplcatve semgroup of the quotent rng off q [x] arxv: v1 [math.co] 6 May 2018 Jun Hao a Haol Wang b Lzhen Zhang a a Department of Mathematcs, Tanjn Polytechnc Unversty, Tanjn, , P. R. Chna b College of Computer and Informaton Engneerng Tanjn Normal Unversty, Tanjn, , P. R. Chna Abstract Let S be a semgroup endowed wth a bnary assocatve operaton. An element e ofss sad to be dempotent f e e=e. The Erdős-Burgess constant of the semgroup S s defned as the smallestl N { } such that any sequence T of terms from S and of length l contans a nonempty subsequence the product of whose terms, n some order, s dempotent. Let q be a prme power, and letf q [x] be the rng of polynomals over the fnte feldf q. Let R=F q [x] K be a quotent rng off q [x] modulo any deal K. We gave a sharp lower bound of the Erdős-Burgess constant of the multplcatve semgroup of the rng R, n partcular, we determned the Erdős-Burgess constant n the case when K s factored nto ether a power of some prme deal or a product of some parwse dstnct prme deals nf q [x]. Key Words: Erdős-Burgess constant; Davenport constant; Multplcatve semgroups; Polynomal rngs Correspondng author s Emal: bjpeuwanghaol@163.com 1

2 1 Introducton LetSbe a nonempty semgroup, endowed wth a bnary assocatve operaton ons, and denote by E(S) the set of dempotents ofs, where x S s sad to be an dempotent f x x= x. P. Erdős posed a queston on dempotent to D.A. Burgess as follows. IfSs a fnte nonempty semgroup of order n, does anys-valued sequence T of length n contan a nonempty subsequence the product of whose terms, n some order, s an dempotent? In 1969, Burgess [1] answered ths queston n the case when S s commutatve or contans only one dempotent. Ths queston was completely affrmed by D.W.H. Gllam, T.E. Hall and N.H. Wllams, who proved the followng stronger result: Theorem A. ([2]) Let S be a fnte nonempty semgroup. Any S-valued sequence of length S E(S) +1 contans one or more terms whose product (n the order nduced from the sequence T) s an dempotent; In addton, the bound S E(S) +1 s optmal. G.Q. Wang [6] generalzed the result n the context of arbtrary semgroups (ncludng both fnte and nfnte semgroups). Theorem B. ([6], Theorem 1.1) LetSbe a nonempty semgroup such that S\ E(S) s fnte. Any sequence T of terms fromsof length T S\E(S) +1 contans one or more terms whose product (n the order nduced from the sequence T) s an dempotent. Moreover, Wang [6] characterzed the structure of extremal sequences of length S\ E(S) and remarked that although the bound S\ E(S) + 1 s optmal for general semgroups S, the better bound can be obtaned for specfc classes of semgroups. Hence, Wang proposed two combnatoral addtve constants assocated wth dempotents. Defnton C. ([6], Defnton 4.1) LetSbe a nonempty semgroup and T a sequence of terms froms. We say that T s an dempotent-product sequence f ts terms can be ordered so that ther product s an dempotent element of S. We call T (weakly) dempotent-product free f T contans no nonempty dempotent-product subsequence, and we call T strongly dempotentproduct free f T contans no nonempty subsequence the product whose terms, n the order nduced from the sequence T, s an dempotent. We defne I(S), whch s called the Erdős- Burgess constant of the semgroups, to be the leastl N { } such that every sequence T of terms fromsof length at leastls not (weakly) dempotent-product free, and we defne SI(S), whch s called the strong Erdős-Burgess constant of the semgroups, to be the least l N { } such that every sequence T of terms fromsof length at leastls not strongly 2

3 dempotent-product free. Formally, one can also defne I(S) = sup{ T + 1 : T takes all dempotent-product free sequences of terms from S} and SI(S)=sup{ T +1 : T takes every strongly dempotent-product free sequences of terms froms}. Very recently, Wang [7] made a comprehensve study of the Erdős-Burgess constant for the drect product of arbtrarly many of cyclc semgroups. As ponted out n [6], the Erdős- Burgess constant reduces to be the famous Davenport constant n the case when the underlyng semgroup happens to be a fnte abelan group. So we need to ntroduce the defnton of Davenport constant below. Let G be an addtve fnte abelan group. A sequence T of terms from G s called a zerosum sequence f the sum of all terms of T equals to zero, the dentty element of G. We call T a zero-sum free sequence f T contans no nonempty zero-sum subsequence. The Davenport constant D(G) of G s defned to be the smallest postve ntegerl such that, every sequence T of terms from G and of length at leastls not zero-sum free. In 2008, Wang and Gao [8] extended the defnton of the Davenport constant to commutatve semgroups as follows. Defnton D. LetSbe a fnte commutatve semgroup. Let T be a sequence of terms from the semgroups. We call T reducble f T contans a proper subsequence T (T T) such that the sum of all terms of T equals the sum of all terms of T. Defne the Davenport constant of the semgroups, denoted D(S), to be the smallestl N { } such that every sequence T of length at leastlof terms fromss reducble. Several related addtve results on Davenport constant for semgroups were obtaned (see [4], [5], [9], [10]). For any commutatve rng R, we denotes R to be the multplcatve semgroup of the rng R and U(S R ) to be the group of unts of the semgroups R. Wth respect to the Davenport constant for the multplcatve semgroup assocated wth polynomal rngsf q [x], Wang obtaned the followng result. Theorem E. ([4]) Let q>2 be a prme power, and letf q [x] be the rng of polynomals over the fnte feldf q. Let R be a quotent rng off q [x] wth 0 R F q [x]. Then D(S R )=D(U(S R )). G.Q. Wang [4] proposed to determne D(S R ) D(U(S R )) for the remanng case that R s a quotent rng off 2 [x]. 3

4 L.Z. Zhang, H.L. Wang and Y.K. Qu partally answered Wang s queston and obtaned the followng. Theorem F. ([10]) LetF 2 [x] be the rng of polynomals over the fnte feldf 2, and let R= F 2 [x] ( f ) be a quotent rng off 2 [x], where f F 2 [x] and 0 R F 2 [x]. Then D(U(S R )) D(S R ) D(U(S R ))+δ f, where δ f = 0 f gcd(x (x+1 F2 ), f )=1 F2 ; 1 f gcd(x (x+1 F2 ), f ) {x, x+1 F2 }; 2 f gcd(x (x+1 F2 ), f )= x (x+1 F2 ). Motvated by the above addtve research on semgroups, n ths manuscrpt we make a study of the Erdős-Burgess constant on the multplcatve semgroups of the quotent rngs of the polynomal rngsf q [x] and obtan the followng result. Theorem 1.1. Let q be a prme power, and letf q [x] be the rng of polynomals over the fnte feldf q. Let R=F q [x] K be a quotent rng off q [x] modulo any deal K. Then I(S R ) D(U(S R ))+Ω(K) ω(k), whereω(k) s the number of the prme deals (repettons are counted) andω(k) the number of dstnct prme deals n the factorzaton when K s factored nto a product of prme deals. Moreover, the equalty holds for the case when K s factored nto ether a power of some prme deal or a product of some parwse dstnct prme deals nf q [x]. 2 Notaton LetSbe a fnte commutatve semgroup. The operaton onswll be denoted by+nstead of. The dentty element ofs, denoted 0 S (f exsts), s the unque element e ofssuch that e+a=afor every a S. IfShas an dentty element 0 S, let U(S)={a S:a+a = 0 S for some a S} be the group of unts ofs. The sequence T of terms from the semgroupsss denoted by T= a 1 a 2... a l = a va(t), 4 a S

5 where v a (T) denotes the multplcty of the element a occurrng n the sequence T. By we denote the operaton to jon sequences. Let T 1, T 2 be two sequences of terms from the semgroups S. We call T 2 a subsequence of T 1 f v a (T 2 ) v a (T 1 ) for every element a S, denoted by T 2 T 1. In partcular, f T 2 T 1, we call T 2 a proper subsequence of T 1, and wrte T 3 = T 1 T 1 2 to mean the unque subsequence of T 1 wth T 2 T 3 = T 1. Let σ(t)=a 1 + a 2 + +a l be the sum of all terms n the sequence T. Let q be a prme power, and letf q [x] be the rng of polynomals over the fnte feldf q. Let R=F q [x] K be the quotent rng off q [x] modulo the deal K, and lets R be the multplcatve semgroup of the rng R. Take an arbtrary element a S R. Letθ a F q [x] be the unque polynomal correspondng to the element a wth the least degree, thus,θ a =θ a + K s the correspondng form of a n the quotent rng R. In what follows, snce we deal wth only the multplcatve semgroups R whch happens to be commutatve, we shall use the termnology dempotent-sum and dempotent-sum free n place of dempotent-product and dempotent-product free, respectvely. 3 Proof of Theorem 1.1 Lemma 3.1. Let q be a prme power, and letf q [x] be the rng of polynomals over the fnte feld F q. Let f be a polynomal nf q [x] and let f= p n 1 1 pn 2 2 pn r r, where r 1, n 1, n 2,..., n r 1, and p 1, p 2,..., p r are parwse non-assocate rreducble polynomals nf q [x]. Let R=F q [x] ( f ) be the quotent rng off q [x] modulo the deal ( f ). Let a be an element n the semgroup of S R. Then a s dempotent f and only fθ a 0 Fq [1, r]. (mod p n ) orθ a 1 Fq (mod p n ) for every 5

6 Proof. Suppose that a s dempotent. Thenθ a θ a θ a (mod f ), whch mples thatθ a (θ a 1 Fq ) 0 Fq (mod p n ) for all [1, r]. Snce gcd(θ a,θ a 1 Fq )=1 Fq, t follows that for every [1, r], p n dvdesθ a or p n dvdesθ a 1 Fq, that s,θ a 0 Fq (mod p n ) orθ a 1 Fq (mod p n ). Then the necessty holds. The suffcency holds smlarly. We remark that n Theorem 1.1, f K =F q [x], then R s a trval zero rng and I(S R )= D(S R )=1 andω(k)=ω(k)=0, and f K s the zero deal then R=F q [x] and I(S R ) s nfnte snce any sequence T of any length such thatθ a s a nonconstant polynomal for all terms a of T s an dempotent-sum free sequence, and thus, the concluson holds trvally for both cases. Hence, we shall only consder the case that K s nonzero proper deal off q [x] n what follows. Proof of Theorem 1.1. Note thatf q [x] s a prncpal deal doman. Say s the prncpal deal generated by a polynomal f F q [x], where K= ( f ) (1) f= p n 1 1 pn 2 2 pn r r, (2) where p 1, p 2,..., p r are parwse non-assocate rreducble polynomals off q [x] and n 1 for all [1, r], equvalently, K= P n 1 1 Pn 2 2 Pn r r s the factorzaton of the deal K nto the product of the powers of dstnct prme deals P 1 = (p 1 ), P 2 = (p 2 ),..., P r = (p r ). Observe that Ω(K)= r n (3) =1 and ω(k)=r. (4) Take a zero-sum free sequence V of terms from the group U(S R ) of length D(U(S R )) 1. Take b S R such thatθ b = p for each [1, r]. Now we show that the sequence V b n 1 =1 s an dempotent-sum free sequence ns R. Suppose to the contrary that V b n 1 contans a =1 nonempty subsequence W, say W= V b β, such thatσ(w) s dempotent, where V s a =1 subsequence of V and β [0, n 1] for all [1, r]. 6

7 It follows that θ σ(w) =θ σ(v )θ σ( =1 =θ b β σ(v ) )p β 1 1 pβ r r. (5) If r β = 0, then W= V s a nonempty subsequence of V. Snce V s zero-sum free n the =1 group of U(S R ), we derve thatσ(w) s a nondentty element of the group U(S R ), and thus, σ(w) s not dempotent, a contradcton. Otherwse,β j > 0 for some j [1, r], say Snce gcd(θ σ(v ), p 1 )=1 Fq, t follows from (5) that gcd(θ σ(w), p n 1 β 1 [1, n 1 1]. (6) 1 )= pβ 1 1. Combned wth (6), we have thatθ σ(w) 0 Fq (mod p n 1 1 ) andθ σ(w) 1 Fq (mod p n 1 1 ). By Lemma 3.1, we conclude that σ(w) s not dempotent, a contradcton. Ths proves that the sequence V s dempotent-sum free ns R. Combned wth (3) and (4), we have that I(S R ) V =1 b n 1 +1=( V +1)+ =1 b n 1 r (n 1)=D(U(S R ))+Ω(K) ω(k). (7) =1 Now we assume that K s factored nto ether a power of some prme deal or a product of some parwse dstnct prme deals nf q [x],.e., ether r=1 or n 1 = =n r = 1 n (2). It remans to show the equalty I(S R )=D(U(S R ))+Ω(K) ω(k) holds. We dstngush two cases. Case 1. r=1n (2),.e., f= p n 1 1. Take an arbtrary sequence T of length T =D(U(S R ))+n 1 1=D(U(S R ))+Ω(K) ω(k). Let T 1 = a and T 2 = TT1 1. Note that all terms of T 2 are from U(S R ). By the a T θ a 0 (mod p 1 ) Pgeonhole Prncple, we see that ether T 1 n 1 or T 2 D(U(S R )). It follows that ether θ σ(t1 ) 0 Fq (mod p n 1 1 ), or T 2 contans a nonempty subsequence T 2 such thatσ(t 2 ) s the dentty element of the group U(S R ). By Lemma 3.1, the sequence T s not dempotent-sum free, whch mples that I(S R ) D(U(S R ))+Ω(K) ω(k). Combned wth (7), we have that I(S R )=D(U(S R ))+Ω(K) ω(k). Case 2. n 1 = =n r = 1 n (2),.e., f= p 1 p 2 p r. Then Ω(K)=ω(K)=r. (8) 7

8 Take an arbtrary sequence T of length T =D(U(S R )). For any term a of T, let ã S R be such that for each [1, r], θã 1 Fq (mod p ) fθ a 0 Fq (mod p ); θ a (mod p ) otherwse. (9) Note that ã U(S R ). Let T = ã. Then T s a sequence of terms from the group U(S R ) wth length T = T = a T D(U(S R )). It follows that there exsts a nonempty subsequence W of T such thatσ( ã) s the a W dentty element of the group U(S R ),.e.,θ σ( ã) 1 Fq a W (mod p ) for each [1, r]. By (9), we derve thatθ σ(w) 0 Fq (mod p ) orθ σ(w) 1 Fq (mod p ) for each [1, r]. By Lemma 3.1, we conclude thatσ(w) s dempotent. Combned wth (8), we have that I(S R ) D(U(S R ))= D(U(S R ))+Ω(K) ω(k). It follows from (7) that I(S R )=D(U(S R ))+Ω(K) ω(k), completng the proof. We close ths paper wth the followng conjecture. Conjecture 3.2. Let q>2 be a prme power, and letf q [x] be the rng of polynomals over the fnte feldf q. Let R=F q [x] K be a quotent rng off q [x] modulo any nonzero proper deal K. Then I(S R )=D(U(S R ))+Ω(K) ω(k). Acknowledgements Ths work s supported by NSFC (grant no , ). References [1] D.A. Burgess, A problem on sem-groups, Studa Sc. Math. Hungar., 4 (1969) [2] D.W.H. Gllam, T.E. Hall and N.H. Wllams, On fnte semgroups and dempotents, Bull. Lond. Math. Soc., 4 (1972) [3] K. Rogers, A Combnatoral problem n Abelan groups, Proc. Cambrdge Phl. Soc., 59 (1963) [4] G.Q. Wang, Davenport constant for semgroups II, J. Number Theory, 153 (2015)

9 [5] G.Q. Wang, Addtvely rreducble sequences n commutatve semgroups, J. Combn. Theory Ser. A, 152 (2017) [6] G.Q. Wang, Structure of the largest dempotent-free sequences n fnte semgroups, arxv: [7] G.Q. Wang, Erdős-Burgess constant of the drect product of cyclc semgroups, arxv: [8] G.Q. Wang and W.D. Gao, Davenport constant for semgroups, Semgroup Forum, 76 (2008) [9] G.Q. Wang and W.D. Gao, Davenport constant of the multplcatve semgroup of the rng Z n1 Z nr, arxv: [10] L.Z. Zhang, H.L. Wang and Y.K Qu, A problem of Wang on Davenport constant for the multplcatve semgroup of the quotent rng off 2 [x], Colloq. Math., 148 (2017)