aa r X i v : . [ m a t h . A C ] S e p Irreducibility of integer-valued polynomials I
Devendra [email protected] of MathematicsIISER-TirupatiTirupati, Andhra PradeshIndia, 517507
Abstract
Let S ⊂ R be an arbitrary subset of a unique factorization domain R and K be the field of fractions of R . The ring of integer-valued polynomialsover S is the set Int( S, R ) = { f ∈ K [ x ] : f ( a ) ∈ R ∀ a ∈ S } . This articleis an effort to study the irreducibility of integer-valued polynomials overarbitrary subsets of a unique factorization domain. We give a methodto construct special kinds of sequences, which we call d -sequences. Wethen use these sequences to obtain a criteria for the irreducibility of thepolynomials in Int( S, R ) . In some special cases, we explicitly constructthese sequences and use these sequences to check the irreducibility ofsome polynomials in Int(
S, R ) . At the end, we suggest a generalization ofour results to an arbitrary subset of a Dedekind domain.
For a given subset S of a domain R the set of polynomialsInt( S, R ) = { f ∈ K [ x ] : f ( S ) ⊂ R } , where K is the field of fractions of R , forms a ring. This ring is termed as thering of integer-valued polynomials over S . A general reference for this topiccould be Cahen and Chabert [3] and some interesting results on the topic canbe found in [4] [5] [6] [8] and [16]. This ring is very rich in properties and ishelpful in constructing examples/counterexamples in commutative algebra. Inthe previous few decades this ring attracted the attention of several mathemati-cians and now the study of this ring has become a major field of specialization.In the case when S = R , we just write Int( R ) instead of Int( R, R ) . In ring theory, one of the most fascinating concepts is irreducibility. Theirreducibility of polynomials has a venerable history but in the case of the ringof integer-valued polynomials, irreducibility has not been explored that much.Only some methods are known so far and they are only for particular rings.1or the interested readers, we give a short summary of articles dealing with theirreducibility of integer-valued polynomials.In 2005, Chapman and McClain [7] gave a criteria for testing the irreducibil-ity of polynomials in Int(
S, R ) where R is a unique factorization domain. Perug-inelli [9] gave a computational method to test the irreducibility of polynomials inInt( Z ) for some special polynomials of Q [ x ]. Antoniou, Nakato and Rissner [1]introduced ‘table method’ to check the irreducibility of polynomials in Int( Z ) . For a summary of work on the irreducibility of integer-valued polynomials werefer to Prasad, Rajkumar and Reddy [12], where a whole section is devoted tothe irreducibility of integer-valued polynomials.The organization of the paper is as follows. In section 2, we present somepreliminaries and fix notations for the whole paper. In section 3, we introduce d -sequences and give some examples. In section 4, we obtain a criteria forthe irreducibility of polynomials in Int( S, R ) in the case when R is a uniquefactorization domain and S is an arbitrary subset. We give some examples toexplain how sometimes d -sequences can be very helpful and viable in testing theirreducibility of polynomials in Int( S, R ) . As per our knowledge there is no criteria known till date to test the irre-ducibility of polynomials in Int(
S, R ) when R is a Dedekind domain and S is anarbitrary subset of R . In section 5, we suggest a generalization of our results toget a criteria in this case for the first time. Finally, we show how sometimes ourresults remain valid for the ring of integer-valued polynomials over an arbitrarysubset of a domain. We start this section by fixing a few notations. Throughout the article R denotesa unique factorization domain (UFD) with the field of fractions K and S denotesan arbitrary subset of R . For a polynomial f ∈ K [ x ] , g denotes the uniquepolynomial in R [ x ] such that f = gd , where d ∈ R is also unique. Recall thatan element u of a ring A is said to be a unit if we can find an element v ∈ A such that uv = 1 . A non-zero non-unit element α of a ring A is said to be an irreducible element if it is not a product of two non-units. Equivalently, if α = α α for α , α ∈ A , then either α is a unit or α is a unit. For brevity, we justcall ‘irreducible’ instead of an irreducible element, where the ring automaticallycomes from the context.Given a subset S ⊂ R and a polynomial f = gd ∈ Int(
S, R ), consider thefollowing subset of
R T f = { f ( a ) = g ( a ) d : a ∈ S } .
2f each element of T f is a multiple of some non-unit element d ′ ∈ R , thenthe polynomial cannot be irreducible since we have the proper factorization f = d ′ . fd ′ in Int( S, R ) . In order to test the irreducibility of a polynomial f , we mustassume that each element of T f is not a multiple of some non-unit element of R. Such a polynomial is said to be ‘image primitive’ . Throughout the article,a polynomial in Int(
S, R ) refers to an image primitive polynomial in Int(
S, R ) . Also, for brevity, an irreducible polynomial refers to an irreducible polynomialin Int(
S, R ), where S and R automatically come from the context. We denotethe highest power of a prime ideal P dividing an ideal I by w P ( I ) . For instance, w (12) = 2 . d -sequences In this section, we construct special kinds of sequences called d -sequences . Beforeintroducing these kinds of sequences we need the notion of π -sequences . Weknow that the ideal generated by some irreducible element π ∈ R is always aprime ideal, hence the ring R ( π ) is a local ring.A given subset S ⊂ R can also be seen as a subset of the local ring R ( π ) for any prime ideal ( π ) ⊂ R. With this assumption we give the definition of π -sequences. Definition.
A sequence { u i } i ≥ of elements of S ⊂ R is said to be a π -sequenceif for each k > u k ∈ S satisfies ( x − u ) ... ( x − u k − )( u k − u ) ... ( u k − u k − ) ∈ Int(
S, R ( π ) ) . In this way we get a sequence of elements { u i } i ≥ in S with arbitrary u .These kinds of sequences were also studied by Bhargava [2] in a slightly differentway to construct his generalized factorials. With this definition in hand, wedefine d -sequences as follows. Definition.
For a given element d ∈ R , let π , π , . . . , π r be all the irreduciblesof R dividing d . Let for 1 ≤ j ≤ r , { u ij } i ≥ be a π j -sequence of S and π e kj j be ( u kj − u j ) . . . ( u kj − u k − j ) viewed as a member of the ring R ( π j ) . Then a d -sequence { x i } ≤ i ≤ k of S of length k is a solution to the following congruences x i ≡ u ij mod π e kj +1 j ∀ ≤ j ≤ r, (1)where 0 ≤ i ≤ k.
3y Chinese remainder theorem we get infinitely many solutions of Eq. (1).We fix a solution of Eq. (1) for each i and get a sequence a , a , . . . , a k of k + 1elements. Such a sequence may be inside S or may not be. We just call ‘a d -sequence’ if the subset S is clear from the context. This sequence is importantthroughout our study. Before proceeding it is apropos to give a few examplesof d -sequences. Example 3.1.
In the case when R = Z and S = a Z + b where a, b ∈ Z , thesequence b, a + b, . . . , ak + b is a d -sequence of length k for every d ∈ Z . This isbecause the sequence b, a + b, . . . is a p -sequence for every prime number p .In fact, any k + 1 consecutive terms of S form a d -sequence of length k forevery d ∈ Z . Example 3.2.
In the case when R = Z and S is the set of square numbersincluding zero, the first k + 1 consecutive terms of S starting from zero form a d -sequence of length k for every d ∈ Z . This can be shown by the same reasoningas in the previous example.Recall that, for a given subset S ⊂ R, the fixed divisor of a polynomial f ∈ R [ x ] over S is the greatest common divisor of the values taken by f over S .This quantity is denoted by d ( S, f ) . Thus, d ( S, f ) = gcd { f ( a ) : a ∈ S } . Classically, this quantity was applied to the problems of the ring of integer-valued polynomials only but recently mathematicians used this quantity to gen-eralize some number theoretic problems (see for instance, [11], [14] and [15])also. For some latest results on fixed divisors we refer to Semwal, Rajkumarand Reddy [13] and for a solid summary of literature on fixed divisors we highlyrecommend Prasad, Rajkumar and Reddy [12] (see [10] also).A sequence of distinct elements { a i } i ≥ of S is said to be a fixed divisorsequence (see Prasad, Rajkumar and Reddy [12]) if for every k > , ∃ l k ∈ Z , such that for every polynomial f of degree kd ( S, f ) = ( f ( a ) , f ( a ) , . . . , f ( a l k )) , and no proper subset of { a , a , . . . , a l k } determines the fixed divisor of all thedegree k polynomials. For instance, the sequence 0 , , , . . . is a fixed divisorsequence in Z with l k = k ∀ k > . Example 3.3.
Let S be a subset of R with a fixed divisor sequence { a i } i ≥ . Iffor every positive integer k, l k = k then a , a , . . . , a k is a d -sequence of length k for every d ∈ R since the sequence a , a , . . . is a π -sequence for all irreducible π ∈ R .In all the examples given so far, d -sequences always belong to the set. Nowwe give an example where this is not the case.4 xample 3.4. Let S be the set of prime numbers in Z and we wish to constructa 6-sequence of length four. We have 2 , , , ,
17 and 2 , , , ,
19 as a 2-sequenceand a 3-sequence respectively of length four. Now we have the following factor-ization (17 − − − −
7) = 2 . − − − −
5) = 3 . . The element 1575 is invertible in Z (2) and so is 15232 in Z (3) . Hence, a firstterm of a 6-sequence is a solution of the congruences x ≡ x ≡ . A solution to the above congruence is a = 290 . Similarly at the last (fifth)step we solve the congruences x ≡
17 (mod 32)and x ≡
19 (mod 9) , to get a solution a = 145 . The readers can compute the other terms to get290 , , , ,
145 as a 6-sequence of length four in which all the elementsare not members of S . Before coming to the main result, we prove an important lemma which is helpfulin proving our main result.
Lemma 4.1.
Let a , a , . . . , a k be a d -sequence of length k for some d ∈ R anda given positive integer k . Then, for any polynomial f ′ = g ′ d ′ ∈ K [ x ] , where d ′ | d , of degree k ′ ≤ k the following holds f ′ ∈ Int(
S, R ) ⇔ f ′ ( a i ) ∈ R ∀ ≤ i ≤ k ′ . Proof.
Observe that, for any π | d , w π (( a i − a )( a i − a ) . . . ( a i − a i − )) = w π (( b i − b )( b i − b ) . . . ( b i − b i − )) ∀ ≤ i ≤ k , where b , b , . . . is a π -sequencein S . Consider the following representation f ′ = g ′ d ′ = k ′ X i =0 c i ( x − a )( x − a ) ... ( x − a i − ) d ′ , (2)where c i ∈ R ∀ ≤ i ≤ k ′ . Now we have the following observation5 ′ ( a i ) ∈ R ∀ ≤ i ≤ k ′ ⇔ c i ( a i − a )( a i − a ) ... ( a i − a i − ) d ′ ∈ R ∀ ≤ i ≤ k ′ , ⇔ c i ( b i − b )( b i − b ) ... ( b i − b i − ) d ′ ∈ R ∀ ≤ i ≤ k ′ . By the definition of π -sequences the above holds iff for any arbitrary element α ∈ S, c i ( α − b )( α − b ) ... ( α − b i − ) d ′ ∈ R ∀ ≤ i ≤ k ′ , which is true iff c i ( α − a )( α − a ) ... ( α − a i − ) d ′ ∈ R ∀ ≤ i ≤ k ′ , for any arbitrary element α ∈ S. i.e., if and only if f ′ ( α ) ∈ R ∀ α ∈ S orequivalently f ′ ∈ Int(
S, R ) . Similar to the case of Z , square-free elements can be defined in any uniquefactorization domain R . When d is a square-free element of R, Lemma 4.1 canbe improved as follows.
Lemma 4.2.
Let f = gd ∈ K [ x ] be a polynomial of degree k and a , a , . . . , a k be a d -sequence, where d is a square-free element of R . Let s π be the numberof elements of S ⊂ R which are not congruent to each other (modulo π ) for anirreducible π , then f ∈ Int(
S, R ) ⇔ f ( a i ) ∈ R ∀ ≤ i ≤ min( s π , k ) , for every divisor π of d .Proof. By Lemma 4.1 we have f ∈ Int(
S, R ) ⇔ f ( a i ) ∈ R ∀ ≤ i ≤ k. If s π > k for a given irreducible π dividing d , then we are done. Hence weassume s π < k. Let S π be the set of elements of S which are not congruent toeach other (modulo π ) for a given irreducible π . Then the elements of S π forma π -sequence of S in any order. Hence, if a , a , . . . , a k is a d -sequence, thenthe first s π elements of this sequence are congruent to a unique element of S π , where s π is the cardinality of the set S π and d is a multiple of π . Observe thatfor any polynomial h ( x ) ∈ R [ x ] hπ ∈ Int(
S, R ) ⇔ hπ ( S π ) ⊂ R, ⇔ hπ ( a i ) ∈ R ∀ ≤ i ≤ s π . Also, if π and π ′ are two different irreducibles, then hπ and hπ ′ are members of Int( S, R ) ⇔ hππ ′ ∈ Int(
S, R ) .
6n particular, this argument can be applied to the polynomial gd . This com-pletes the proof.Sometimes this lemma may reduce so much calculation. For instance, seeEx. (4.5). Now we prove our main theorem.
Theorem 4.3.
Let f = gd ∈ Int(
S, R ) be a polynomial of degree k and a , a , . . . , a k be a d -sequence. Then f is irreducible iff the following holds:for any factorization g = g g and a divisor π of d such that e k is themaximum integer satisfying π e k | g ( a i ) ∀ ≤ i ≤ deg( g ) , there exists aninteger j satisfying ≤ j ≤ deg( g ) and w π ( dπ ek ) ∤ g ( a j ) . Proof.
For a given polynomial f = gd ∈ Int(
S, R ), suppose for every factorization g = g g there exists a divisor π of d satisfying π e k | g ( a i ) ∀ ≤ i ≤ deg( g )and π w π ( d ) − e k ∤ g ( a j ) for some non-negative integer j ≤ deg( g ) . Let us assumecontrary that f is reducible. Hence, there exists a factorization f = h d h d , such that h d and h d are members of Int( S, R ) . If for a divisor π of d , w π ( d ) = π e k = w π ( d ) , then this is contradiction to the assumption since π | h ( a ) ∀ a ∈ R. Similarly, w π ( d ) cannot be π . Hence we assume that w π ( d ) is a properdivisor of w π ( d ) . In this case by assumption π w π ( d ) − e k ∤ h ( a j ) for some positiveinteger j satisfying j ≤ deg( h ) . By Lemma 4.1 it follows that h d cannot bea member of Int( S, R ) , which is again a contradiction. Hence, the polynomialmust be irreducible.Now we assume that f = gd ∈ Int(
S, R ) is irreducible. For any factorization g = g g we can find suitable d and d such that f = h d h d , where h d is a member of Int( S, R ) and h d is not. Since h d does not belongto Int( S, R ) , hence by Lemma 4.1 there exists a divisor π of d , such that w π ( d ) does not divide h ( a i ) for some 0 ≤ i ≤ deg( h ) . Clearly w π ( dd ) divides h ( a j ) ∀ ≤ j ≤ deg( h ) completing the proof.We give some examples to illustrate our theorem. Example 4.4.
Let us check the irreducibility of the polynomial f = 19 ( x + 4 x − x + 20 x + 4 x + 24 x + 27)in Int( Z ) . In this case we have only the following way of factorization f = 19 ( x − x + 2 x + 3)( x + 6 x + 2 x + 9) . , , , d -sequence for any integer d of length three. Hence, we check the values of onepolynomial at these points. Let f = x +6 x +2 x +9 , then one is the maximumpositive integer such that 3 | f ( i ) for i = 0 , , , . Taking f = x − x +2 x +3,it can be seen that 3 − does not divide f (1) = 4. Hence, the polynomial isirreducible.Sometimes if the factorization is known then it may be possible to predictthe irreducibility of an integer-valued polynomial with a minuscule amount ofinformation. We give an example to illustrate this. Example 4.5.
Let us test the irreducibility of the polynomial f = 16 ( x − x + 205 x − x + 3195 x − x + 6247 x − x + 720 x +18 x − Z ) with g = x − x + 42 x − x + 26 x − g = x − x +42 x − x + 30 x + 3 as known polynomials such that f = g g . We can start with the prime 3 since g (0) = 2 is not a multiple of three.In view of Lemma 4.2, we need to find a non-negative integer 0 ≤ j ≤ ≤ j ≤ ∤ g ( j ) . Observe that 3 ∤ g (1), hence f is irreducible.In the above example we determined the irreducibility by merely checkingat three points. This could be the case even if the degree of polynomial is veryhigh. In such cases our method becomes very easy and practical. In the previous sections we tested the irreducibility of a polynomial f = gd ∈ Int(
S, D ) by using the unique factorization of the element d ∈ R . In this sec-tion, we suggest a generalization of the Theorem 4.3 for some special domains.Assume the ideal generated by d in a domain D factors uniquely as a product ofprime ideals. Then we can use the similar reasoning to get a d -sequence in thissetting as well. Recall that each ideal in a Dedekind domain factors uniquely asa product of prime ideals. Hence, we can generalize Theorem 4.3 to an arbitrarysubset S of a Dedekind domain D . For the sake of completeness we state theresult (whose rigorous proof will be supplied in one of the subsequent articles)which can be proved by using essentially the same technique. Theorem 5.1.
Let S be an arbitrary subset of a Dedekind domain D and f = gd ∈ Int(
S, D ) be a polynomial of degree k. If a , a , . . . , a k is a d -sequence, then f is irreducible iff the following holds:for any factorization g = g g and a prime ideal P dividing d such that e k is the maximum integer satisfying P e k | g ( a i ) ∀ ≤ i ≤ deg( g ) , there existsan integer j satisfying ≤ j ≤ deg( g ) and w P ( dP ek ) ∤ g ( a j ) . S of a domain D , Theorem 4.3 can be applied veryeasily to test the irreducibility of any polynomial f = gd ∈ Int(
S, D ), if d has aunique factorization (into irreducibles or prime ideals). For instance, when D is a quotient of a Dedekind domain, we can rely on the Theorem 4.3. In thecase when the ideal generated by d factors uniquely into prime ideals, we mustassume that the underlying domain is Noetherian since we can factor d as afinite product of irreducibles in this case. In conclusion, when ‘ d ’ has a uniquefactorization, Theorem 4.3 remains valid for an arbitrary subset S of a domain D . For instance, we have the following corollary Corollary 5.2.
Let S be an arbitrary subset of a domain D and f = gd ∈ Int(
S, D ) be a polynomial of degree k where d is an irreducible element. Let a , a , . . . , a k be a d -sequence, then f is irreducible iff the following holds:for any factorization g = g g there exist integers i, j satisfying ≤ i ≤ deg( g ) and ≤ j ≤ deg( g ) such that w d ( g ( a i )) = w d ( g ( a j )) = d . In practice, whenever d is irreducible (or ‘square-free’) Lemma 4.2 can beused to get more sharper results. Hence, results similar to Corollary 5.2 can beimproved further.In conclusion, we would like to emphasize that this article is an initial stepto test the irreducibility of integer-valued polynomials over arbitrary subsets ofa domain. We believe that concepts similar to d -sequences would be helpful intesting the irreducibility of integer-valued polynomials over arbitrary subsets ofa domain in near future. This seems a very fertile area of research, which hasnot been explored so far. Acknowledgement
We thank Dr. A. Satyanarayana Reddy and Dr. Krishnan Rajkumar for theirhelpful suggestions. We also thank the referee for a careful reading of themanuscript and giving several suggestions to correct misprints and improvereadability.
References [1] Austin Antoniou, Sarah Nakato, and Roswitha Rissner. Irreducible poly-nomials in Int( Z ). In ITM Web of Conferences , volume 20, page 01004.EDP Sciences, 2018.[2] Manjul Bhargava. P -orderings and polynomial functions on arbitrary sub-sets of Dedekind rings. J. Reine Angew. Math. , 490:101–127, 1997.[3] Paul-Jean Cahen and Jean-Luc Chabert.
Integer-valued polynomials , vol-ume 48 of
Mathematical Surveys and Monographs . American MathematicalSociety, Providence, RI, 1997. 94] Paul-Jean Cahen and Jean-Luc Chabert. What you should know aboutinteger-valued polynomials.
Amer. Math. Monthly , 123(4):311–337, 2016.[5] Jean-Luc Chabert. Integer-valued polynomials: looking for regular bases(a survey). In
Commutative algebra , pages 83–111. Springer, New York,2014.[6] Jean-Luc Chabert and Paul-Jean Cahen. Old problems and new questionsaround integer-valued polynomials and factorial sequences. In
Multiplica-tive ideal theory in commutative algebra , pages 89–108. Springer, New York,2006.[7] Scott T. Chapman and Barbara A. McClain. Irreducible polynomials andfull elasticity in rings of integer-valued polynomials.
J. Algebra , 293(2):595–610, 2005.[8] Sophie Frisch. Integer-valued polynomials on algebras: a survey.
Actes desrencontres du CIRM , 2(2):27–32, 2010.[9] Giulio Peruginelli. Factorization of integer-valued polynomials with square-free denominator.
Comm. Algebra , 43(1):197–211, 2015.[10] Devendra Prasad.
Fixed Divisors and Generalized Factorials . PhD thesis,Shiv Nadar University, Greater Noida, 2019.[11] Devendra Prasad. A generalization of selfridge’s question. arXiv preprint ,2019.[12] Devendra Prasad, Krishnan Rajkumar, and A Satyanarayana Reddy. Asurvey on fixed divisors.
Confluentes Mathematici , 11(1):29–52, 2019.[13] Krishnan Rajkumar, A Satyanarayana Reddy, and Devendra PrasadSemwal. Fixed divisor of a multivariate polynomial and generalized fac-torials in several variables.
J. Korean Math. Soc. , 55(6):1305–1320, 2018.[14] Marian Vˆajˆaitu. An inequality involving the degree of an algebraic set.
Rev. Roumaine Math. Pures Appl. , 43(3-4):451–455, 1998.[15] Marian Vˆajˆaitu and Alexandru Zaharescu. A finiteness theorem for a classof exponential congruences.
Proc. Amer. Math. Soc. , 127(8):2225–2232,1999.[16] Nicholas J. Werner. Integer-valued polynomials on algebras: a survey ofrecent results and open questions. In