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Factoring Polynomials in the Ring of Formal Power Series Over Z

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Factoring Polynomials in the Ring of Formal Power Series Over Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B. GIL, AND MICHAEL WEINER Abstract. We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over Z[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers. 1. Introduction In this paper we consider polynomials with integer coefficients and discuss their factorization as elements of Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility in Z[[x]]. For polynomials of degree two or three these conditions are also necessary. If the constant term of the polynomial is not a prime power, the reducibility discussion is straightforward. We briefly address these cases in Section 2. On the other hand, if the constant term of the polynomial is a nontrivial prime power, say pn with p prime and n ≥ 2, the question of reducibility in Z[[x]] leads in some cases (when n is no greater than twice the p-adic valuation of the linear coefficient) to the discussion of p-adic integers. In this context, our main result is that, if such a polynomial has a root in Zp whose p-adic valuation is positive, then it is reducible in Z[[x]]. This particular case is presented in Section 3. Our proofs are constructive and provide explicit factorization algorithms. All the needed material regarding the ring Zp of p-adic integers can be found in [5, 6]. It is important to note that irreducible elements in Z[x] and in Z[[x]] are, in general, unrelated. For instance, 6+x+x2 is irreducible in Z[x] but can be factored in Z[[x]], while 2 + 7x + 3x2 is irreducible in Z[[x]] but equals (2 + x)(1 + 3x) as a polynomial. Observe that the latter is not a proper factorization in Z[[x]] since 1 + 3x is an invertible element. More examples and a few remarks concerning the hypotheses of our main theorem are given in Section 4. A first study of the factorization theory of quadratic polynomials in Z[[x]] was presented in [3]. In that paper we established necessary and sufficient conditions for a quadratic polynomial to be irreducible in Z[[x]]. In particular, for polynomials whose constant term is a power of a prime p, we showed in [3] that reducibility in Z[[x]] is related to reducibility in Zp[x]. Some of the basic ideas from [2] and [3] are summarized in Section 2. 2010 Mathematics Subject Classification. 13P05;13F25,11Y05,11D88. Key words and phrases. Factorization, integral polynomials, power series, p-adic numbers. D. Birmajer would like to acknowledge the support and hospitality of the Universidad Nacional de San Luis, San Luis, Argentina. 1 2 DANIEL BIRMAJER, JUAN B. GIL, AND MICHAEL WEINER The results of this paper contain and expand those obtained in [3] to polynomials in Z[x] of arbitrary degree. The special case of cubic polynomials is explicitly and fully discussed in Section 4. Recently, J. Elliott posted a preprint [4] that discusses the irreducibility and factoring of polynomials in the ring of formal power series over a principal ideal domain. His methods are different from ours and his results rely on a generalization of the p-adic Weierstrass preparation theorem. Finally, we would like to acknowledge our correspondence with J.-P. B´ezivin.We appreciate his initiative to share with us an early draft of his unpublished work. We also thank the IJNT referee for all the useful suggestions. 2. Preliminaries We start by reviewing some of the basic properties of the factorization theory of power series over Z. For a more extensive discussion, we refer the reader to [2]. d Let f(x) = f0 + f1x + ··· + fdx be a polynomial with integer coefficients. It is easy to check that f(x) is invertible in Z[[x]] if and only if f0 = ±1. In particular, the reducibility of f(x) is equivalent to the reducibility of −f(x). Also observe that if f0 = 0, then f(x) is either an associate of x or f(x) has x as a proper factor. For this reason, it suffices to look at the cases when f0 > 0. We say that f(x) is reducible in Z[[x]] if there exist power series, 1 1 X k X k A(x) = akx and B(x) = bkx with ak; bk 2 Z; k=0 k=0 such that a0 6= ±1, b0 6= ±1, and f(x) = A(x)B(x). A proof of the following basic proposition can be found in [2]. Proposition 2.1. Let f(x) be a non-invertible polynomial in Z[[x]] with f0 > 0. (a) If f0 is prime, then f(x) is irreducible in Z[[x]]. (b) If f0 is not a prime power, then f(x) is reducible in Z[[x]]. n (c) If f0 = p with p prime, n ≥ 2, and p 6 j f1, then f(x) is irreducible in Z[[x]]. As an immediate consequence, we have: Proposition 2.2 (Linear polynomials). A polynomial n f(x) = p + f1x with n ≥ 2 is reducible in Z[[x]] if and only if p j f1. It remains to examine the reducibility of nonlinear polynomials of the form n m 2 d f(x) = p + p γ1x + γ2x + ··· + γdx ; γd 6= 0; (2.3) with p prime, n ≥ 2, m ≥ 1, gcd(p; γ1) = 1 or γ1 = 0, and gcd(p; γ2; : : : ; γd) = 1. As it turns out, the reducibility of such polynomials depends on the relation between the parameters n and m in (2.3) and on their factorization properties as elements of Zp[x]. Accordingly, we divide our study into two cases: The case when n > 2m, see Proposition 2.5, and the case when n ≤ 2m, which is more involved and will be investigated in the next section. Remark 2.4. If a polynomial of the form (2.3) has no linear term, i.e. if γ1 = 0, we can assume m as large as needed and can therefore discuss its reducibility as for the case when n ≤ 2m. FACTORING POLYNOMIALS IN Z[[x]] 3 Proposition 2.5. If n > 2m and gcd(p; γ1) = 1, then the polynomial f(x) in (2.3) is reducible in both Z[[x]] and Zp[x]. Proof. First of all, observe that f(pn) ≡ 0 (mod pn), and f 0(pn) ≡ 0 (mod pm) because n > m. Moreover, since n > 2m ≥ m + 1 and gcd(p; γ1) = 1, we have 0 n m+1 n f (p ) 6≡ 0 (mod p ). Then, by Hensel's lemma, p lifts to a root of f(x) in Zp. Thus f(x) is reducible in Zp[x]. To show the reducibility of f(x) in Z[[x]], we give an inductive procedure to find ak; bk 2 Z such that m 2 n−m 2 f(x) = p + a1x + a2x + ··· p + b1x + b2x + ··· : First of all, observe that n > 2m implies n − m > m. As a first step, we need m m n−2m p γ1 = p (b1 + p a1) m n−2m (2.6) γ2 = p s2 + a1(γ1 − p a1) n−2m n−2m 2 with s2 = b2 + p a2. If we let h(x) = γ2 − γ1x + p x , then solving the last m equation is equivalent to finding a1 2 Z such that h(a1) ≡ 0 (mod p ). Since the discriminant of h(x) is a square mod p, there exists r 2 Zp such that h(r) = 0. We m let a1 2 Z be the reduction of r mod p . With this choice of a1 we can now use n−2m m (2.6) to determine that b1 = γ1 − p a1 and s2 = h(a1)=p . With the notation n−2m sj = bj + p aj; j = 2; 3;::: (2.7) the rest of the equations become m n−2m γ3 = p s3 + a2(γ1 − 2p a1) + a1s2 m n−2m γ4 = p s4 + a3(γ1 − 2p a1) + a1s3 + a2b2 . m n−2m γd = p sd + ad−1(γ1 − 2p a1) + a1sd−1 + a2bd−2 + ··· + ad−2b2 m n−2m 0 = p sd+1 + ad(γ1 − 2p a1) + a1sd + a2bd−1 + ··· + ad−1b2 . m n−2m 0 = p sk+1 + ak(γ1 − 2p a1) + a1sk + a2bk−1 + ··· + ak−1b2 . n−2m Since γ1 − 2p a1 is not divisible by p, there are integers a2 and s3 such that the equation for γ3 is satisfied. Now that a2 has been chosen, we combine it with s2 in (2.7) to determine b2. For each of the above equations, the same argument can be used to solve inductively for the pair ak; sk+1 2 Z. At each iteration, equation (2.7) then gives bk. In the next section, we will make use of the following elementary result: Lemma 2.8 (Theorem 1.42 in [5]). A polynomial with integer coefficients has a k root in Zp if and only if it has an integer root modulo p for any k ≥ 1. 4 DANIEL BIRMAJER, JUAN B. GIL, AND MICHAEL WEINER 3. Factorization in the presence of a p-adic root In this section, we finish our discussion of reducibility in Z[[x]] for polynomials with integer coefficients. Our results rely on the existence of a root % 2 pZp, and the nature of the factorization depends on the multiplicity of % and on its p-adic valuation, denoted by vp(%).
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