An Infinite Family of Cubics with Emergent Reducibility at Depth 1
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AN INFINITE FAMILY OF CUBICS WITH EMERGENTREDUCIBILITY AT DEPTH 1
JASON I. PRESZLER
Abstract.
A polynomial f ( x ) has emergent reducibility at depth n if f ◦ k ( x )is irreducible for 0 ≤ k ≤ n − f ◦ n ( x ) is reducible. In this paper we provethat there are infinitely many irreducible cubics f ∈ Z [ x ] with f ◦ f reducibleby exhibiting a one parameter family with this property. Introduction
Given a polynomial f ( x ) ∈ Q [ x ], one can construct the sequence of iterates f ◦ f ( x ) , f ◦ f ◦ f ( x ) , . . . , f ◦ n ( x ). Such sequences form dynamical systems and havebecome the focus of considerable scrutiny in recent years. In the 1980’s, Odoniproved fundamental facts about the behavior of the discriminant and resultant [4],proved instances where entire sequences consisted of irreducible polynomials [5],gave examples of sequences with irreducible initial terms but reducible terms aftera certain point [5], and showed that the Galois groups of f ◦ n ( x ) embed into the n -fold iterated wreath product of Gal( f ), the Galois group of f [4]. More recently,[1] showed that there are finitely many irreducible quadratic polynomials where f ◦ n is reducible if n ≥
2. In [2] it was already shown that there are infinitely manyirreducible quadratics f with f ◦ f reducible. It should be noted that once a termin the sequence is reducible all subsequent terms will be reducible. Additionally,Hindes [3] has shown that the Galois group of f ◦ n can fail to be the full n -folditerated wreath product even when f ◦ n is always irreducible.The phenomena that we focus on in this paper will be called emergent reducibility(or “newly reducible” in [1]). Definition 1.1 (Emergent Reducibility at Depth n ) . We say a polynomial f ( x ) ∈ K [ x ] has emergent reducibility at depth n if and only if f ◦ i ( x ) isirreducible over K [ x ] for ≤ i ≤ n − and f ◦ n ( x ) is reducible over K [ x ] . Notethat n counts the number of compositions so f ◦ ( x ) = f ( x ) . We will prove
Theorem 1.2.
There are infinitely many irreducible cubics f ∈ Z [ x ] with emer-gent reducibility at depth 1. To do this, we show that the family f a ( x ) = − ax − (8 a + 2) x + (4 a − x + a is irreducible for infinitely many integers a and that f a ◦ f a ( x ) is the product of acubic and sextic polynomial for all a , namely g a ( x ) = 32 a x + (32 a + 16 a ) x + ( − a + 12 a + 2) x + ( − a − a + 1) and h a ( x ) = 128 a x + (256 a + 32 a ) x + 32 ax + ( − a − a − x − (4 a + 2) x + (16 a + 1) x + 2 a . (1) 2. The Irreducibility of f a Since reducible polynomials will have reducible iterates, it is crucial to showthat infinitely many polynomials of the form f a ( x ) for a ∈ Z are irreducible. Thiscan be accomplished, somewhat unsatisfactorily, by use of Hilbert’s IrreducibilityTheorem. We can consider our parameterized family as a family of polynomialsin two variables, f a ( x ) = f ( a, x ) ∈ Q [ a, x ]. Since f ( a, x ) is linear in a and the“coefficients” have no common factors in Q [ x ], we know that f ( a, x ) is irreduciblein Q [ a, x ]. Hilbert’s Irreducibility Theorem[6] ensures that for infinitely many valuesof a ∈ Z the specialization f a ( x ) is irreducible in Q [ x ]. Unfortunately, this fails togive any specific values of a such that f a ( x ) is irreducible.The following theorem gives a more effective determination of when f a ( x ) isirreducible. We note that computational evidence suggests that f a ( x ) is irreduciblefor every non-zero a (verified for 0 < | a |≤ ) but for our purposes the followingis sufficient. Theorem 2.1.
For a ∈ Z with ∤ a , the polynomial f a ( x ) = − ax − (8 a +2) x + (4 a − x + a is irreducible over Z / and therefore irreducible in Z [ x ] .Proof. Since 3 ∤ a , we have f a ( x ) = − ax − (8 a + 2) x + (4 a − x + a ≡ ( x + 2 x + 1 if a ≡ x + 2 x + 1 if a ≡ Z / (cid:3) Considering a = 3 t for t ∈ Z , reduction modulo other small primes shows thateven more values of a will result in irreducible polynomials, matching computationalevidence. 3. The Reducibility of f a ◦ f a The required reducibility is easily verified. The composition of f a with itself is f a ◦ f a ( x ) = 4096 a x + (12288 a + 3072 a ) x + (6144 a + 7680 a + 768 a ) x +( − a + 2560 a + 1408 a + 64 a ) x + ( − a − a + 64 a + 32 a ) x +(3072 a − a − a − a − x + (1216 a + 1024 a − a − a − x +( − a + 240 a + 40 a + 16 a ) x + ( − a − a + 16 a − a + 1) x +( − a − a + 2 a ) . (3)With the cubic g a ( x ) and the sextic h a ( x ) defined as g a ( x ) = 32 a x + (32 a + 16 a ) x + ( − a + 12 a + 2) x + ( − a − a + 1) N INFINITE FAMILY OF CUBICS WITH EMERGENT REDUCIBILITY AT DEPTH 1 3 and h a ( x ) = 128 a x + (256 a + 32 a ) x + 32 ax + ( − a − a − x − (4 a + 2) x + (16 a + 1) x + 2 a (4)then g a ( x ) h a ( x ) = f a ◦ f a ( x ) . (5)Thus, f a ◦ f a is reducible and together with 2.1 we have proven: Theorem 3.1.
For ∤ a ∈ Z , the polynomials f a ( x ) = − ax − (8 a + 2) x +(4 a − x + a are irreducible with emergent reducibility of depth . Theorem 1.2 is a direct consequence of the existence of the family f a ( x ). Wealso note that computational evidence suggests g a and h a are irreducible over Z [ x ],but this in ancillary to our requirements. In the next section we show that thereare a number of other examples of this behavior.4. Other Examples
There are many examples of non-monic cubics with depth 1 emergent reducibil-ity, including other parameterizable families. The more interesting situation is thecase of monic integral cubics where there seems to be only a finite number withdepth one emergent reducibility. The following is a list of all known examples wherethe absolute value of the coefficients are less than 500. x ± x + 23 x ± x ± x + 11 x ± x ± x − x ∓ x ± x + 3 x ∓ Conjecture 4.1.
There are only finitely many monic cubics in Z [ x ] with depthone emergent reducibility. References [1] K. Chamberlin, E. Colbert, S. Frechette, P. Hefferman, R. Jones, and S. Orchard,
Newlyreducible iterates in families of quadratic polynomials , ArXiv e-prints (October 2012), availableat .[2] Lynda Danielson and Burton Fein,
On the irreducibility of the iterates of x n − b , Proc. Amer.Math. Soc. (2002), no. 6, 1589–1596 (electronic).[3] W. Hindes, Galois Uniformity in Quadratic Dynamics over Rational Functions Fields , ArXive-prints (May 2014), available at .[4] R.W.K. Odoni,
The galois theory of iterates and composites of polynomials , Proc. LondonMath. Soc. (1985), no. 3, 385–414 (electronic).[5] , On the prime divisors of the sequence w n +1 = 1 + w . . . w n , J. London Math. Soc. (1985), no. 2, 1–11 (electronic).[6] Jean-Pierre Serre, Lectures on the mordell-weil theorem , Vieweg, 1990.
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