Root separation for reducible monic quartics

Root Separation for Reducible Monic Quartics

ANDREJDUJELLA(*) - TOMISLAVPEJKOVICÂ(**)

ABSTRACT- We study root separation for reducible monic integer polynomials of degree four. If H(P) is the height and sep(P) the minimal distance between two distinct roots of a separable integer polynomialP(x), and sep(P)ˆH(P) ^e(P), we show that lim supe(P)ˆ2, where limsup is taken over all reducible monic in- teger polynomialsP(x) of degree4.

1. Introduction.

The height H(P) of an integer polynomialP(x) is the maximum of the absolute values of its coefficients. For an integer polynomialP(x) of degree d2 and with distinct rootsa₁;. . .;a_d, we set

sep(P):ˆ min

1i5jdja_i a_jj and definee(P) by

sep(P):ˆH(P) ^e(P):

For an infinite setSof integer polynomials containing polynomials of ar- bitrary large height, we define

e(S)ˆ lim sup

P(X)2S;H(P)!‡1e(P):

(*) Indirizzo dell'A.: Department of Mathematics, University of Zagreb, BijenicÏka cesta 30, 10000 Zagreb, Croatia.

E-mail: [email protected]

(**) Indirizzo dell'A.: Tomislav PejkovicÂ, Department of Mathematics, Uni- versity of Zagreb, BijenicÏka cesta 30, 10000 Zagreb, Croatia.

E-mail: [email protected]

The authors were supported by the Ministry of Science, Education and Sports, Republic of Croatia, grant 037-0372781-2821 and by the French-Croatian bilateral COGITO projectDiophantine approximations.

In this note we will be concerned with reducible monic polynomials of de- gree four with integer coefficients. Therefore, we introduce notationRMdfor the set of all reducible monic polynomials of degreedwith integer coefficients.

First, we briefly summarize what is known about bounds one(S) ifSis some class of integer polynomials of small degree. A classical result of Mahler [5] asserts that ifScontains only polynomials of degreed, thene(S)d 1.

The case of quadratic polynomials is almost trivial and won't be dis- cussed further:

dˆ2 general monic

irreducible eˆ1 eˆ0 reducible eˆ1 eˆ0

For cubic polynomials, the case of general (i.e. nonmonic) polynomials was first solved by Evertse [4] and later SchoÈnhage [6] gave an easier constructive proof. In the monic case Bugeaud and Mignotte [3] proved the lower bounde(M3)3

2, whereM3 is the set of monic cubic polynomials with integer coefficients. They also showed thate(M3)ˆ3

2is equivalent to Hall conjecture. Proving thate(RM3)ˆ1 is not hard when we notice that a polynomial from this set is a product of a linear and a quadratic poly- nomial, both monic and with integer coefficients because of Gauss's Lemma. In the next table we summarize known results fordˆ3:

dˆ3 general monic

irreducible eˆ2 e3 2 reducible eˆ2 eˆ1

Until now no exact values when dˆ4 were known, just the lower bounds given in the following table:

dˆ4 general monic

irreducible e13

6 e3

2 reducible e7

3 e2

The bound for nonmonic irreducible case arises from a general construction by Bugeaud and Dujella [2] which in this special case givese((P4;n(x))_n2N)ˆ 13

6, where

P4;n(x)ˆ(20n⁴ 2)x⁴‡(16n⁵‡4n)x³‡(16n⁶‡4n²)x²‡8n³x‡1:

For nonmonic reducible polynomials, a recent unpublished result by Bu- geaud and Dujella, shows that the sequence

Pe4;n(x)ˆ (2n‡1)x³‡(2n 1)x²‡(n 1)x 1

(n²‡3n‡1)x (n‡2) givesee((Pe4;n(x))_n2N)ˆ7

3. The bound for monic irreducible polynomials e3

2is deduced by looking at the sequence

Pb_4;n(x)ˆ(x² nx‡1)² 2(nx 1)²; n2N

(see Bugeaud and Mignotte [3]). Finally, for reducible monic polynomials, it follows from a general case discussed in [3] that e(RM₄)2. While the proof from [3] is nonconstructive, in Section 2 we establish the same in- equality by exhibiting a setS RM4such thate(S)ˆ2. In Section 3 we prove thate(RM4)2. By putting together the results from Sections 2 and 3, we obtain the main result of this paper, which gives the first exact value in the above table fordˆ4.

THEOREM1. It holds that e(RM4)ˆ2.

Furthermore, in Section 4, we show that if the coefficients of poly- nomials in the sequence Sˆ(P_n(x))_n2N RM₄ grow polynomially inn, we must have a strict inequality e(S)52. But we also show that we can choose such a sequence so thate(S) is arbitrarily close to 2. More precisely, we prove the following theorem.

THEOREM2. If Sˆ(P_n(x))_n2N RM₄ is a sequence of polynomials whose coefficients are polynomials in n, then e(S)52. For anye>0, there is a a sequence of polynomials Sˆ(P_n(x))_n2N RM₄whose coefficients are polynomials in n such that e(S)>2 e.

A survey of results on separation of roots for integer polynomials of general degree can be found in the paper by Bugeaud and Mignotte [3] (see also [2]).

2. The constructive proof ofe(RM4)2.

We want to find a sequence of polynomialsSˆ(Pn(x))_n2N RM4such thate(S)ˆ2. We look at integer polynomials of the type

P(x)ˆ(x²‡rx‡s)(x²‡ax‡b);

whererandsare fixed whileaandbdepend on them and onnsuch that one root of the polynomial in the first bracket is very close to a root of the polynomial in the second bracket.

Choose r and s such that the roots l1;l2 of the polynomial R(x)ˆ x²‡rx‡s2Z[x] satisfylˆl₁>1>l₂>0. Also, let (a_n)_n2N be an in- creasing sequence of positive integers that satisfies the recurrence an‡2‡ran‡1‡sanˆ0 whose characteristic polynomial isR(x). Hence,

a_nˆc₁lⁿ₁ ‡c₂lⁿ₂ ˆc₁lⁿ‡c₂ s lⁿ; for some constantsc₁;c₂.

Assume thatl‡eis a root of the polynomialx²‡ax‡b2Z[x]. Then we have

(l‡e)²‡a(l‡e)‡bˆ0 e²‡(2l‡a)e‡(a r)l‡(b s)ˆ0:

Therefore 2eˆ (2l‡a) 

(2l‡a)² 4 (a r)l‡(b s) q

. If we have 2l‡a>0 and j4 (a r)l‡(b s)

j5(2l‡a)²; …1†

then we get a smallerjejfor the‡sign, so j2ej ˆ

4 (a r)l‡(b s)

(2l‡a) 

(2l‡a)² 4 (a r)l‡(b s)

q

(a r)l‡(b s) 2l‡a

…2†

(hereMNstands forMNandNM, where the implicit constants depend only onrands). At this point we see that by choosing

a rˆa_n; r 1; b sˆ a_n‡1; sˆ1;

conditions onl ,l , (a ) and inequalities (1) are fulfilled, while from (2)

we have

sep(Pn)ˆ jej

a_nl a_n‡1

2l‡an‡r

ˆ

c1l^n‡1‡_ln^c21 c1l^n‡1 _ln‡1^c2

2l‡c1lⁿ‡_l^c2n‡r 1

l²ⁿmaxf1;jaj;jbjg ²H(P_n) ² and thus

e((Pn)_n2N)ˆ2;

where

Pn(x)ˆ(x²‡rx‡1) x²‡(r‡an)x‡(1 an‡1) : This shows thate(RM₄)2.

Note that we could have takensˆ 1 before and if we were trying to approach the smaller root i.e. l2, we would get a similar family of poly- nomials

P_n(x)ˆ(x²‡rx 1) x²‡(r a_n‡1)x (a_n‡1)

; and after substitutionx7! x, we would get

Pn(x)ˆ(x² rx 1) x²‡( r‡an‡1)x (an‡1) : In case ofa₁ˆ1,a₂ˆ1,rˆ 1, the above polynomial is

P_n(x)ˆ(x²‡x 1) x²‡(1‡F_n‡1)x (F_n‡1)

where (Fn)_n2N is the Fibonacci sequence. This last sequence of poly- nomials, which was first obtained by numerical experiments, was the mo- tivating factor for this study.

3. The proof ofe(RM4)2.

Let us prove thate(RM₄)2. In other words, the best separation of roots we can get in the case of areducibleseparable monic quartic poly- nomialP(x)2Z[x] is H(P) ₂

. (All the constants implied in;;in this section are absolute.)

We have to look at two cases: when the polynomial has a cubic irreducible factor and when the polynomial has a quadratic irreducible factor. Because of Gauss's Lemma all the divisors inQ[x] ofP(x) will actually be fromZ[x].

Therefore, the case whenP(x) is a product of linear factors is trivial.

If we haveP(x)ˆ(x k)(x³‡ax²‡bx‡c), wherea;b;c;k2Z, then by the result of Mahler we know that the roots ofQ(x)ˆx³‡ax²‡bx‡c can be no closer than ( maxf1;jaj;jbj;jcjg) ². Because of Gelfond's Lemma (see e.g. [1, p. 221]), we have

1

16maxf1;jkjgmaxf1;jaj;jbj;jcjg …3†

H(P)16 maxf1;jkjgmaxf1;jaj;jbj;jcjg;

so sep(Q)H(P) ². There only remains to check whether we can have a root of Q(x) close to k. Let us take Q(k‡e)ˆ(k‡e)³‡a(k‡e)²‡ b(k‡e)‡cˆ0 where without loss of generality we can supposejej51. It is obvious thatjk‡ej5jaj ‡ jbj ‡ jcj ‡1 must hold, otherwise we get a con- tradiction. Thus, from (3) we getjkj H(P)¹⁼². SinceP(x) does not have multiple roots andQ(x)2Z[x] we have

1 jQ(k)j ˆ jQ(k‡e) Q(k)j ˆ jQ⁰(t)j jej;

wheret2(k;k‡e)(k 1;k‡1). But, using (3) andjkj H(P)¹⁼², we get jQ⁰(t)j ˆ j3t²‡2at‡bj 3(jkj ‡1)²‡2jaj(jkj ‡1)‡ jbj H(P):

Finally, we arrive atjej 1=jQ⁰(t)j H(P) ¹.

If P(x)ˆQ₁(x)Q₂(x), where Q₁(x);Q₂(x)2Z[x] are two quadratic polynomials, then we have from Gelfond's Lemma

1

16H(Q₁)H(Q₂)H(P)16H(Q₁)H(Q₂):

…4†

Since for quadratic polynomials we have sep(Q_i)H(Q_i) ¹, we only have to check the proximity of the rootsaandbofQ1(x) andQ2(x), respectively.

Theorem A.1 from [1, p. 223] states that in our separable case

ja bj 2 ¹3 ⁵⁼²H(Q1) ²H(Q2) ²maxf1;jajgmaxf1;jbjg H(P) ²: Hence, we proved thate(RM₄)2, which concludes the proof of Theo- rem 1.

4. Polynomial growth of coefficients.

In Section 2 we exhibited a family of reducible monic polynomialsP_n(x) whose coefficients grow exponentially innsuch that sep(Pn)H(Pn) ².

We will show that this is not possible if the coefficients grow poly- nomially. More precisely, let P (x)ˆP(n;x)2Z[n;x] be a polynomial

which is monic of degree 4 inxand such that for every positive integern⁰, polynomialPn⁰(x)2Z[x] is reducible. This is the exact meaning of condi- tions in the first statement of Theorem 2. Hilbert's Irreducibility Theorem (see e.g. Zannier [7]) implies that

Pn(x)ˆQn;1(x)Qn;2(x);

whereQn;1(x) andQn;2(x) are monic polynomials inxwhose coefficients are integer polynomials inn. Note that because of the previous section, the case of a reducible monic polynomial with a linear factor is not very interesting. Therefore, we will assume that Q_n;1(x) and Q_n;2(x) are ir- reducible quadratic polynomials in xwithout common roots, so

Qn;1(x)ˆx²‡r(n)x‡s(n); Qn;2(x)ˆx²‡a(n)x‡b(n);

where r(n);s(n);a(n);b(n)2Z[n]. For the sake of simplicity, we will most often omit n. As already mentioned, we can assume that the clo- sest roots of P are a root of Q₁ and a root of Q₂. So, without loss of generality, let us take

2sep(P)ˆ2eˆ r‡ 

r² 4s

p ‡a‡ 

a² 4b

p :

After some manipulation we get thatesatisfies the following equality …5† e⁴ 2(a r)e³‡(r²‡a² 3ra‡2s‡2b)e²

(a r)( ra‡2s‡2b)e‡(s²‡b² rsa rab 2bs‡sa²‡br²)ˆ0:

Notice that the last term is just the resultant Res_x(Q₁;Q₂) of polynomialsQ₁ andQ2:

Res(Q1;Q2)ˆRes(Q1;Q2 Q1)ˆ(b s)²‡(a r)(as br):

Let us suppose that eH ², where by Gelfond's Lemma HˆH(P)H(Q1)H(Q2). It can be mentioned here that all the constants in O, , , in the first part of this section depend at most on the coefficients of r;s;a;b. Since P(x) is a separable integer polynomial, it follows that Res(Q₁;Q₂) is an integer polynomial innandjRes(Q₁;Q₂)j 1.

Now we get from (5) and (4) that

H ²e jRes(Q1;Q2)j j|{z}e³

O(H ⁶)

2(a r)e²

|‚‚‚‚‚‚{z‚‚‚‚‚‚}

O(H ³)

‡(r|‚‚‚{z‚‚‚}²‡a²

O(H²)

3ra‡2s‡2b

|‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚}

O(H)

)e

|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}

O(1)

(a r)( ra‡2s‡2b)j

and

H ²e jRes(Q1;Q2)j jO(1) 2as|‚‚‚‚‚‚{z‚‚‚‚‚‚}‡2rb

O(H)

‡ra² r²a‡2rs 2abj: …6†

Because of Gelfond's Lemma,jrj;jsj;jaj;jbj H andjarj H which implies thatjaj H¹⁼²orjrj H¹⁼². Without loss of generality we can suppose that jaj H¹⁼². Thus we getjra²j ˆ jraj jaj H³⁼²andjabj ˆ jaj jbj H³⁼². We also havej r²a‡2rsj ˆ jrj jra 2sj ˆ jrjO(H) so the inequality (6) becomes

H ²e 1

maxfO(H³⁼²);jrjO(H)g:

It implies that jrj H, so from jrj H, we get jrj H. Also, jRes(Q1;Q2)j ˆ O(1). Since r;s;a;b are polynomials in n and jraj H, jrbj H, we conclude thataandbare constants.

If we now have deg_ns5deg_nrthen

deg_nRes(Q1;Q2)ˆdeg_n (b s)²‡(a r)(as br)

deg_nr‡deg_ns;

sojRes(Q1;Q2)j H, which leads to a contradiction. Therefore, deg_nsˆ deg_nrand hencejsj jrj H! 1.

The leading coefficient of Res(Q1;Q2) as a polynomial in n, i.e. the coefficient that belongs to the monomial of degree 2deg_nrˆ2deg_ns, is the leading coefficient of s² ars‡br², i.e. k²_s ak_rk_s‡bk²_r, where k_s, k_r are leading coefficients of s and r, respectively. If it were 0, then ks=kr2Qwould be a root of x²‡ax‡bwhich is impossible, since by our assumption this polynomial is irreducible. Thus deg_nRes(Q1;Q2)ˆ 2 deg_nr2 and this is in contradiction with the conditionjRes(Q₁;Q₂)j ˆ O(1).

We conclude that sep(P_n)H(P_n) ²cannot hold in this case, and this proves the first statement of Theorem 2.

Although the previous result of this section shows that we cannot have a family of reducible monic quartic integer polynomials with polynomial growth of coefficients that has the best possible exponent for root se- paration in this case, i.e. 2, we can still construct families with the ex- ponent as close to 2 as we like. The construction that follows is similar to the one in Section 2.

We look at the family of polynomials P_k;n(x) indexed with n2N in variablex. As before, we will usually omitnand write simply P (x). We

define

P_k(x)ˆ(x|‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚}²‡nx‡1)

Qk(x)

(x²‡nx‡1‡A_k‡1x‡A_k)

|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}

Rk(x)

ˆ(x²‡|{z}n

r

x‡|{z}1

s

) x²‡(A|‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚}k‡1‡n)

a

x‡(A|‚‚‚‚‚{z‚‚‚‚‚}k‡1)

b

; where A_k(n)

k2N0is defined recursively by

A0(n)ˆ1; A1(n)ˆn; Ak‡1(n)ˆnAk(n) Ak 1(n) forn2:

It is easy to see that deg_nA_k ˆk, so we get (implied constants are absolute from now on)

H(Pk)n^k‡2: Let us look at the resultant:

Res_x(Q_k;R_k)ˆ(b s)² r(b s)(a r)‡s(a r)²

ˆA²_k nA_kA_k‡1‡A²_k‡1

ˆA²_k‡A_k‡1(A_k‡1 nA_k)

ˆA²_k Ak‡1Ak 1

ˆA²_k (nA_k A_{k 1})A_{k 1}

ˆA_k(A_k nA_{k 1})‡A²_{k 1}

ˆA²_{k 1} A_kA_{k 2}

ˆ. . .ˆA²₁ A2A0ˆn² (n² 1)1ˆ1:

The roots ofQk(x) are

a₁ˆ n 

n² 4 p

2 ; a₂ˆ n‡ 

n² 4 p

2 ;

and the roots ofR_k(x) are

b₁ ˆ (A_k‡1‡n) 

(A_k‡1‡n)² 4(A_k‡1) q

2 ;

b₂ ˆ (Ak‡1‡n)‡ 

(Ak‡1‡n)² 4(Ak‡1) q

2 :

Therefore,

a₁ n; a₂ 1

n; b₁ n^k‡1; b₂ˆA_k‡1

b₁ 1

n ;

so we have

1ˆRes(Q_k;R_k)ˆ1²1²ja|‚‚‚‚‚{z‚‚‚‚‚}₁ b₂j

n

ja₁ b₁j

|‚‚‚‚‚{z‚‚‚‚‚}

n^k‡1

ja₂ b₁j

|‚‚‚‚‚{z‚‚‚‚‚}

n^k‡1

sep(P_k);

and it follows that

sep(P_k)n ^{2k 3}ˆn ^2(k‡2)nH(P_k) ^2‡k‡21: Hence, we proved the last statement of Theorem 2.

Acknowledgements. The authors would like to thank Yann Bugeaud for introducing them to this topic and for useful comments on the previous version of this paper.

REFERENCES

[1] Y. BUGEAUD, Approximation by algebraic numbers. Cambridge Tracts in Mathematics, Cambridge, 2004.

[2] Y. BUGEAUD- A. DUJELLA,Root separation for irreducible integer polyno- mials, Bull. Lond. Math. Soc., to appear.

[3] Y. BUGEAUD- M. MIGNOTTE,Polynomial root separation, Intern. J. Number Theory,6(2010), pp. 587-602.

[4] J.-H. EVERTSE, Distances between the conjugates of an algebraic number, Publ. Math. Debrecen,65(2004), pp. 323-340.

[5] K. MAHLER,An inequality for the discriminant of a polynomial, Michigan Math. J.,11(1964), pp. 257-262.

[6] A. SCHOÈNHAGE,Polynomial root separation examples, J. Symbolic Comput., 41(2006), pp. 1080-1090.

[7] U. ZANNIER,On the Hilbert irreducibility theorem, Rend. Semin. Mat. Univ.

Politec. Torino,67, no. 1 (2009), pp. 1-14.

Manoscritto pervenuto in redazione il 28 ottobre 2010.