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HEBBIIHEBCKHH CBOPHHK Tom 9 BbinycK 1 (2008)
SIMULTANEOUS DIOPHANTINE APPROXIMATION OF INTEGRAL POLYNOMIALS IN THE DIFFERENT METRICS
Natalia Budarina and Detta Dickinson Throughout, let
P(f) = a.nfn + a-n-1 fn 1 + • • • + af + ao
be an integer polynomial of degree deg P ^ n and height H = H(P) = max1^j^n |aj|. In this paper we will consider a problem of Diophantine approximation on such polynomials in the real, complex and p-adic fields simultaneously. That is, we will study the approximation of zero by the values of |P(x)|, |P(z)| and |P(w)|p, where x £ R, z £ C, W £ Qp.
Let ^ be a monotonically decreasing function. In [9] it is shown that if the volume sum C=1 V(T) converges then the set of points (x, z, w) £ R x C x Qp which satisfy the inequalities |P(x)| ^ H-Vl¥Al (H), |P(z)| ^ H-V2¥A2(H) and |P(w)|p ^ H-v3¥A3 (H), where v1 + 2v2 + v3 = n — ^d A1 + 2A2 + A3 = 1, for infinitely many P
A more specialised result is that of V.N. Borbat in [8] who showed that the system of inequalities
{ |P(x)| < H-n+v',
\ |P'(x)| < H1-v'-e', 0 ^ v' <1,
for any e1 > 0 has infinitely many solutions P £ Z[x] only for a set of measure zero. Borbat’s result allows us to find a lower bound for the HausdorfF dimension of the x
the derivative of the minimal polynomial is relatively small.
In the present paper, we generalize this result to simultaneous approximation on R x C x Qp and consider some applications.
Before we proceed, some notation is needed. Let ^1 (A) be the Lebesgue measure of a measurable set A C R, f^2(A) the Lebesgue measure of a measurable set A C C and f^3(A) the Haar measure of a measurable set A C Qp. Using these definitions, define the measure ^ on a set A C R x C x Qp by ^(A) = ^1(A)^2(A)^3(A).
Let Ln(v) denote the set of points lying in a parallelepiped T = I x K x D, where
I is an interva 1 in R, K is a disc in C and D is a cylinder in Qp, for which the system of inequalities
max(|P(x)|, |P(z)|, |P(w)|p|) < H-n-3 +v¥?(H),
max(|P'(x)|, |P'(z)|) < H1-v, (1)
|P '(w)|p < H-v,
has infinitely many solutions P £ Z[x].
Theorem 1. If n ^ 3 and Xh=i ^(H) < oo then f^(Ln(v)) = 0 with 0 ^ v ^
0.027.
For n = 3 this theorem is easily proved. Hereafter, only the case n ^ 4 will be considered.
As ^ is monotonic and the series L H= .i y(H) converges it is easy to show that on average ^(H) < CiH-1, where c1 is independent of H. Therefore, instead of the first inequality of (1) the weaker inequality
max(|P(x)|, |P(z)|, |P(w)|p|) << H-^ +v, (2)
may be considered at some stages for simplicity. Here and throughout A < B means
that there exists a constant C > 0 such that A ^ CB.
n c(n)
Where necessary these constants will be numbered ck(n), k = 1,2,_______________
It is shown in [1, 7, 10] and [11] that, without loss of generality, it is enough to
P
and also satisfy
H(P) = |aj, |ajp >p-n. (3)
Let Pn(H) denote this set and define Pn = UH=1 Pn(H).
Let P £ Pn(H) have roots ai, a2, • • •, «4 in C and roots y1,y2, • • • ,Yn in Qp,
where Qp is the smallest field containing Qp and all algebraic numbers. From (3) it
is shown in [3] and [5] that
|ai| ^ 2, |Yi|p <pn, i = 1, • • •, n
From among the roots ai choose a real root a1 and a non-real root aj which will
hereafter be denoted by Pi. Order the ro ots ai according to their distan ce from a1 or Pi as follows:
|ai — a2| ^ ••• ^
|Pi — N ^ ••• ^
|Yi — Y2|p ^ ••• ^ with n1 + n2 = n, and define the sets
S1(a1) = {x £ R : |x — a1| = min |x — a^},
S2O1) = {z £ C : |z — Pi| = min |z — Pj|},
1<j<n2
Sp(Yi) = {w £ Qp : |w — Yi|p = min |w — Yj|p|
1<j<n
|ai — ani |,
|pi — pn2 |,
|Yi — Yn|p,
For example, Sp(Yi) is the set of those points w £ Qp, for which Yi is the nearest
root.
Fix £ > 0 where £ is sufficiently small and suppose that £1 = £N-1 where N = N(n) > 0 is sufficiently large and let T = [£-1]. For a polynomial P define the real numbers p— i = 1,2,3, and the integers k- l- m- by
|ai — aj| |pi — I— |Yi — Yj|
= H-P1
and
kj — 1 kj lj — 1 lj
— < pij <T, — « p2j < t,
mj — 1 T
P3j <
mj
T'
Further define the numbers qi5 ri5 Si by
qi =
ri =
Si =
ki+1 + • • • + kn
T ,
li+1 + • • • + ln
T ,
mi+i + • • • + mn
T
, (1 ^ i ^ n1 — 1 )
, (1 ^ i ^ n2 — 1 )
, (1 ^ i ^ n — 1 )•
Each polynomial P £ Pn(H) is now associated with three integer vectors q = (k2, • • •, kni), r = (l2, • • •, ln2) and s = (m2, • • •, mn) and the number of these vectors is finite (and depends only on n, ^d T). Let Pn(H, q, r, s) denote the set of polynomials P £ Pn(H) with the same triple of vectors (q, r, s).
Fix 6i > 0. Any complex number z lying in the parallelepiped T with |Imz| < 61 will be excluded. As 6i is arbitrary this can be done without loss of generality. Hence, from now on we assume that |Im z| ^ 6i. Later, there will be inequalities of the kind |z — p| < H(P)-V v > 0; as the RHS tends to zero it will follow that there exists a root P such that |Im p| > 261. ^n this case there is also a conjugate root P of P such that |p — p| > 61, and for any real root a of P the inequalities | P — a| = |p — a| > i61 hold. Collecting this information, we have
|Im p | > ^6i,
|Imz| ^ 61, | P — P| > 61,
| p — a| > 26i-
(4)
1 Preliminary Results
From now on it will be assumed without loss of generality that x £ S1 (a1), z £ S2(P i), w £ Sp(Yi)- In many places in the proof of the theorem values of polynomials will be estimated by means of a Taylor series. To obtain an upper bound on the terms in the Taylor series (and for other purposes) the following two lemmas (proved in [4] and [10]) will be used.
Lemma 1. If P G Pn(H) then
|u — a| ^ 2n|P(u)||P/(a)|-1, |w — Yil ^ |P(w)|p|P/(yi)|p1,
|u — a| ^ min (2nj|P(u)||P/(a)| 1 TT |a — ak|) ,
2<Kn V Li )
_ 1
|w — Yi|p ^ min ( |P(w)|p|P/(Yi)|p1 n |Y1 — Yk|p)
2<j<^ i=2 )
where u represents x or z and a is ai or |3 1 as required.
Lemma 2. Let P G Pn(H, q, r, s). Then
|P(1)(ai)| < c(n)H1-qi+(n-l)£1,
|P(l)( 3 i)| < c(n)H1-ri+(n-l)£1,
|P(1) (Yi) |p < c(n)H
-si +(n-l)e1
The next lemma is proved in [12].
Lemma 3. Let G(v) be the set of points (x, z, w) for which the inequality |P(x)||P(z)|2|P(w)|p <H-v, n = degP ^ 3, H = H(P), has infinitely many solutions P G Z[x]. Then, for v > n — 2
M-(G(v)) = 0.
The following lemma is proved in [6]. At several points in the proof of the theorem there are various cases (of different types of polynomial) to consider; usually the existence of one case is disproved by finding a contradiction to the final inequality in the lemma below.
Lemma 4. Let P1 and P2 be two integer polynomials of degree at most n with no common roots and max(H(Pi), H(P2)) ^ H. Let 5 > 0 and n
>0
i = 1,2,3. Lei I C R be an interval, K C C be a disk and D C Qp be a cylinder with m(I) = H-r|1, diam K = H-r|2 and |Xp(D) = H-r|3. If there exist T1 > — 1, T2 > —1 and T3 > 0 such that for all (x, z, w) G I x K x D
max(|Pi(x)|, |P2(x)|) < H-T1,
xeI
max(|Pi(z)|, |P2(z)|) < H
zeK
max(|Pi(w)|p, |P2(w)|p) < H"
weD
-T2
T3
then
T1 +2t2+t3+3+2 max(T1 + 1 —n1, 0)+4 max(T2+1 —n2,0)+2 max(T3—n3, 0) < 2n+5.
Finally, we state two classical results. The first is proved in [2] and is an adaptation of Cauchy’s Condensation Test. The second is the convergence half of the Borel-Cantelli Lemma which will be used throughout the proof of the theorem.
Lemma 5. Let ^(H), H = 1,2,..., be a rnonotonieally decreasing sequence of positive numbers. If the series XH=i ^(H) converges, then for any number c > 0 the series £“o 2k^(c2k )
Lemma 6 (Borel-Cantelli). Let (Q, j) be a measure space with j(Q) finite and let A-t, i G N be a family of measurable sets. Let
A = (w G Q : w G At for infinitely many i G N}
and suppose the sum D=i j(A t) < oo. Then j(A) = 0.
2 Proof of the Theorem
Since |at| ^ 2, |Yt|p < pn for 1 ^ i ^ n and |w|p ^ 1 it follows from Lemma 1 (using j = n and H ^ Ho) that the set of points (x, z, w), for which (1) is satisfied,
is a subset of the set T = I x K x D, where I = [—3,3], K = (z : |z| ^ 3},
D = (w : |w|p < 1}.
The proof of the theorem will consist of a series of propositions. As a reminder,
P
Let
Pt = Pt(n, q, r, s) = y Pn(H, q, r, s)
2* <H<2* +1
and suppose that the polynomials P G P* are irreducible and satisfy (3). In much of what follows system (2) will be used rather than (1). A polynomial is called
(ii, i2, i3)-linear if for ij = 0, j = 1,2,3, the system of inequalities
i -r-1 n + 2
qi + k2T 1 < —4--------v
n + 2
ri + I2T-1 < —4-------v, (5)
—1 n — 2
si + m2T < —-------v,
ij = 1 j = 1, 2, 3 (0, 1, 1 ) <
third have ^.Denote by Pt(i1,i2,i3) C P* , ij = 0,1, j = 1,2,3, the class of (ii, i2, i3Hinear polynomials. As there are only 8 kinds of linearity we shall consider them in turn.
We will use the constants
di = qi + 2ri + si, d2 = (k2 + 2I2 + m2)T—1
heavily for the rest of the proof with different ranges of d1 + d2 considered separately.
Proposition 1. If Y.H=1 ^(H) < to then j(Ln(v)) = 0 when the polynomials are restricted, to the subclass P*(0,0,0) for which d1 + d2 > n + £.
Proof. By Lemma 1, all u = (x, z, w) G S(a1) x S( 3 1) x S(y1) satisfying (2)
belong to the parallelepiped cr(P) defined as the set of points u satisfying
|x — ai| < 2—1(^—qi —v),
|z — 3 i| < 2—1(^—T1 —v), (6)
|w — Yi |p < 2—1(^—si —v).
The initial parallelepiped T is divided into smaller parallelepipeds M = IM x KM x DM such that
ji (Im) = 2—tk2T— 1, diam (Km) = 2—tl2T— 1, jp(DM) = 2—tm2T— 1. (7)
PM
exists u G M such that (2) holds; we will denote this by P(u) G M. Let P(u) G M and develop P as a Taylor series on M remembering that P(a1) = P( 3 1) = P(yi ) = 0 to obtain
n
P(t) ^(jirV'HZiMx — Ci)j (8)
j=1
for t = x,z,w and Zi = a1, 3i,Yi repectively. An upper bound for |P(u)| is found
using (7) and Lemma 2. As an example we will show how to estimate |P(z)|. The
following inequalities obtained from the definitions of rj and IjT—1 are used:
rj + jl2To 1 = rj + ^2To 1 + (j — 1 )l2T0 1 ^ rj + ^2T0 1 + (l2 + • • • + lj—1 )To 1 = r1 + ^2To 1 •
These imply
|P'(3 1 )||z — 3 1| < 2t(1—Ti +(n—1)£i —l2T— 1) < 2—t(ri +l2T— 1 —1—(n—1)£i),
|P(j)(3 1 )||z — 3 1|j < 2t(1—Tj +(n—j)£1 —jl2T—1) < 2—t(ri +l2T—1 —1—(n—1)£1), 2 < j ^ n.
Clearly these further imply that |P(z)| ^ 2—t(Ti +l2T 1 —1—(n—1)£i). It is not difficult to acquire similar estimates for |P(x)| and |P(w)|p so that
|P(x)| ^ 2—*(qi +k2T— 1 —■i—(n—1 )ei)
|P(z)| < 2—t(Ti +l2T— 1 —1—(n—1)£i), (9)
|P(w)|p < 2—t(si +m2T— 1 —(n—1)£i).
We now consider the case where at most one polynomial belongs to each parallelepiped M. The number of such polynomials is at most c(n)2t(k2 +212 +m2)T = c(n)2td2.
uGM
^ c(n)2—t(n+1—di —d2 —4v)
From (5) it follows that d1 + d2 < n + 1 — 4v so the series HH=1 2—t(n+1—di —d2 —4v) converges and the proposition follows from the Borel-Cantelli lemma.
MP
n
2t+1. For two such polynomials P1, P2 G M the system of inequalities (9) holds. Using Lemma 4, with t1 = q1 + k2T—1 — 1 — (n — 1 )e1, t2 = r1 + l2T—1 — 1 —(n — 1 )e1, t3 = s1 + m2T—1 — (n — 1 )e1, n = k2T—1, n2 = l2T—1, n3 = m2T—1, we obtain
3qi + k2T 1 + 6ri + 2I2T 1 + 3s 1 + m.2T 1 — 12(n — 1)£1 < 2n + 6.
Replacing q1 by k2T—\ 2^ by 2l2T—^d s1 by m2T—1 gives
2(d1 + d2) — 12(n — 1 )e1 < 2n + 6,
which for 6 = £^d £ > 6n£1 contradicts the condition in Proposition 1. This completes the proof.
Proposition 2. If Y-h=1 ^(H) < to then j(Ln(v)) = 0 when the polynomials are restricted, to the subclass P*(0,0,0) for which d1 + d2 <4 — £.
Proof. We denote by Ln(v) the set of solutions (x,z, w) of the system of inequalities
max(|P(x)|, |P(z)|, |P(w)|p|) < H—^ +v¥1 (H),
Ha9—v < max(|P'(x)|, |P'(z)|) < H1—v, (10)
H—a1—v < |P'(w)|p < H—v,
Denote by Ln(v) the set Ln(v) \ Ln(v). Then for all (x, z, w) G Ln(v) we have
max(|P(x)|, |P(z)|, |P(w)|p|) < H—^ +v¥1 (H),
max(|P'(x)|, |P'(z)|) < Ha9—v, (11)
|P'(w)|p < H—a1—v.
We replace ¥(H) by H—1 in (11). Further, we use the method which was introduced by Borbat [8] to get that the new system of inequalities has infinitely many solutions ( x, z, w)
Now we investigate the set Ln(v). By Lemma 1, all solutions (x,z,w) for a
fixed P G P* satisfying (1) are contained in the parallelepiped c2(P) defined by the
inequalities
|x — ai| < 2—*(^ —v)^(2t)1/4|P/(ai)|—1,
|z — 3i| < 2—*(^ —v)^(2*)1/4|P/(3i)|—1,
|w — Yi |p « 2—*(nf1—v)^(2*)1/4|P/(yi ) |—1.
c 4(P)
qualities
|x — ai| < 2—*( 1 —v)|P/(ai)|—1,
|z — 3i| << 2—*( 1 —v)|P/(3i)|—1, (13)
|w — Yi |p << 2—*(2 —v)|P/(yi )|—1.
Clearly, c2(P) C c4(P).
P c 4(P)
P(x) = P/(ai)(x — ai) + 1/2P"(hi)(x — ai)2, h G (ai,x).
Estimating each term in the last equality individually gives
|P/(ai)||x — ai| < 2—*(2~v),
|P//(ai)||x — ai|2 < 2—*(9 —4v).
For 3v < 1.3 we obtain that |P(x)| < 2—1(0•5—v) for x G c4(P). It is easy to do the same for |P(z)| and |P(w)|p so that for v < 0.1
|P(x)| < 2—*( 1 —v),
|P(z)| < 2—*(1—v), (14)
|P(w)|p << 2—(2—v).
We similarly estimate P/(x) = P/(a1) + P"(£2)(x — a1), h2 G (a-^x) on c4(P). As before, each term is estimated individually so that
|P/(ai)| < 2—*(v—1),
|P(")(h2)||x — a1| < 2—*(—1+0^5—v+1—v—0^1) < 2—*(—2v+0^4).
Hence, |P/(x)| ^ 2|P/(a1)| < 2 *(v 1) for v < 0.1. From this and similar inequalities
for P/(x) the following inequalities hold on c4(P) for v < 0.1
|P/(z)| < 2—*(v—1),
|P/(w)|p << 2—*v. (15)
Fix the vector d = (a6, a7,..., an), |aj| ^ 2*+1 and let P^ denote the set of polynomials P G P* with the same vector d. The parallelepiped c4(P1) is called
P2 G Pd
j(C4(Pi) n C4(P2)) < 2j(c4(Pi)).
If, on the other hand, there exists P2 G P* such that
^(a4(Pi) n 04^2)) ^ 2^(ff4(Pi)),
then the parallelepiped a4(P1) is called inessential.
First, assume that a4(P1) is essential. Then, it follows that
Y_ ^(MPi)) << ^(t).
pi
Also, from (12) and (13),
^(ff2(Pi)) << ^(a4(Pi))2*(—n+5)¥(2*).
Since the number of classes P* is at most 0^)2*^°'5) from the above two displayed inequalities we have
II ^(ff2(Pi)) << 2*¥(2*)^(T).
d Pi ePd
By Lemma 5, the series
£«=i 2*^(2*
)
intervals can be completed using the Borel-Cantelli Lemma.
Now, assume that cr4(P1) is inessential so that the re exists P2 G Pd such that
ff(Pi,P2) = C4(Pi) n C4(P2), ^(c(Pi,P2)) ^ 1^(C4(Pi)).
The systems of inequalities (14) and (15) hold simultaneously on c(P1,P2) for both P^d P2. Hence, if R(f) = P2(f) — P1 (f) = b5f5 + ... + b1f + b0 then R satisfies
|R(x)| <
|R(z)| <
|R(w)|p <
|R/(x)| <
|R/(z)| <
|R/(w)|p <
< 2—*( 1—v), (16)
* — i
)—W
If 01,..., 05 are the roots of R then
R(f) = b5(f — 0i)(f — 02)... (f — 05),
R/(0i) = b5(0i — 02)... (0i — 05).
From (4) and (16) it follows that there must be another real root close to the real root a. By the same argument, the complex root 3 has another complex root which is close to it, and similarly, for its conjugate p. Hence, there is a contradiction as R cannot have 6 roots.
Proposition 3. If Yh=1 ^(H) < to then n(Ln.(v)) = 0 when the polynomials are restricted, to the subclass P*(0,0,0) for which
4 — £ ^ d1 + d2 ^ n + £. (17)
Proof. Instead of system (1) we use system (2). Exactly as in (7) the parallelepiped T is divided into parallelepipeds M. Let P G M and develop P as a Taylor series to obtain (9). For some 0 > 0 consider only parallelepipeds which contain at most c(n)2*9 polynomials. Then, by Lemma 1, the measure of the set of points u G T which satisfy (2) is at most the measure of the parallelepiped c(P) (defined in (6)) multiplied by the number of parallelepipeds M and c(n)2*9, that is
c(n)2 —t(n+1—di —d2—9—4v)
If 0 < n +1 — d1 — d2—4v then the series 2—t(n+1—di —d2—9—4v) converges and the
Borel-Cantelli Lemma can be used to complete the proof. Thus, from now on, we assume that 0 ^ u = n +1 — d1 — d2—4v. From (17), 1 — 4v — £ ^ u ^ n — 3 — 4v + £. Let u1 = u — d where d = 0.14. Writing u1 as a sum of integer and fractional parts [u1] + {u1} calculate
p = n — [ui] = di + d2 — 1 + {ui} + d + 4v. (18)
According to the Dirichlet box principle, there are at least k = 0^)2*^+^}) polynomials P1,..., Pk among these c(n)2*u polynomials whose first [u1] highest coefficients are the same. Consider the k — 1 polynomials Rj(f) = Pj(f) — Pi(f) for
2 ^ j ^ k. It can be readily verified that
|Rj(x)| < 2t(1—qi—k2T—1 +(n—1)£i),
|Rj(z)| < 2t(1—ri—l2T—1 +(n—1)£1), (19)
|Rj(w)|p < 2*(—si—m2T—1 +(n—1)£1),
with 2 ^ j ^ k, deg Rj ^ n — [u1] and H(R) ^ 2*+2. The polynomials Rj(f) = bn— [ui ]fn— [ui ] + • • • + bif + b0 are now divided into sets. In each set the values of the coefficients bn— [Ui ],..., b1 lie in an interval of length 2t(1—hi) where h1 = {u1}(n — [ui])—1. Again apply Dirichlet’s box principle to obtain that there are at least L = c(n)2*d polynomials Rj in one such set. These will be renumbered
R1,..., Rl. Develop the Rj(f) as a Taylor series on M and consider the polynomials
Si(f) = Ri(f) — R1(f), which satisfy
|Si(x)| < 2t(1—qi—k2T—1 +(n—1)£i), |Si(x)| < 2t(1—qi +(n—1)£i)
|Si(z)| < 2^Ti—l2T—1 +(n—1)£i), |Si(z)| < 2t(1—ri +(n—1)£i) (20)
|Si(w)|p < 2*(—si—m2T—1 +(n—1)£i), |Si(w)|p < 2*(—S1 +(n—1)£i),
with 2 ^ i ^ L deg Si ^ n—[u1 ^d H(Si) < 2t(1—hi \ Note that min(q1,r1,s1) ^ v in this case.
Si
have the form i1 S, i2S,..., iLS for some fixed polynomial S. Then i/ = maxi^L |ij| ^ c(n)2*d and (20) holds for i% with H(S0) < 2^hi —d). By (20),
|S0(x)||S0(z)|2|S0(w)|p << 2*(3—di —d2 —3d+4(n—1)£1). (21)
Then we apply for the system (20) the strengthening of the Lemma 3 which we can get by using the induction method in the Sprindzuk’s theory of essential and inessental domains [11]. The proof will be complete if it can be shown that
|S0(x)||S0(z)|2|S0(w)|p < H(S0)2—degS0 +4v—£2. (22)
S
d1 + d2 — 3 + 3d — 4(n — 1 )e1 > (n — [u1] — 2 — 4v + e2)(1 — h1 — d),
pd — 4vd — 2{u1}/p — 4v{u1}/p — 4(n — 1 )e1 — e2(1 — {u1}/p — d) >0,
This is true for d = 0.14, v ^ 0.027, p ^ 4 and e1, e2 sufficiently small.
For the second case, assume that one of the polynomials Si5 1 ^ i ^ L (say, S0),
is reducible, i.e. S0 = S01)S0'2^. Then, for one of these, for example S01)(f) the system
(20) holds and degS01)(f) ^ n — [u1] — 1. In this case Lemma 3 can be applied if it can be proved that the inequalities
di + d2 — 3 — 4(n — 1)£ 1 > (di + d2 — 4 + {ui} + d)(1 — hi), , ,
1 — 4v — d — 4( n — 1 ) £1 — 3{u1 }/p > 0
hold. It is not difficult to show that this is true for d = 0.14, v ^ 0.027 p ^ 4 and £1 sufficiently small.
Finally assume that among the Si there are at least two polynomials (say S1 and
S2 Si
and apply Lemma 4 with h = 1 — h1. Then,
t1 = (q1 + k2T—1 — 1 — (n — 1 )£1)h—1, n1 = k2T—1 h—1, t2 = (r1 + l2T—1 — 1 — (n — 1 )£1)h—1, n2 = l2T—1h—1, t3 = (s1 + m2T—1 — (n — 1)£1)h—1, n3 = m2T—1h—1,
deg S ^ n — [u1],
and the inequality
3qi + k2T 1 + 6ri + 2I2T 1 + 3si + ^T2T 1 — 12(n — 1)£1 — 9hi < 2(n — [ui])h + 5 must hold. Reduce the LHS by replacing q1 with k2T—\ 2r1 with 2l2T—^d s1 with
m2T—1
5 >2 — 2d — 8v — 9{u} — 12(n — 1)£ 1. p
If n — [u1] ^ 6 then the above inequality is a contradiction for d = 0.14, v ^ 0.027 and sufficiently small 5 and £i. Hence, the set of (x,z, w) for which the inequalities
Si
empty.
The proof when n — [u1] = 4 and n — [u1] = 5 can be done exactly as in [9]. The proof of the proposition is complete.
Now we consider the case when each coordinate is equal to 1 in the vector (ii, i2, i3) of the definition of linearity. So the system
_— i n + 2
qi + k2T ^ —4------v
n + 2
ri + I2T—1 ^ —------v, (24)
-1 n — 2
s1 + m2T 1 ^ —------v,
4
holds together with system (2).
Proposition 4. If Hh=1 ^(H) < to then ^(Ln(v)) = 0 when the polynomials are restricted, to the subclass P*(1,1,1).
Proof. Using (2) and Lemma 1 we obtain
^——v )
A 2 ) = 2—-tm
|x — ai| < 2 V / = 2
|z — pi| < 2 ( 2 ) = 2—tH2, (25)
—— s2 — v )
|w — Yi|p << 2 v 2 ) = 2—tH3.
Let cr5(P) be the parallelepiped defined by these inequalities. Divide the parallelepiped T into smaller parallelepipeds M with sidelengths 2—t(w—y)? 2—t(H2—y) and
2—t(^3—'y) where y = ^ P ^ M and develop it as a Taylor series on M. As
before, obtain an upper bound for all the terms in the series. The estimates for the real coordinate are presented below.
|P/(ai)||x — ai| < 2tY|P/(ai)2—t^11 < 2t(1—qi +Y+(n—1)£i +v/2+q2/2— (n+2)/8)
^ 2t(v+2y+(2—n)/4+(n—1)£i )
|P"(£i)||x — ai|(2) < 22tY|P"(ai)2—2t^11 < 2t(v+2Y+(2—n)/4+(n—1)£i).
Obtain similar estimates for |P(z)| and |P(w)|p so that the inequalities
|P(x)| < 2t(v+2y+(2—n)/4+(n—1)£i )
|P(z)| < 2t(v+2Y+(2—n)/4+(n—1)£1), (26)
|P(w)|p < 2t(v+2Y+(2—n)/4+(n—1)£i)
P
M
c(n)2t(w +2^2 +^3—'4y) so the measure of the set of u E M satisifying (2) and (24) (using (25)) is
c(n)2—t(^i +2^2 +H3 —m —2m-2 —M-3 +4y) = c(n)2 —4ty
Clearly the series ^^=0 c(n)2—4ty is convergent which is enough to complete the proof in this case.
Now assume that the parallelepipeds M contain two or more polynomials P1 and
P2 P1 P2
the system of inequalities (26) holds and they do not have common roots. Use Lemma 4, with
2—n
t1 = t2 = t3 = —v — 2y----------------4-(n — 1 )£1,
1 —n — 2
ni =— 2(v + q2 +—4—)— Y,
1 —n — 2
n2 = — ^(v + r2 +—4—) — Y,
1 , 2 — n.
ni =— 2(v + s2 + )— Y,
to obtain
2 + 2n — 8v — 16y — 12(n — 1 )£1 + (q2 + 2r2 + s2) < 2n + 5,
so that
5 > 2 — 8v — 16y — 12(n — 1 )£1 + (q2 + 2r2 + s2).
If 16y + 12(n — 1)£1 < 0.5 then 5 > 1.5 — 8v. Hence, for 5 = 0.^d v < 0.175
M
more irreducible polynomials and Proposition 4 is proved.
In the cases when one or two coordinates are equal to 1 in the linearity vector
(ii, i2, i3) we must combine the calculation for the subclass P-(0,0,0) and P*(1,1,1). Putting all the propositions together completes the proof of the theorem (more details are in [9]).
We indicate the following important applications of Theorem 1.
First, we adapt Theorem 1 to the problem for polynomials with small discrimi-
nant. The discriminant of the polynomial can be written as the determinant
n (n -1 )
D(P)=(—1) 2
1 an-1 a2 a1 ao ■ ■ 0
0 an an-1 a2 a1 ■ 0
0 0 an an-1 a - 2 ■ ao
n (n — 1) an-1 2a2 a1 0 ■ 0
0 nan (n — 1) an-1 2a2 a1 ■ 0
0 0 0 nan (n — 1) an-1 ■ ■ a1
or as the product of squares of root differences
D(P) = a.nn-2 n (^1 - <Xj)2.
1 <j<i<n
(27)
(28)
From (28)it follows that D(P) = 0 if and only if the polynomial P has multiple roots. By (27), we obtain that if D(P) = 0 then D(P) ^ 1.
If the first coefficient an of the polynomial P(f) is a sufficiently large integer and the inequality min1^j<i^n|ai — aj > b > 0 holds for the roots of P(f), then
2n-2 "m *
|D(P)| > c(b)a; Further, let Q be a sufficiently large number with
H(P) < Q*
(29)
Denote by Pn the set of polynomials satisfying (29). From (27) - (29) it can be seen that all the values of D(P) belong to the interval
[—c(n)Q2n-2, c(n)Q2n-2]*
2n-2i
(30)
By (29), we also note that the set Pn contains exactly (2Q + 1)n+1 polynomials (including the zero polynomial).
For some prime number q, positive integer ^d p > 0 denote by Pn(Q, q,l, p) the subset of polynomials P G Pn, for which
D(P) ^ Q2n-2-2p, (31)
ql II D(P). (32)
Here ql II d(p ) means that q11 D(P) and ql+1 / D(P). The question of how many polynomials satisfy (31) or (32) or both together is a natural problem in the theories of Diophantine approximation and the theory of Diophantine equations.
From (28) we obtain that (31) holds if the distance between two roots of the
polynomial Pn(f) decreases as Q increases. In particular, it will hold if for some j,
1 ^ j ^ n, the derivative
|P'(oj)| = |an(aj — a1) ■ ■ ■ (a, — aj-1)(aj — j) ■ ■ ■ (a, — an)I
tends to 0 as Q ^ ro. If (32) holds then the p-adic norm IP'(ai)Ip is small for some
i, 1 ^ i ^ n.
From Theorem 1 we can obtain the lower bounds for the derivatives of the polynomial P in R x C x Qp for the set of the point u1 = (x, z,w) G Ti, Ti C T, for which ^(T1) > By Lemma 1, for every point u1 G T1 there exists a point
P1
in the algebraic coordinate for every metric will satisfy the system of inequalities
(1) if < is replaced by This gives that the inequalities (31) and (32) hold for the parameters p and l. These parameters depend on v because the discriminant contains the derivative of the polynomial at the roots. Then, we can choose a point u2 G T1 for which there exists a polynomial P2 ^ P-|. Such a point u2 exists because ^(Ti ) > This procedure allows us to construct a large number of polynomials
satisfying conditions (31) and (32).
As a second application of a Theorem 1, we would like to investigate the more general question when the first inequality in (1) is
max(IP(x)I, IP(z)I, IP(w)IpI) < H-4 +t+v^1-t(H), 0 < t < 1*
Acknowledgements. This work was supported by the Science Foundation Ireland Grant RFP06/MAT0015.
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Received 21.09.2008.
